Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line
Brandon Gontmacher, Benjamin Hutz, Grayson Jorgenson, Srinjoy Srimani, Simon Xu
AAUTOMORPHISM LOCI FOR DEGREE 3 AND DEGREE 4ENDOMORPHISMS OF THE PROJECTIVE LINE
BRANDON GONTMACHER, BENJAMIN HUTZ, GRAYSON JORGENSON, SRINJOY SRIMANI,AND SIMON XU
Abstract.
Let f be an endomorphism of the projective line. There is a natural conjugationaction on the space of such morphisms by elements of the projective linear group. The group ofautomorphisms, or stabilizer group, of a given f for this action is known to be a finite group.We determine explicit families that parameterize all endomorphisms defined over ¯ Q of degree 3and degree 4 that have a nontrivial automorphism, the automorphism locus of the moduli spaceof dynamical systems. We analyze the geometry of these loci in the appropriate moduli space ofdynamical systems. Further, for each family of maps, we study the possible structures of Q -rationalpreperiodic points which occur under specialization. Introduction
Let K be a field and P the projective line. Throughout, K is a finite extension of Q . Anendomorphism of P of degree d can be represented as a pair of homogeneous polynomials of degree d with no common factors. The space of all such maps is denoted as Rat d . There is a naturalconjugation action on Rat d by PGL , the automorphisms of P , given as f α := α − ◦ f ◦ α for f ∈ Rat d and α ∈ PGL . The quotient by this action is a geometric quotient in terms of geometric invariant theory andforms the moduli space of degree d dynamical systems on P , M d := Rat d / PGL [42]. We denotea conjugacy class as [ f ] ∈ M d and a representation of a conjugacy class as f ∈ Rat d . Our primaryobjects of study are those conjugacy classes [ f ] ∈ M d for which there is a nontrivial α ∈ PGL sothat f α = f . Such an α is called an automorphism of f , and the set of all such automorphismsforms a group Aut( f ) := { α ∈ PGL : f α = f } . In additional to being special from the existence of these extra symmetries, conjugacy classes withnontrivial automorphisms are exactly the singular points of the moduli space M d for d ≥ f must be a finite subgroup of PGL .Sharper bounds than this permutation bound on the size of an automorphism group in terms of d can be obtained but do not concern us here [25]. Key to this work is the (classical) classification incharacteristic zero of the finite subgroups of PGL restated in modern notation by Silverman [41].It is important to note that Aut( f ) is well defined on conjugacy classes. In particular, given α ∈ PGL , the action on Aut( f ) defined by σ (cid:55)→ α − ◦ σ ◦ α provides a group isomorphismAut( f ) ∼ = Aut( f α ) . The conjugacy class of Aut( f ) in PGL is, thus, an invariant of [ f ] rather than Mathematics Subject Classification. a r X i v : . [ m a t h . D S ] J u l ust f . Denote by A d ⊂ M d the set of all conjugacy classes with a nontrivial automorphism. LetΓ ⊂ PGL be a finite subgroup with representation ρ : Γ → SL . Denote the set of conjugacyclasses whose automorphism group contains a subgroup isomorphic to Γ by A d (Γ). Similarlydenote A d ⊂ Rat d as the set of rational maps with nontrivial automorphism group with ρ (Γ) asa subgroup, i.e., A d (Γ) = { f ∈ A d : Aut( f ) ⊇ Γ } . It is important to note that while every finitesubgroup of PGL has only one inequivalent representation, the choice of group representationaffects the representation in homogeneous coordinates of the map. In particular, questions aboutthe field of definition for elements of A d (Γ) are heavily dependent on the choice of representationof Γ, e.g., [7, 41].The goals of this work are two-fold. The first is to give explicit parameterizations for all mapsin A d for 3 ≤ d ≤
4. The case d = 2 is well known [15]. The case d = 3 can be derived from theunpublished work of West [47], but he focuses on the parameterizations of M as a whole and aside result is the ability to determine which elements lie in A . However, it is nontrivial to movebetween West’s parameterization of M and elements of Rat . Further, his methods are not easilyapplicable in degree d >
3. The methods here can be used for any degree, and we produce explicitfamilies in Rat and Rat , ( M and M ). The second goal is to study other arithmetical dynamicalproperties of these families with nontrivial automorphisms focusing on the structure of the set ofrational preperiodic points. The motivation for this portion is the uniform boundedness conjectureof Morton and Silverman. Conjecture 1.1 ([34]) . Fix integers d ≥ N ≥
1, and D ≥
1. There is a constant C ( d, N, D )such that for all number fields K/ Q of degree at most D and all morphisms f : P N → P N of degree d defined over K , f, P N ( K )) ≤ C ( d, N, D ) . This conjecture is equivalent to a uniform bound on the number of rational preperiodic graphstructures, where the vertices are rational preperiodic points and edges connect a point Q to itsforward image f ( Q ) [11]. While an unconditional bound remains out of reach, we use Poonen [38]and Manes [28] as our model and classify graph structures assuming an upper bound on the periodof a rational periodic point.We now give a summary of the main results and an outline of the article. Section 2 gives param-eterizations of the automorphism locus A ⊂ M . Taking the classification of finite subgroups ofPGL given in Silverman [41] combined with the known dimensions of A (Γ) from Miasnikov-Stout-Williams [31], we find families in the parameter space Rat that map finite-to-one onto families in M of the appropriate dimension. Theorem 1.2. (1) A ( C ) = A ( D ) is a single conjugacy class in M given by f ( z ) = z . (Corollary 2.3).(2) A ( A ) is a single conjugacy class in M given by f ( z ) = z − − z (Proposition 2.5).(3) A ( C ) is an irreducible curve in M given by f a ( z ) = z + aaz , a (cid:54) = 0 (Proposition 2.8).(4) A ( D ) consists of two irreducible curves given by f a ( z ) = az − z − az , a (cid:54) = ± g a ( z ) = az + 1 z + az , a (cid:54) = ± , which intersect at the single point A ( D ) = A ( C ) (Proposition 2.9).(5) The locus A ( C ) is the union of two irreducible surfaces given by f a,b ( z ) = z + azbz + 1 , ab (cid:54) = 1 and g a,b ( z ) = az + 1 z + bz , ab (cid:54) = 1which intersect in a curve that is the g a component of A ( D ) (Proposition 2.10, Proposition2.12). A ( C ) and A ( C ) intersect at a single point in the moduli space, and the point of inter-section is A ( A ) (Proposition 2.11).The methods are a combination of invariant theory as utilized in deFaria-Hutz [7], explicit formsfor maps with cyclic or dihedral automorphism groups from Silverman [41], and explicit calculationusing the generators of the finite subgroups.In the process of studying A , we were able to complete the rational realization problem over Q started in [8]. Theorem 1.3.
Every finite subgroup of PGL can be realized as the automorphism group of amap defined over Q .Section 3 studies these parameterizations as families in moduli space giving explicit maps to setsof periodic point multiplier invariants that are finite-to-one. To construct the multiplier invariants,recall that to each fixed point Q we can compute an algebraic number called the multiplier λ Q =˜ f (cid:48) ( Q ), where ˜ f is a dehomogenization and (cid:48) represents the derivative. The multiplier is conjugationinvariant and the set of fixed points is invariant (as a set) under conjugation. So, taking theelementary symmetric polynomials evaluated on the set of multipliers produces invariants of themoduli space [42]. We can similarly construct invariants from the set of periodic points (or formalperiodic points) of any period. We denote these invariants σ ( n ) i , where n denotes the period ofthe points used and 1 ≤ i ≤ (deg( f )) n + 1. These invariants were first studied by Milnor [33] toconstruct an isomorphism M ∼ = A . In higher degrees, we no longer get an isomorphism to anaffine space, but utilizing enough multiplier invariants does produce a finite-to-one map [30]. Inall families but one component of A ( D ), using the fixed points multipliers is sufficient; in thatcomponent the multiplier invariants of the 2-periodic points are needed. Theorem 1.4 summarizesthe results. The details of the embeddings into affine space via the multiplier invariants that areused to analize the geometry can be found within the referenced propositions. Theorem 1.4. (1) The map A \ { } → M , a (cid:55)→ (cid:20) azaz + 1 (cid:21) is one-to-one. The locus A ( C ) is an irreducible curve of genus zero with one singularpoint corresponding to A ( A ). (Proposition 3.1)(2) The map A \ {± } → M , a (cid:55)→ (cid:20) az + 1 z + az (cid:21) is two-to-one. This component of A ( D ) is a smooth irreducible curve of genus zero.(Proposition 3.2)(3) The map A \ {± } → M , a (cid:55)→ (cid:20) az − z − az (cid:21) is six-to-one. This component of A ( D ) described by the image is a smooth irreduciblecurve of genus zero. (Lemma 3.3, Proposition 3.4)(4) The map A \ { ab = 1 } → M , ( a, b ) (cid:55)→ (cid:20) z + azbz + 1 (cid:21) is two-to-one. This component of A ( C ) is an irreducible rational singular surface. (Propo-sition 3.6)
5) The map A \ { ab = 1 } → M , ( a, b ) (cid:55)→ (cid:20) az + 1 z + bz (cid:21) is four-to-one. This component of A ( C ) is an irreducible singular surface. (Lemma 3.8,Proposition 3.9)Section 4 gives parameterizations of the automorphism locus A ⊂ M . The methods are similarto Section 2. Theorem 1.5. (1) The loci A ( C ) = A ( D ) is the single conjugacy class given by f ( z ) = z . (Proposition4.2)(2) The locus A ( C ) is given by either 1-parameter family f k ( z ) = z +1 kz or g k ( z ) = kzz +1 for k (cid:54) = 0. (Proposition 4.3)(3) The locus A ( C ) is given by the family f k ,k ( z ) = z + k zk z +1 for k k (cid:54) = 1. (Proposition 4.4)(4) The locus A ( D ) is given by the family f k ( z ) = z + kzkz +1 for k (cid:54) = ±
1. (Proposition 4.5)(5) The locus A ( C ) is given by the 3-parameter family f k ,k ,k ( z ) = z + k z +1 k z + k z for k + k (cid:54) = k k k . (Proposition 4.6).Section 5 studies these parameterizations as families in moduli space giving explicit maps to setsof periodic point multiplier invariants that are finite-to-one. In all families but A ( C ), using thefixed point multipliers is sufficient; in that family the multiplier invariants of the formal 2-periodicpoints are needed. The details of the embeddings into affine space via the multiplier invariants thatare used to analize the geometry can be found within the referenced Propositions. Theorem 1.6. (1) The map A \ { } → M , k (cid:55)→ (cid:20) z + 1 kz (cid:21) is one-to-one. The locus A ( C ) is an irreducible curve of genus zero with one singularpoint. (Lemma 5.1, Proposition 5.2)(2) The map A \ {± } → M , k (cid:55)→ (cid:20) z + kzkz + 1 (cid:21) is one-to-one. The locus A ( D ) is an irreducible curve of genus zero with one singularpoint. (Lemma 5.3, Proposition 5.4)(3) The map A \ { k k = 1 } → M , ( k , k ) (cid:55)→ (cid:20) z + k zk z + 1 (cid:21) is one-to-one. The locus A ( D ) is a singular irreducible surface. (Proposition 5.6)(4) The map A \ { k + k = k k k } → M , ( k , k , k ) (cid:55)→ (cid:20) z + k z + 1 k z + k z (cid:21) is two-to-one. (Proposition 5.7)Having given parameterizations of A and A , we turn to studying rational preperiodic struc-tures. Section 6 classifies the Q -rational preperiodic structures that occur for maps in the locus A given by these parameterizations. The graph structures are specified in the theorems of Section 6and summarized in the following theorem. heorem 1.7. (1) Single conjugacy classes A ( C ) = A ( D ), and A ( A ): Each has one possible rationalpreperiodic graph structure. (Theorem 6.3)(2) A ( C ): Assuming there are no points of period 4 or higher, then there are four possible Q -rational preperiodic graph structures for f a ( z ) = z + aaz . Each of these graph structuresoccurs infinitely often. (Theorem 6.5).(3) A ( D ):(a) Assuming there are no points of period 4 or higher, then there are four possible Q -rational preperiodic graph structures for f a ( z ) = az +1 z + az . Each of these graph structuresoccurs infinitely often. (Theorem 6.11).(b) Assuming there are no points of period 4 or higher, then there are four possible Q -rational preperiodic graph structures for f a ( z ) = az − z − az . Each of these graph structuresoccurs infinitely often. (Theorem 6.14).(4) A ( C ):(a) The family f a,b ( z ) = az +1 z + bz has at least 23 different Q -rational preperiodic graph struc-tures. (Section 6.5).(b) The family f a,b ( z ) = z + azbz +1 has at least 33 different Q -rational preperiodic graph struc-tures. (Section 6.6).The methods of this section are to first classify the occurrence of Q -rational periodic points upto some bound, then to classify the rational preimages of the possible cycle structures. Further, wemust classify when distinct connected components can occur for the same choice of parameter. Inall these cases, the problems come down to finding all the rational points on a curve. In the caseof rational or elliptic curves, this set can be infinite. In the case of higher genus curves, there canonly be finitely many rational points (Falting’s Theorem). For the two A ( C ) components thatare dimension two, the task moves to finding all rational points on surfaces. The tools availablefor such problems are much more limited, so instead we perform a census of Q -rational preperiodicgraph structures.In Section 7 we study the Q -rational preperiodic structures that occur for maps in the locus A given by these parameterizations. The graph structures are specified in the Theorems of Section 7and summarized in the following theorem. Theorem 1.8. (1) Single conjugacy classes A ( C ) = A ( D ) have one possible rational preperiodic graphstructure. (Theorem 7.1)(2) A ( C ): Assuming there are no points of period 3 or higher, then there are four possible Q -rational preperiodic graph structures for f a ( z ) = z +1 az . Each of these graph structuresoccurs infinitely often. (Theorem 7.4).(3) A ( D ): Assuming there are no points of period 3 or higher, then there are five possible Q -rational preperiodic graph structures for f a ( z ) = z + azaz +1 with possibly finitely many ex-ceptional values of the parameter a . Each of these graph structures occurs infinitely often.(Theorem 7.7).(4) A ( C ): The family f k ,k ( z ) = z + k z k z +1 has at least 20 different Q -rational preperiodicgraph structures. (Section 7.4).(5) A ( C ): The family f k ,k ,k ( z ) = z + k z +1 k z + k z +1+1 has at least 55 different Q -rational preperi-odic graph structures. (Section 7.5). he methods parallel those in Section 6, but the increase in degree causes an increase in thecomplexity of the curves. The finitely many possible exceptions for A ( D ) are the rational pointsof two explicitly given genus 6 curves.Finally, in Section 8 we further examine the restrictions on rational preperiodic structures createdby automorphism groups with rational elements. Specifically, we provide computational evidencethat the number of rational elements in the automorphism group affects how similarly a mapbehaves to a random mapping on a finite field.2. Automorphism loci in M In this section we determine A , the conjugacy classes of degree 3 endomorphisms of P thathave a nontrivial automorphism. We will always be working over Q .We know that if Γ ⊂ Aut( f ) for some f ∈ Rat d , then Γ must be one of the following groups [41]: • Cyclic group of order n , denoted as C n . • Dihedral group of order 2 n , denoted as D n . • Tetrahedral group A (or alternating group on 4 elements). • Octahedral group S (or symmetric group on 4 elements). • Icosahedral group A (or alternating group on 5 elements.Also, from the classification of finite subgroups of PGL , all isomorphic subgroups are in fact con-jugate to each other [31, Lemma 2.3]. Thus, given a group, we can simply fix an SL representationand work only with this representation.From previous work of Miasnikov, Stout, and Williams [31] (henceforth abbreviated as MSW),we are able to determine the dimension of A (Γ) for any Γ. Lemma 2.1.
When deg( f ) = 3, Aut( f ) must be C , C , C , D , D or A . The dimensions of A (Γ) for these groups are given by(1) dim A ( C ) = 2 , dim A ( C ) = 1, and dim A ( C ) = 0.(2) dim A ( D ) = 1, and dim A ( D ) = 0.(3) dim A ( A ) = 0.Moreover,(4) A ( D ) ⊂ A ( C ) ⊂ A ( C ).(5) A ( D ) ⊂ A ( C ).(6) A ( A ) ⊂ A ( C ), A ( A ) ⊂ A ( C ), and A ( A ) ⊂ A ( D ). Proof.
The groups that occur and the dimensions are calculated in Section 2 of MSW [31]. Thesecond half follows from the observation that if G is a subgroup of H , then A d ( H ) ⊆ A d ( G ). (cid:3) To determine A (Γ), we proceed in two steps. First we find a family in A ⊂ Rat whichparameterizes all maps that have Γ contained in their automorphism group. Then we find a normalform for elements in this family. By normal form, we mean that the projection π : Rat → M is anisomorphism on this family, i.e., every member of the family represents a distinct conjugacy class.However, in practice we typically end up with a finite-to-one projection so that the dimension ofthe family in parameter space is the same as the dimension in moduli space. Stated more precisely,given a group Γ, we want to find a k -parameter family in Rat such that k = dim A (Γ) and for allmaps f ∈ A (Γ), f is conjugate to some member of this k -parameter family.2.1. A ( C ) and A ( D ) . From Silverman’s classification of maps with cyclic automorphism groups[41, Proposition 7.3], we know that if a degree 3 map f has a C symmetry, then f must be of theform az for some a ∈ Q . This family of maps is actually a family of twists corresponding to asingle conjugacy class in M . roposition 2.2. Let f ∈ Rat . If C ⊆ Aut( f ), then f is conjugate to z . Proof.
Let ϕ ( z ) = z and f a ( z ) = az for any nonzero a ∈ Q . Consider the M¨obius transform α ( z ) = pz , where p = a . We can compute ϕ α = ( α − ◦ ϕ ◦ α )( z ) = pz ) p = 1 p z = 1 az = f a . (cid:3) Note that the automorphism group of z is actually all of D . By the containments from Lemma2.1, we have the following corollary. Corollary 2.3.
The automorphism locus A ( C ) = A ( D ) is a single point in M given by theconjugacy class of z .2.2. A ( A ) . To tackle the A case, we appeal to the invariant theory techniques formalized in [8].For a finite group G acting on a polynomial ring R , we say F ∈ R is invariant for G (relative to thegroup character χ ) if g · F = χ ( g ) F for all g ∈ G . The connection to automorphisms comes from aclassical result of Klein [22] with the complete statement given by Doyle and McMullen [10]. Lemma 2.4. [10, Theorem 5.2] A homogeneous 1-form θ is invariant if and only if θ = F λ + dG, where F and G are invariant homogeneous polynomials, F = 0 or deg G = deg F + 2, and λ =( xdy − ydx ) / Proposition 2.5.
The locus A ( A ) is a single point in moduli space represented by f ( z ) = √− z − z −√− z . Proof.
By Lemma 2.4, to produce a degree 3 map with A automorphisms we need either a singleinvariant of degree 4 or two invariants (from the same character) of degree 4 and 2, respectively.Looking at the Molien series associated to each (linear) character for A :1 + t + 2 t + t + O ( t ) t + t + 2 t + t + O ( t ) t + t + t + 2 t + 2 t + O ( t ) t + t + t + 2 t + 2 t + O ( t ) t + t + t + 2 t + 2 t + O ( t ) t + t + t + 2 t + 2 t + O ( t ) , we see that there are no degree 2 invariants and only two characters have a degree 4 invariant.These invariants are x + ( − ν − ν − ν − x y + y x + (8 ν + 12 ν + 32 ν + 14) x y + y , where ν is a root of x + 2 x + 5 x + 4 x + 1. Applying the Doyle-McMullen construction (Lemma2.4) to these invariants results in the two maps( x : y ) (cid:55)→ ( −√− x y − y : x + √− xy )( x : y ) (cid:55)→ ( √− x y − y : x − √− xy ) . These maps are conjugate by α = (cid:18) − (cid:19) , so A ( A ) is a single point in moduli space. (cid:3) s mentioned in the introduction, the choice of representation affects the field of definition andit is natural to ask whether a map defined over Q has A as automorphism group. Hutz-deFaria[7] proved that the “standard” representation of A given by Silverman [41] does not have a mapdefined over Q with tetrahedral automorphism group. However, when computing the intersectionof A ( C ) and A ( C ) in Section 2.6, we discovered that the map f ( z ) = z − − z has tetrahedral automorphism group (and is conjugate to the maps in Proposition 2.5), which canbe verified with direct computation by the algorithm of Faber-Manes-Viray [14] as implemented inSage. This example completes the construction started in Hutz-deFaria to show that every finitesubgroup of PGL can be realized as the automorphism group of a map defined over Q . Theorem 2.6.
Every finite subgroup of PGL can be realized as the automorphism group of amap defined over Q .2.3. A ( C ) . From Silverman’s classification of maps with cyclic automorphism groups [41, Propo-sition 7.3], we know that if f ∈ Rat has a C symmetry, then f belongs to one of the followingtwo families: f ( z ) = az + bcz or f ( z ) = azbz + c . Since all parameters must be nonzero to be a degree 3 map, we can divide through by a and b ,respectively, to get(1) f ( z ) = z + a a z or f ( z ) = b zz + b , where a , a , b , b ∈ Q \{ } . These two families in parameter space are exactly A ( C ), and wenext show they project onto the same irreducible curve in moduli space. Lemma 2.7. An k -parameter family in Rat d induces a map F : U → M d for an open subset U ⊆ A k and this map is a morphism of varieties. In particular, the image isirreducible and so is its closure. Proof.
Formally, a k -parameter family of rational degree d maps is a set of rational maps P → P of the form ( x : y ) (cid:55)→ ( F x d + F x d − y + . . . + F d y d : F d +1 x d + . . . + F d +1 y d )for some chosen polynomials F , . . . , F d +1 in indeterminants u , . . . , u k , where we may require inaddition that some of the F j never vanish and the two defining polynomials of the rational maphave no common factors for all ( u , . . . , u k ) considered.These F j determine a morphism of varieties φ : A k → A d +2 defined as( u , . . . , u k ) (cid:55)→ ( F ( u , . . . , u k ) , . . . , F d +1 ( u , . . . , u k )) . The complement of the origin A d +2 \ { } is an open subset and, thus, U (cid:48) := φ − ( A d +2 \ { } )is also open in A k . So restricting φ gives a morphism U (cid:48) → A d +2 \ { } .There is a projection morphism A d +2 \ { } → P d +1 defined( a , . . . , a d +1 ) (cid:55)→ ( a : . . . : a d +1 ) . Composing the restriction of φ with this projection yields a morphism U (cid:48) → P d +1 . y definition, Rat d is the complement of a hypersurface in P d +1 and is open itself. Therefore,its inverse image by the composite map from U (cid:48) is also an open subset U of A k .One may further refine U as needed by intersecting it with the inverse image of the complementof the hyperplane in P d +1 defined by the vanishing of a given a j . This amounts to requiring thecorresponding coefficient of the rational maps in the k -parameter family to be nonzero.Finally, geometric invariant theory can be used to show the quotient map Rat d → M d [43,Theorem 4.36] is a morphism; thus, altogether we obtain the desired map F : U → M d .From point set topology, nonempty open subsets of varieties are irreducible, the closure of anirreducible subset of a larger topological space is irreducible, and the continuous image of anirreducible set is irreducible. Therefore, the image of F and its closure are both irreducible. (cid:3) For example, with the notation from Lemma 2.7, the 1-parameter family of rational maps of theform f a ( z ) = azaz +1 written projectively as( x : y ) (cid:55)→ ( axy : ax + y ) , where a (cid:54) = 0 corresponds to taking F = F = F = F = F = 0 F = F = aF = 1 . We have refined the open subset U from the proof by intersecting it with the inverse image of thecomplement of the hyperplane defined by a = 0 (or, equivalently, that defined by a = 0), if thecoordinates for P are ( a : a : a : . . . : a ). Lemma 2.7 then shows that the set of all conjugacyclasses of maps of the form f a ( z ) = azaz +1 is an irreducible subset of M .Let f ∈ Rat d . To each periodic point z of period n for f , we can compute the multiplier as λ z := ( ˜ f n ) (cid:48) ( z ) for a dehomogenization ˜ f of f . The set of all multipliers of a given period is invariantunder conjugation (but may be permuted) so applying the elementary symmetric polynomials tothis collection of multipliers produces a set of complex numbers that are invariants of the conjugacyclass—these multiplier invariants have also been called Milnor parameters . See Silverman [43, § >
1. The key fact we need is thatthey are invariants of the conjugacy class. In particular, if f, g ∈ Rat d have different multiplierinvariants for any fixed n , then they cannot be conjugate. However, the converse is not true; havingthe same multiplier invariants for some (or all) n does not make two functions conjugate. Proposition 2.8. A ( C ) is an irreducible curve in M given by either one-parameter family, f a ( z ) = azaz + 1 or g b ( z ) = z + bbz . Proof.
We first show that we can conjugate the two-parameter families in equation (1) to the one-parameter families given in the statement. We first consider f . Let β = a a , and consider theM¨obius transform α ( z ) = β z so that α − ( z ) = βz . We can compute f α = (cid:16) β z (cid:17) + a a (cid:16) β z (cid:17) β = (cid:32) z β + a a z β (cid:33) β = βz + a β a βz = z + a β a z = z + a a z . Thus, the two-parameter family z + a a z is conjugate to the one-parameter family g b . Similarly for f , we can set γ = b , and consider the M¨obius transform α ( z ) = γz and its inverse α − ( z ) = γ z . e can compute f α = (cid:18) b γzγ z + b (cid:19) γ = b zγ z + b = b b z b b z + 1 = bzbz + 1 , where b = b b . Thus, the two-parameter family b zz + b is conjugate to the one-parameter family f a .Next, consider the M¨obius transform α ( z ) = z . We can compute g αb = 1 z + bb z = 1 bz +1 bz = bzbz + 1 = f b . Thus, the two one-parameter families f a and g b are conjugate and we can consider either one. Inparticular, every f with a C symmetry is conjugate to some member of the family g a = z + aaz . (cid:3) A ( D ) . The representation of D we will be working with is (cid:26)(cid:20) (cid:21) , (cid:20) − i i (cid:21) , (cid:20) ii (cid:21) , (cid:20) −
11 0 (cid:21)(cid:27) , where (cid:20) − i i (cid:21) and (cid:20) ii (cid:21) are the two generators. They correspond to M¨obius transforms α = − z and α = z . Proposition 2.9. A ( D ) consists of two irreducible curves which intersect at the single point A ( D ) = A ( C ). Proof.
Let f ( z ) = a z + a z + a z + a b z + b z + b z + b be a degree 3 map. If f has a D symmetry, we can obtainrestrictions on the coefficients from the equations f = f α and f = f α .We get 14 equations and compute the irreducible components of the variety generated by theseequations as a subvariety of P . There are 9 irreducible components, but only 4 of them correspondto degree 3 maps. Thus, f ∈ Rat has a D symmetry (with this representation) if and only if ithas one of the following four forms: f ( z ) = k z − z − k z f ( z ) = k z + 1 z + k zf ( z ) = z + k zk z + 1 f ( z ) = z − k zk z − , where none of the parameters can be 1. We will show that f , f , and f are all conjugate to eachother, whereas f is not in general conjugate to the rest. First, consider the families f and f .Conjugating f by the element α = (cid:20) − i − i (cid:21) , we get f α ( z ) = z + k +3 k − z k +3 k − z + 1 . Thus, f is conjugate to the map f ( z ) = z + k zk z +1 , where k = k +3 k − . Now we show that f isconjugate to f . For every k (cid:54) = 1, set k = k +3 k − and work with f ( z ) = k k − z − z + k k − . Conjugatingthis map by α = (cid:20) i i − (cid:21) , we get f α ( z ) = z − k zk z − = f ( z ). Thus, f , f , and f are all conjugateto each other. o see that f is not conjugate to the rest, we compute the multiplier invariants for the fixedpoints of f . Interestingly, they are independent of the parameter k : σ = − σ = 54 σ = − σ = 81 . However, the multiplier invariants for the fixed points of f are dependent on the parameter k : σ ( k ) = 2 k + 6 k + 1 σ ( k ) = k − k + 10 k + 6 k + 9 k + 2 k + 1 σ ( k ) = − k + 6 k − k + 18 k k + 2 k + 1 σ ( k ) = k − k + 9 k k + 2 k + 1 . Solving the equations σ ( k ) = − , σ ( k ) = 54 , σ ( k ) = − , σ ( k ) = 81 , we see that the only possible conjugacy occurs for k = 3. Setting k = 3, we compute themultiplier invariants for the periodic points of period 2 for f and f , denoted σ (2) i for 1 ≤ i ≤ f the invariants depend on k , and for f at k = 3 they are constants. Solving the equations σ (2) i ( f ) = σ (2) i ( f ), we see that the three values k ∈ { , , − } are all conjugate to f with k = 3.Further, note that the three elements of the f family with k ∈ { , , − } are all in the sameconjugacy class as z . These are the only conjugacies in the families f and f .In summary, every degree 3 map f with D symmetry is conjugate to some member of the family f ( z ) = k z − z − k z or the family f ( z ) = k z +1 z + k z , and we can conclude that A ( D ) has two irreduciblecomponents which intersect at the calculated conjugacy class. (cid:3) A ( C ) .Proposition 2.10. The locus A ( C ) is the union of two irreducible surfaces. Proof.
From Silverman’s classification of maps with cyclic automorphism groups [41, Proposition7.3], we know that if f is degree 3 with a C symmetry; then f ( z ) = k z + k zk z + k or f ( z ) = l z + l l z + l z , where k k (cid:54) = 0 and l l (cid:54) = 0. Thus, we can divide through by k and l , respectively, to obtain: f ( z ) = z + a za z + a or f ( z ) = b z + b z + b z . We first reduce these two three-parameter families to two-parameter families. Consider the firstfamily f ( z ) = z + a za z + a . Conjugating it by the M¨obius transform α ( z ) = √ a z , we get f α ( z ) = 1 √ a (cid:18) ( √ a z ) + a √ a za a z + a (cid:19) = a z + a za a z + a = z + a a za z + 1 . Thus, for every choice of a , a , a , we can set k = a a and k = a ; note that a (cid:54) = 0 since k (cid:54) = 0.Thus, the family f ( z ) = z + a za z + a is conjugate to the family ϕ ( z ) = z + k zk z +1 . imilarly, given f ( z ) = b z + b z + b z , we can set β = b and consider the M¨obius transform α ( z ) = β z . We can compute f α ( z ) = β (cid:32) b z β + b z β + b zβ (cid:33) = b β z + b β z + b β z = b β z + 1 z + b β z . Thus, for any choice of b , b , b , we can set l = b β and l = b β . So the family f ( z ) = b z + b z + b z is conjugate to the family φ ( z ) = l z +1 z + l z .Finally, we need to determine if these two-parameter families are conjugate to each other. Takinga generic PGL element α = (cid:18) a bc d (cid:19) and conjugating, we can setup a system of equations for ϕ α tobe of the form φ by equating the known coefficients (up to scalar multiple). Eliminating the othervariables using a lexicographic Groebner basis computation, we determine that ϕ is conjugate tosome member of φ if and only if k = k . So, in general, the two families are not conjugate to eachother, and A ( C ) is the union of two irreducible surfaces. (cid:3) Intersection and containment.
While we have computed several of the intersections ofcomponents of A along the way, there are a few final cases to address. Proposition 2.11. A ( C ) and A ( C ) intersect at a single point in the moduli space, and thepoint of intersection is A ( A ). Proof.
To determine this intersection, we use a Groebner basis calculation similar to the end of theproof of Proposition 2.10. Recall that A ( C ) is given by the family g ( z ) = z + kkz .Consider the component of A ( C ) given by f k ,k ( z ) = z + k zk z +1 . Taking a generic element ofPGL , α = (cid:18) a bc d (cid:19) , we consider the system of equations obtained from the coefficients of f αk ,k = tg for some nonzero constant t . Note that f α is degree 3, so that if f α = g , then its numerator anddenominator differ from those of g only by a constant multiple. The resulting equations have nosolutions.Now consider the component given by f k ,k = k z +1 z + k z . Again we set f α = tg and solve theresulting system of equations. In this case, there are two solutions k = − , ( k , k ) = ( ±√ , ∓√ . These two choices of ( k , k ) are in fact conjugate to each other. Furthermore, the automorphismgroup of f k ,k has order 12 at these values. The only group with order 12 in our list of possibleautomorphism groups is A . Thus, A ( C ) and A ( C ) intersect at a single point in the modulispace, and the point of intersection is A ( A ). (cid:3) Proposition 2.12.
The two irreducible surfaces comprising A ( C ) intersect in a curve that is thecomponent of A ( D ) given by g a = az +1 z + az . Proof.
To determine this intersection, we use several Groebner basis calculations similar to the onesin Proposition 2.11. Recall that A ( C ) is given by the two families f k ,k ( z ) = z + k zk z + 1 and g k ,k ( z ) = k z + 1 z + k z aking a generic element of PGL , α = (cid:18) a bc d (cid:19) , we consider the system of equations obtained fromthe coefficients of f αk ,k = tg k ,k for some nonzero constant t . Examining the last element in a lexicographic Groebner basis weconclude that k = k . Repeating the Groebner basis calculation assuming k = k results in − k + k ( k + 9) = 0 . If k = k , we get exactly the curve in A ( D ) given by g a = az +1 z + az . If k = ±√−
9, then we get( k + 3)( k + (2 / v − / k + ( − / v − / , where v = ±√−
9. This results in the four possible maps f = vx + 1 x + vx , f = vx + 1 x − vx ,f = − vx + 1 x + vx , f = − vx + 1 x − vx . The two maps f , f are conjugate and clearly in the component of A ( D ) given by g a . The othertwo maps f , f are also conjugate and are, in fact, conjugate to z , which is A ( D ). This pointis in the intersection of the two components of A ( D ) so must be in the component given by g a .Therefore, the intersection of the two components of A ( C ) is exactly the component of A ( D )given by g a ( z ) = az +1 z + az . (cid:3) This completes the proof of Theorem 1.2.
Remark 2.13.
It is worth noting that for each of the cyclic automorphism groups, the sameprocedure was able to produce a family in moduli space. Fix an integer m ≥
2. The followingalgorithm will produce a k -parameter family in A d ( C m ) where k = dim( A d ( C m )).(1) Start with the normal form zψ ( z m ) from Silverman [41, Proposition 7.3]. The number ofparameters of this normal form is dim A ( C m ) + 2. Moreover, there are two parameters thatare nonzero.(2) Divide through by one of the nonzero parameters.(3) Apply a matrix of the form (cid:20) a
00 1 (cid:21) to eliminate another parameter.3.
Geometry of certain automorphism loci in M In this section, we examine the geometry of automorphism loci A (Γ) ⊂ M , such as smoothnessand genus. The method in general is to embed M into an affine space with a collection of multiplierinvariants, i.e., τ n : M → A k [ f ] (cid:55)→ ( σ ( n )1 , . . . , σ ( n )3 n +1 ) . We can then examine the image τ n ( A (Γ)) ⊆ A k and talk about the geometry of the resultingvariety. .1. A ( C ) . Recall that every map with a C symmetry is conjugate to some map of the form f a ( z ) = z + aaz Proposition 3.1.
Define f a ( z ) = azaz +1 . The map A \ { } → M a (cid:55)→ [ f a ]is one-to-one.The locus A ( C ) is an irreducible curve of genus zero with one singular point corre-sponding to A ( A ). Proof.
We compute the multiplier invariants associated to the fixed points for the family f a ( z ) asfunctions of a as σ ( a ) = a − a + 9 aσ ( a ) = − a + 21 a − a + 27 a σ ( a ) = 12 a − a + 63 a − a + 27 a σ ( a ) = − a + 36 a − a + 27 a . Then, to see which choices of parameters a and b have f a and f b with the same invariants, we solvethe system of equations σ i ( a ) = σ i ( b ) 1 ≤ i ≤ . The only solutions occur with a = b . In particular, since for each choice of a , f a has distinctmultiplier invariants, each choice of a corresponds to a distinct conjugacy class.Because every choice of a provides a distinct set of fixed point multiplier invariants and, hence,a distinct conjugacy class in M , we can use the fixed point multiplier invariants to parameterizethis curve in M . In particular, we have a map τ : A ( C ) → A defined by f a (cid:55)→ ( σ , σ , σ ) . Wecan omit σ because it is dependent on ( σ , σ , σ ) through the standard index relation (see Hutz-Tepper [19] or Fujimura-Nishizawa [15, Theorem 1]). Then we can consider the ideal generatedby a − a + 9 − aσ , − a + 21 a − a + 27 − a σ , (12 a − a + 63 a − a + 27) − a σ . We compute the saturation with the ideal ( a ) to avoid the vanishing of a and compute the Gr¨obnerbasis of the resulting ideal using lexicographic ordering to eliminate a . We obtain the followingrelations among the multiplier invariants:0 = σ + 16 σ σ + 52 σ − σ − σ −
60 = σ σ + 12 σ σ + 36 σ + 16 σ + 92 σ − σ σ − σ − σ − σ −
540 = σ σ + 8 σ − σ − σ −
120 = σ + 12 σ σ + 28 σ + 272 σ σ + 324 σ − σ σ − σ σ − σ − σ − σ . These relations define a curve in A . This curve is irreducible over Q and has genus 0. It is notsmooth, and the only singular point is ( − , , − a = − nd, thus, to the rational map f − ( z ) = z − − z . This rational map has a tetrahedral automorphismgroup. (cid:3) A ( D ) . There are two components in the locus A ( D ). We examine each separately. Proposition 3.2.
Define f a ( z ) = az +1 z + az . The map A \ {± } → M a (cid:55)→ [ f a ]is two-to-one. The component of the locus A ( D ) described by the image in M of the family f a is a smooth irreducible curve of genus zero. Proof.
We first compute the fixed point multiplier invariants for this family as functions of a as σ ( a ) = 4 a + 12 a − σ ( a ) = 6 a + 4 a + 54 a − a + 1 σ ( a ) = 4 a − a − a − a + 1 . To show the map is two-to-one, we fix a and determine how many b satisfy(2) ( σ ( a ) , σ ( a ) , σ ( a )) = ( σ ( b ) , σ ( b ) , σ ( b ))and then show that both solutions are conjugate. We start with the ideal defined by equation (2)and exclude the cases a = 1 and b = 1 through saturation. The resulting ideal is ( a − b ).We conclude that f a and f b have the same fixed point multiplier invariants if and only if a = ± b .We know that f a ( z ) = az +1 z + az and f − a ( z ) = − az +1 z − az are conjugate to each other via the M¨obiustransform α ( z ) = − iz : f αa ( z ) = − i · a ( − iz ) + 1( − iz ) + a ( − iz ) = − i · − az + 1 iz − aiz = − az + 1 z − az = f − a ( z ) . Furthermore, this shows we can parameterize the image of the family f a in M by the fixed pointmultiplier invariants. To study its geometry, consider the ideal generated by(4 a + 12) − ( a − σ , (6 a + 4 a + 54) − ( a − a + 1) σ , (4 a − a − − ( a − a + 1) σ . After saturation by ( a − a to obtain the relations0 = σ − σ − σ + 120 = 4 σ + 4 σ σ − σ + σ − σ + 216 . These define a curve in A . The curve is smooth, irreducible, and has genus 0. (cid:3) The other component of A ( D ) is given by the family f a ( z ) = az − z − az . Recall that this familyhas fixed point multiplier invariants that are independent of the parameter a . We consider the ultiplier invariants of the periodic points of period 2: σ (2)1 = 2 a + 36 a + 18 a + 72 a − a + 1 σ (2)2 = a + 76 a + 514 a + 1228 a + 9 a + 2268 a − a + 6 a − a + 1 σ (2)3 = 40 a + 1208 a + 7304 a + 32528 a + 30744 a + 51192 a + 40824 a − a + 6 a − a + 1 σ (2)4 = 636 a + 11232 a + 85806 a + 335448 a + 785376 a + 927288 a + 459270 a − a + 6 a − a + 1 σ (2)5 = 5080 a + 74700 a + 624024 a + 2354184 a + 6805944 a + 7637004 a + 3306744 a − a + 6 a − a + 1 σ (2)6 = 21286 a + 365112 a + 2597184 a + 11914776 a + 25286094 a + 32122656 a + 14880348 a − a + 6 a − a + 1 σ (2)7 = 45720 a + 1030968 a + 6945912 a + 29498256 a + 47711592 a + 63772920 a + 38263752 a − a + 6 a − a + 1 σ (2)8 = 51516 a + 979776 a + 11396457 a + 12045996 a + 86093442 a + 57395628 a + 43046721 a − a + 6 a − a + 1 σ (2)9 = 29160 a + 13122 a − a + 40389516 a + 86093442 a a − a + 6 a − a + 1 σ (2)10 = 6561 a − a + 3188646 a − a + 43046721 a a − a + 6 a − a + 1 . Lemma 3.3.
The map ϕ : A \ {± } → M a (cid:55)→ [ f a ]is six-to-one. The map ˜ τ : ϕ ( A ) → A [ f a ] (cid:55)→ ( σ (2)1 , σ (2)2 , σ (2)3 )is injective. Proof.
The ideal generated by ( σ (2)1 , . . . , σ (2)10 ) is, in fact, generated by ( σ (2)1 , σ (2)2 , σ (2)3 ), so we justneed to focus on these three invariants. Thinking of the invariants as functions of a , we need tofind all b so that (cid:16) σ (2)1 ( a ) , σ (2)2 ( a ) , σ (2)3 ( a ) (cid:17) = (cid:16) σ (2)1 ( b ) , σ (2)2 ( b ) , σ (2)3 ( b ) (cid:17) . Taking the ideal generated by these equations and saturating by the ideals ( a −
1) and ( b − − a + b )( a + b )( ab − a − b − ab − a + b + 3)( ab + a − b + 3)( ab + a + b − . We check that the six parameter values b ∈ (cid:26) ± a, ± a + 3 a − , ± a − a + 1 (cid:27) produce functions that are all conjugate. f b = − a , f a is, in fact, conjugate to f b via the M¨obius transform α ( z ) = izf αa = 1 i · a ( iz ) − iz ) − a ( iz ) = 1 i · − az − − iz − aiz = − az − z + az = f − a . If b = a +3 a − , then f a is conjugate to f b via the M¨obius transform α = (cid:18) − (cid:19) . This calculation isstraightforward, but omitted.If b = a − a +1 = − − a +3 − a − , then f − a is conjugate to f b since − a +3 − a − = ( − a )+3( − a ) − . Thus, this choice of b isalso conjugate to a .If b = − a − a − = − a +3 a − , then f b is conjugate to f a +3 a − and f a .If b = − a +3 a +1 = a − − a − = − a − a +1 , f b is conjugate to f a − a +1 and f a as well.The six parameter values that produce the same triple ( σ (2)1 , σ (2)2 , σ (2)3 ) are all conjugate, so ϕ issix-to-one and ˜ τ is injective on the image ϕ ( A ). (cid:3) Proposition 3.4.
The curve given by the component f a ( z ) = az − z − az of A ( D ) is a smooth irre-ducible curve of genus 0. Proof.
By Lemma 3.3 we can parameterize the curve with the coordinates ( σ (2)1 , σ (2)2 , σ (2)3 ) in A .In terms of the parameter a , this gives the equations(2 a + 36 a + 18 a + 72) − ( a − a + 1) σ = 0 , ( a + 76 a + 514 a + 1228 a + 9 a + 2268) − ( a − a + 6 a − a + 1) σ = 0 , (40 a + 1208 a a + 32528 a + 30744 a + 51192 a + 40824) − ( a − a + 6 a − a + 1) σ = 0 . Saturating with respect to ( a −
1) and eliminating a yields the relations0 = 916 σ − σ + σ − σ − σ σ + σ + 859456 σ − σ − . These relations define a smooth, irreducible curve of genus 0 in A . (cid:3) A ( C ) . We start with the family f a,b ( z ) = z + azbz +1 . Lemma 3.5.
The map ϕ : A \ { ab = 1 } → M ( a, b ) (cid:55)→ [ f a,b ]is two-to-one. The map τ : ϕ ( A \ { ab = 1 } ) → A [ f a,b ] (cid:55)→ ( σ , σ , σ )is injective. roof. We calculate the fixed point multiplier invariants as functions of ( a, b ) as σ ( a, b ) = a b + ab − ab + 3 a + 3 b − ab − σ ( a, b ) = a b − a b − a b + 4 a b + 7 a b + 4 ab − a b − ab + 7 ab − a − b + 9( ab − σ ( a, b ) = 1( ab − (cid:16) − a b + 5 a b + 5 a b − a b − a b − ab + 4 a + 14 a b + 14 ab + 4 b − a − ab − b + 9 a + 9 b (cid:17) , where we have omitted σ as usual due to its dependence from the standard index relation. Todetermine the degree of ϕ , we consider the equations( σ ( a, b ) , σ ( a, b ) , σ ( a, b )) = ( σ ( c, d ) , σ ( c, d ) , σ ( c, d )) . We saturate the resulting ideal with respect to ( ab −
1) and ( cd − A as( c, d ) = ( a, b ) or ( c, d ) = ( b, a ) . The maps f a,b and f b,a are conjugate by α ( z ) = z .In no other situation are the fixed point multiplier invariants equal, so τ is injective on theimage of ϕ . (cid:3) Proposition 3.6.
The component of A ( C ) given by f a,b = z + azbz +1 is an irreducible rational surfacedefined by0 = 36 σ − σ σ + σ σ − σ σ − σ σ + 8 σ σ σ − σ σ σ + 4 σ σ − σ − σ σ + 64 σ σ − σ σ + 8 σ + 4 σ σ + 60 σ σ σ − σ σ σ + 4 σ σ − σ σ σ + σ + 318 σ σ + 180 σ σ − σ + 261 σ σ − σ σ σ − σ σ − σ σ + 54 σ σ + 27 σ + 684 σ − σ σ − σ − σ σ + 108 σ − σ + 864 σ + 432 σ + 1728whose projective closure is parameterized by the map φ : P → S ⊂ P ( x : y : z ) (cid:55)→ ( φ : φ : φ : φ )for φ = 112 x y + 6 x z + 2 x yz + 24 x z + 18 x yz + 72 x yz + 288 x z + 108 xyz − xz φ = 1324 x y + 9 x z + 374 x yz − x z + 51 x yz + 216 x z + 90 x yz − x z + 54 xyz − xz + 324 yz + 1296 z φ = x y + 232 x yz + 36 x yz + 36 x yz − yz φ = x z + 232 x z + 36 x z + 36 x z − z . This surface is singular with singular locus given by the conjugacy classes described by { ( a, b ) : a = b } ∪ { ( a, b ) : a + b = − } ∪ { ( a, b ) : b = 3 a + 2 } . roof. To obtain the surface equations, take the fixed point multiplier invariants and consider theideal generated by( a b + ab − ab + 3 a + 3 b − − ( ab − σ , ( a b − a b − a b + 4 a b + 7 a b + 4 ab − a b − ab + 7 ab − a − b + 9) − ( ab − σ , ( − a b + 5 a b + 5 a b − a b − a b − ab + 4 a + 14 a b + 14 ab + 4 b − a − ab − b + 9 a + 9 b ) − ( ab − σ . We saturate by the ideal ( ab −
1) to avoid the parameters where the family is degenerate andcompute the elimination ideal to eliminate the variables a and b . This results in the single equationin ( σ , σ , σ ) in the statement.The parameterization was computed in Magma and is easily checked by substituting ( σ , σ , σ ) =( φ /φ , φ /φ , φ /φ ) into the surface equation.The singular locus is also computed in Magma and its irreducible components computed in Sageare given by S : σ = 4 σ = 6 σ = 4 S : (cid:40) σ − σ − σ + 12 = 04 σ + 4 σ σ + σ − σ − σ + 216 = 0 S : (cid:40) σ − σ σ − σ + 12 σ + 36 = 06 σ + σ σ + 15 σ − σ − σ −
36 = 0 . To analyze the components, we proceed similarly through elimination of variables. Starting withthe ideal defined by the equations of the component along with the defining equations of the fixedpoint multiplier invariants we eliminate ( σ , σ , σ ) to get an ideal in a and b . Then we saturate by( ab −
1) to avoid the degenerate elements. The component S corresponds to the degenerate case a = b = 1.The component S results in the components( a + b + 6)( b − a ) = 0 . When a = b , this is the component of A ( D ) from Proposition 3.2.The component S results in the components( ab + 2 b − ( ab + 2 a − = 0 . These corresponds to a = b +2 and b = a +2 . Since f a,b and f b,a are conjugate by α ( z ) = z , this is asingle component in the moduli space. (cid:3) Now we move to the next family with a C symmetry, f a,b ( z ) = az +1 z + bz . Lemma 3.7.
For the family f a,b ( z ) = az +1 z + bz , the image of the map τ : [ f a,b ] (cid:55)→ ( σ , σ , σ ) is acurve given by 0 = σ − σ − σ + 120 = 4 σ + 4 σ σ + σ − σ − σ + 216 . roof. The fixed point multiplier invariants give the equations0 = (4 a − ab + 4 b + 12) − σ ( ab − a − a b + 22 a b − ab + 4 b + 28 a − ab + 28 b + 54) − σ ( a b − ab + 1)0 = ( − a + 28 a b − a b + 28 ab − b − a + 96 ab − b − − σ ( a b − ab + 1) . Saturating with respect to ( ab −
1) to avoid degeneracy and eliminating a and b give the statedequations. (cid:3) To study the geometry of this family, we need to use the multiplier invariants of the periodicpoints of period 2.
Lemma 3.8.
The map ϕ : A \ { ab = 1 } → M ( a, b ) (cid:55)→ [ f a,b ]is four-to-one. The map τ : ϕ ( A \ { ab = 1 } ) → A [ f a,b ] (cid:55)→ ( σ (1)1 , σ (1)2 , σ (1)3 , σ (2)1 , σ (2)2 )is injective. Proof.
To compute the degree of ϕ , we consider the ideal generated by (cid:16) σ ( a, b ) , σ ( a, b ) , σ ( a, b ) , σ (2)1 ( a, b ) , σ (2)2 ( a, b ) (cid:17) = (cid:16) σ ( c, d ) , σ ( c, d ) , σ ( c, d ) , σ (2)1 ( c, d ) , σ (2)2 ( c, d ) (cid:17) as an ideal in K [ a, b ] where K is the function field Q ( c, d ). This forms a zero dimensional varietyand Singular (via Sage) computes the degree of its projective closure as 4. For (almost) every choiceof ( c, d ), we have the four pairs ( a, b ) ∈ { ( c, d ) , ( d, c ) , ( − c, − d ) , ( − d, − c ) } . The functions f a,b ( z ) and f b,a ( z ) are conjugate via α ( z ) = z , and f a,b ( z ) and f − a, − b ( z ) are conjugate via α ( − z ) = iz . Sothe four points with the same A coordinates all are from the same conjugacy class. (cid:3) Proposition 3.9.
The component of A ( C ) given by f a,b ( z ) = az +1 z + bz is an irreducible surfacedefined by σ − σ − σ + 120 = 4 σ + 4 σ σ + σ − σ − σ + 2160 = 1632 σ σ + 792 σ + 3648 σ σ σ (2)1 + 1760 σ σ (2)1 + 1248 σ σ ( σ (2)1 ) + 596 σ ( σ (2)1 ) − σ σ ( σ (2)1 ) − σ ( σ (2)1 ) − σ σ ( σ (2)1 ) − σ ( σ (2)1 ) + 96 σ ( σ (2)1 ) + 40 σ ( σ (2)1 ) + ( σ (2)1 ) − σ σ − σ − σ σ σ (2)1 − σ σ (2)1 − σ σ ( σ (2)1 ) − σ ( σ (2)1 ) + 54720 σ σ ( σ (2)1 ) + 28048 σ ( σ (2)1 ) + 1176 σ ( σ (2)1 ) + 1480 σ ( σ (2)1 ) − σ (2)1 ) + 5248 σ σ σ (2)2 + 2512 σ σ (2)2 + 6784 σ σ σ (2)1 σ (2)2 + 3200 σ σ (2)1 σ (2)2 − σ σ ( σ (2)1 ) σ (2)2 − σ ( σ (2)1 ) σ (2)2 − σ ( σ (2)1 ) σ (2)2 − σ ( σ (2)1 ) σ (2)2 − σ (2)1 ) σ (2)2 + 18436608 σ σ + 9945984 σ + 16035200 σ σ σ (2)1 + 9804928 σ σ (2)1 + 1465472 σ σ ( σ (2)1 ) + 1204512 σ ( σ (2)1 ) − σ ( σ (2)1 ) − σ ( σ (2)1 ) − σ (2)1 ) − σ σ σ (2)2 − σ σ (2)2 − σ σ σ (2)1 σ (2)2 − σ σ (2)1 σ (2)2 + 9984 σ ( σ (2)1 ) σ (2)2 + 4736 σ ( σ (2)1 ) σ (2)2 + 4992( σ (2)1 ) σ (2)2 + 3968 σ σ ( σ (2)2 ) + 1824 σ ( σ (2)2 ) + 1536 σ σ (2)1 ( σ (2)2 ) + 640 σ σ (2)1 ( σ (2)2 ) + 48( σ (2)1 ) ( σ (2)2 ) − σ σ − σ − σ σ σ (2)1 − σ σ (2)1 − σ ( σ (2)1 ) − σ ( σ (2)1 ) + 1890944( σ (2)1 ) + 7087104 σ σ σ (2)2 + 5656320 σ σ (2)2 + 1651968 σ σ (2)1 σ (2)2 + 2407680 σ σ (2)1 σ (2)2 − σ (2)1 ) σ (2)2 − σ ( σ (2)2 ) − σ ( σ (2)2 ) − σ (2)1 ( σ (2)2 ) − σ (2)2 ) + 1187592192 σ σ + 1880381952 σ + 265006080 σ σ (2)1 + 812934144 σ σ (2)1 + 32237568( σ (2)1 ) − σ σ (2)2 − σ σ (2)2 − σ (2)1 σ (2)2 + 345600( σ (2)2 ) − σ − σ − σ (2)1 + 123669504 σ (2)2 + 11418402816 = 1224 σ + 3792 σ σ (2)1 − σ σ ( σ (2)1 ) − σ ( σ (2)1 ) − σ σ ( σ (2)1 ) − σ ( σ (2)1 ) + 1680 σ σ ( σ (2)1 ) − σ ( σ (2)1 ) + 27600 σ σ ( σ (2)1 ) + 13988 σ ( σ (2)1 ) − σ ( σ (2)1 ) − σ ( σ (2)1 ) − σ (2)1 ) + 670152 σ + 26777472 σ σ σ (2)1 + 14457920 σ σ (2)1 + 27403200 σ σ ( σ (2)1 ) + 13581308 σ ( σ (2)1 ) − σ σ ( σ (2)1 ) − σ ( σ (2)1 ) − σ σ ( σ (2)1 ) − σ ( σ (2)1 ) + 3338448 σ ( σ (2)1 ) + 1331704 σ ( σ (2)1 ) + 78251( σ (2)1 ) + 120768 σ σ σ (2)2 + 66096 σ σ (2)2 + 204416 σ σ σ (2)1 σ (2)2 + 114464 σ σ (2)1 σ (2)2 − σ σ ( σ (2)1 ) σ (2)2 − σ ( σ (2)1 ) σ (2)2 − σ σ ( σ (2)1 ) σ (2)2 − σ ( σ (2)1 ) σ (2)2 + 58512 σ ( σ (2)1 ) σ (2)2 + 28460 σ ( σ (2)1 ) σ (2)2 + 264( σ (2)1 ) σ (2)2 − σ σ − σ − σ σ σ (2)1 − σ σ (2)1 − σ σ ( σ (2)1 ) − σ ( σ (2)1 ) + 1631518720 σ σ ( σ (2)1 ) + 743089904 σ ( σ (2)1 ) + 44906280 σ ( σ (2)1 ) + 48051640 σ ( σ (2)1 ) − σ (2)1 ) + 53976320 σ σ σ (2)2 + 28303472 σ σ (2)2 + 128916096 σ σ σ (2)1 σ (2)2 + 63843648 σ σ (2)1 σ (2)2 + 5531008 σ σ ( σ (2)1 ) σ (2)2 + 2665024 σ ( σ (2)1 ) σ (2)2 − σ ( σ (2)1 ) σ (2)2 − σ ( σ (2)1 ) σ (2)2 − σ (2)1 ) σ (2)2 + 352512 σ σ ( σ (2)2 ) + 184416 σ ( σ (2)2 ) + 278272 σ σ σ (2)1 ( σ (2)2 ) + 149184 σ σ (2)1 ( σ (2)2 ) − σ ( σ (2)1 ) ( σ (2)2 ) − σ ( σ (2)1 ) ( σ (2)2 ) − σ (2)1 ) ( σ (2)2 ) + 516653715456 σ σ + 263589240192 σ + 484361825920 σ σ σ (2)1 + 278148291968 σ σ (2)1 + 48513480448 σ σ ( σ (2)1 ) + 38458271328 σ ( σ (2)1 ) − σ ( σ (2)1 ) − σ ( σ (2)1 ) − σ (2)1 ) − σ σ σ (2)2 − σ σ (2)2 − σ σ σ (2)1 σ (2)2 − σ σ (2)1 σ (2)2 + 294902016 σ ( σ (2)1 ) σ (2)2 + 92512768 σ ( σ (2)1 ) σ (2)2 + 179583360( σ (2)1 ) σ (2)2 + 98552320 σ σ ( σ (2)2 ) + 46508640 σ ( σ (2)2 ) + 55121664 σ σ (2)1 ( σ (2)2 ) + 22765952 σ σ (2)1 ( σ (2)2 ) + 3316752( σ (2)1 ) ( σ (2)2 ) + 221952 σ ( σ (2)2 ) + 114240 σ ( σ (2)2 ) + 1408 σ (2)1 ( σ (2)2 ) − σ σ − σ − σ σ σ (2)1 − σ σ (2)1 − σ ( σ (2)1 ) − σ ( σ (2)1 ) + 62583639424( σ (2)1 ) + 225002594304 σ σ σ (2)2 + 166476388608 σ σ (2)2 + 55674839808 σ σ (2)1 σ (2)2 + 76642562304 σ σ (2)1 σ (2)2 − σ (2)1 ) σ (2)2 − σ ( σ (2)2 ) − σ ( σ (2)2 ) − σ (2)1 ( σ (2)2 ) − σ (2)2 ) + 38553253060608 σ σ + 58922313076224 σ + 8872947753984 σ σ (2)1 + 26426257744896 σ σ (2)1 + 1107358166016( σ (2)1 ) − σ σ (2)2 − σ σ (2)2 − σ (2)1 σ (2)2 + 11296544256( σ (2)2 ) − σ − σ − σ (2)1 + 4111788303360 σ (2)2 + 380265217671168 . This surface is singular with singular locus given by the conjugacy classes described by { ( a, b ) : a − ab + b + 3 = 0 } ∪ { ( a, b ) : a = − b } . Proof.
We compute the fixed point multiplier invariants and the first two multiplier invariants forthe points of period 2: σ = 4 a − ab + 4 b + 12 ab − σ = 4 a − a b + 22 a b − ab + 4 b + 28 a − ab + 28 b + 54 a b − ab + 1 σ = − a + 28 a b − a b + 28 ab − b − a + 96 ab − b − a b − ab + 1 σ (2)1 = 2 a b + 16 a + 4 a b + 16 b + 18 ab + 72 a b − ab + 1 σ (2)2 = 1( ab − ( a b + 32 a b + 12 a b + 32 a b + 96 a + 16 a b + 290 a b + 16 a b + 96 b + 512 a b + 204 a b + 512 ab + 80 a − a b + 80 b + 2268) . We clear denominators to create the associated ideal. We saturate by the ideal ( ab −
1) to avoiddegeneracy and eliminate the variables a and b . This results in the surface defined by the fourequations in the statement. n Magma the irreducible components of the singular locus are calculated. For each component,we eliminate variables to have equations in ( σ , σ , σ ) to get S : σ − σ − σ + 1080 = σ σ + 36 σ + 6 σ − σ − σ − σ + 6480 = σ − σ − σ + 12 S : σ − σ − σ + 135000 = σ σ + 6408 σ + 2886 σ − σ − σ − σ + 602640 = σ − σ − σ + 12 S : σ + 549 σ − σ − σ + 148191120 = σ − / σ + 102 σ + 238 / σ − /
50 = σ σ + 29 / σ − σ − / σ + 3078 /
50 = σ − σ − σ + 12 . To determine the pairs ( a, b ) for each of these components, we add in the equations defining theinvariants in terms of a and b and eliminate σ , σ , σ . This results in the defining equations S : ( a − ab + b + 3) = 0 S : ( a + b ) = 0 S : ( a + b ) = 0 . Note that the component(s) with a = − b is the component of A ( D ) from Proposition 3.4. (cid:3) Automorphism loci in M As for A we utilize Miasnikov, Stout, and Williams [31] to determine the possible componentsof A and their dimensions. Unlike in M , in M there are no automorphism groups other thancyclic and dihedral ones. Lemma 4.1.
When deg( f ) = 4, Aut( f ) must either be C , C , C , C , D , or D . The dimensionsof A (Γ) for these groups are given by(1) dim A ( C ) = 3 , dim A ( C ) = 2, dim A ( C ) = 1, and dim A ( C ) = 0.(2) dim A ( D ) = 1, and dim A ( D ) = 0.Moreover,(4) A ( C ) ⊂ A ( C ).(5) A ( D ) ⊂ A ( C ).(6) A ( D ) ⊂ A ( C ). Proof.
Which groups occur and the dimensions are calculated in Section 2 of MSW [31]. The secondhalf follows from the observation that if G is a subgroup of H , then A d ( H ) ⊆ A d ( G ). (cid:3) The analysis in M for functions whose automorphism group contains cyclic and dihedral groupsis more or less the same as in M – in particular, we made use of the fact that for a function f , ifAut( f ) ⊇ C n then the equivalence class of f in M can be written as f = z · ψ ( z n ), where ψ is arational function [41, Proposition 7.3]. .1. A ( C ) and A ( D ) .Proposition 4.2. The loci A ( C ) and A ( D ) both are the single conjugacy class given by f ( z ) = z . Proof.
The only form ψ can take such that z · ψ ( z ) is degree four is when ψ ( z ) = abz , where ab (cid:54) = 0.We can divide through by a to write z · ψ ( z ) = cz ; and since this is conjugate to f ( z ) = z , weknow that A ( C ) is just a point in M . Furthermore, Aut( f ) = D so that A ( C ) = A ( D ). (cid:3) A ( C ) .Proposition 4.3. The locus A ( C ) is given by either 1-parameter family f ( z ) = z +1 kz or f ( z ) = kzz +1 for k (cid:54) = 0. Proof.
Functions of the form f ( z ) = z · ψ ( z ) have degree 4 only when f ( z ) = az + bcz or f ( z ) = bzcz + d , where all the coefficients must be nonzero (or else we have a drop in degree due to cancellation).Thus, in both cases we can divide through by the coefficient of z to have two-parameter families.Then in the first case, we can conjugate f ( z ) = z + k k z via the matrix α = (cid:20) cb /
00 1 (cid:21) to get a one-parameter family f (cid:48) ( z ) = z +1 kz , which is what we should expect since the dimension of this locus is1 by Lemma 4.1. In the second case, we can conjugate f ( z ) = k zz + k by the matrix β = (cid:20) cd /
00 1 (cid:21) to get a one-parameter family f (cid:48) ( z ) = kzz +1 . These two one-parameter families are conjugate via (cid:20) (cid:21) , so they are the same family in the moduli space. (cid:3) A ( C ) and A ( D ) .Proposition 4.4. The locus A ( C ) is given by the family f k ,k ( z ) = z + k z k z +1 . Proof.
By Lemma 4.1 the locus A ( C ) is dimension 2 and functions of the form f ( z ) = z · ψ ( z )have degree 4 only when f ( z ) = az + bzcz + d , where a (cid:54) = 0 and d (cid:54) = 0. Thus, dividing through by a , we get the 3-parameter family f ( z ) = z + k zk z + k , where k (cid:54) = 0. Conjugating via the matrix α = (cid:20) √ k
00 1 (cid:21) , we get f α ( z ) = ( √ k z ) + k √ k zk ( √ k z ) + k · √ k = k z + k zk k z + k = z + k /k zk z + 1 . Renaming k = k /k and k = k , we see that f α ( z ) = z + k zk z +1 . Thus, every degree 4 rational mapwith a C automorphism is conjugate to a map of the form f k ,k ( z ) = z + k zk z + 1 . (cid:3) ecall that D is generated by α ( z ) = ζ z and α ( z ) = 1 /z , where ζ is some third root ofunity. Maps with automorphism group containing ζ z are described in Proposition 4.4, so we canstart with that family and see which members additionally have α as automorphism. Proposition 4.5.
The locus A ( D ) is given by the family f k ( z ) = z + kzkz +1 . Proof.
Let f k ,k ( z ) = z + k zk z +1 . We compute f α k ,k ( z ) = z + k zk z + 1 . So for f k ,k ( z ) = f α k ,k ( z ) we must have k = k . (cid:3) A ( C ) .Proposition 4.6. The locus A ( C ) is given by the 3-parameter family f k ,k ,k ( z ) = z + k z +1 k z + k z . Proof.
By Lemma 4.1 we know the automorphism loci of C in the moduli space has dimension 3.Furthermore, we know that a degree 4 map f has a C automorphism if and only if it is of the form f ( z ) = az + bz + cdz + ez or f ( z ) = az + bzcz + dz + e . First observe that for the family f ( z ), we can conjugate by the matrix β = (cid:20) (cid:21) to obtain f β ( z ) = c + dz + ez az + bz . Thus, the families f and f are the same in moduli space, so we consider only f ( z ). For f ( z ) tobe degree 4, we need a (cid:54) = 0 and c (cid:54) = 0. Dividing through by a , we get f ( z ) = z + k z + k k z + k z , where k (cid:54) = 0. Conjugating via the matrix α = (cid:20) √ k
00 1 (cid:21) , we get f α ( z ) = ( √ k ) z + k ( √ k ) z + k k ( √ k ) z + k √ k z · √ k = z + k √ k z + 1 k z + k √ k z . Renaming k = k √ k , and k = k , and k = k √ k , we see that every f map is conjugate to a mapof the form f k ,k ,k ( z ) = z + k z + 1 k z + k z . (cid:3) Geometry of certain automorphism loci in M In this section, we examine the geometry of automorphism loci A (Γ) ⊂ M . Similar to themethods in Section 3, we will be studying the embedding via multiplier invariants τ n : M → A k [ f ] (cid:55)→ ( σ ( n )1 , . . . , σ ( n )4 n +1 )and the geometry of the image of this embedding. .1. A ( C ) . We start with the family f k ( z ) = z +1 kz for k (cid:54) = 0. Lemma 5.1.
The map ϕ : A \ { } → M k (cid:55)→ [ f k ]is injective, and thus so is the map τ : ϕ ( A \ { } ) → A [ f k ] (cid:55)→ ( σ (1)1 , σ (1)2 , σ (1)3 , σ (1)4 ) . Proof.
Consider the ideal generated by( σ ( k ) , σ ( k ) , σ ( k ) , σ ( k )) = ( σ ( k ) , σ ( k ) , σ ( k ) , σ ( k ) . Computing its lexicographic Groebner basis, we get two generators: k + 8 k − k − k ,k k + 8 k k + 16 k − k − k − k . These factor as ( k − k )( k + 4) , ( k − k )( k + k + 8) . The first says that for f k and f k to have the same fixed point multiplier invariants, we must have k = k or k = −
4. However, if k = −
4, then the second generator tells us that k = − k = k . Thus, the map ϕ and the map τ are one-to-one. (cid:3) Proposition 5.2.
The curve in A given by the image of τ on the family f k ( z ) = z +1 kz of A ( C )is given by the system of equations0 = 36 σ + 3 σ σ + 222 σ − σ σ − σ + 240 σ − σ − − σ − σ σ − σ + 32 σ + 6 σ + 400 = 144 σ − σ + 288 σ − σ + 36 σ − . It is a singular irreducible curve of genus 0 and the singularity corresponds to the rational map f = z + 1 − z . Proof.
The fixed point multiplier invariants give the equations0 = ( k − k + 16) − kσ − k + 70 k − k + 96) − k σ k − k + 528 k − k + 256) − k σ − k + 513 k − k + 1120 k − k + 256) − k σ . Saturating by the ideal ( k −
1) and looking at the generators gives the stated equations. UsingSage, we can determine that these relations define a singular, irreducible curve of genus 0 in A .The only point of singularity is ( − , , − , (cid:3) .2. A ( D ) . We now move on to the family f k ( z ) = z + kzkz +1 with k (cid:54) = ± Lemma 5.3.
The map ϕ : A → M k (cid:55)→ [ f k ]is injective, and thus so is the map τ : ϕ ( A ) → A [ f k ] (cid:55)→ ( σ (1)1 , σ (1)2 , σ (1)3 , σ (1)4 ) . Proof.
We want to compute the ideal generated by( σ ( k ) , σ ( k ) , σ ( k ) , σ ( k )) = ( σ ( k ) , σ ( k ) , σ ( k ) , σ ( k )) . This gives the ideal generated by k + 8 k − k − k ,k k + 8 k k + 16 k − k − k − k . This ideal is the same as the ideal considered in the proof of Lemma 5.1, so the result follows atonce. (cid:3)
Proposition 5.4.
The curve given by the image of τ on the family f k ( z ) = z + kzkz +1 of A ( D ) isdefined by 0 = σ − σ − σ σ + 67 σ + 44 σ σ + 12 σ − σ − σ + 12000 = σ + 2 σ − σ σ − σ − σ + 9 σ + 600 = − σ + 5 σ + 8 σ σ − σ − σ + 9 σ + 240 . It is a singular irreducible curve of genus 0, and the singularity ( − , , − , f = z − z − z + 1 . Proof.
The fixed point multiplier invariants give the equations0 = (2 k − k + 12) − ( k + 1) σ k − k + 25 k − k + 48) − ( k + 2 k + 1) σ − k + 24 k − k + 60 k + 64) − ( k + 3 k + 3 k + 1) σ k − k + 96 k − k + 128 k ) − ( k + 3 k + 3 k + 1) σ . Eliminating k gives the stated equations involving only the fixed point multiplier invariants. Choos-ing an appropriate monomial ordering, we arrive at the stated relations. These relations define asingular, irreducible curve of genus 0 in A . The only point of singularity is ( − , , − , (cid:3) .3. A ( C ) . Now we move on to study the family f k ,k ( z ) = z + k zk z +1 , with k k (cid:54) = 1. Lemma 5.5.
The map ϕ : A \ { k k = 1 } → M ( k , k ) (cid:55)→ [ f k ,k ]is two-to-one. The map τ : ϕ ( A \ { k k = 1 } ) → A [ f k ] (cid:55)→ ( σ (1)1 , σ (1)2 , σ (1)3 )is injective. Proof.
To compute the degree of ϕ , we consider the ideal generated by( σ ( k , k ) , σ ( k , k ) , σ ( k , k )) = ( σ ( t , t ) , σ ( t , t ) , σ ( t , t ))as an ideal in K [ k , k ], where K is the function field ¯ Q ( t , t ). This forms a zero dimensionalvariety, and Singular (via Sage) computes the degree of its projective closure as 2. For (almost)every choice of ( t , t ) we have two pairs ( k , k ) ∈ { ( t , t ) , ( t , t ) } and one can easily verify that f k ,k and f k ,k are conjugate via α ( z ) = z . (cid:3) It is interesting to note that σ (1)4 is determined uniquely by ( σ (1)1 , σ (1)2 , σ (1)3 ). This is not typicalfor elements of M . We do have the standard linear relationship between ( σ (1)1 , σ (1)2 , σ (1)3 ) and σ (1)5 for all maps in M , but the dependence of σ (1)4 is special to this family. Proposition 5.6.
The surface that is the image of τ of the family f k ,k ( z ) = z + k zk z +1 of A ( C ) isgiven by0 = 972 σ − σ σ − σ σ − σ σ + 324 σ σ + 162 σ σ σ − σ σ σ + 27 σ σ + 648 σ − σ σ + 1404 σ σ + 480 σ σ + 192 σ + 540 σ σ − σ σ σ − σ σ σ + 32 σ σ − σ σ σ + 1188 σ − σ σ + 7776 σ σ − σ + 5364 σ σ − σ σ σ − σ σ + 1404 σ σ + 648 σ σ + 108 σ + 2160 σ − σ σ + 3840 σ + 4320 σ σ − σ σ + 2160 σ − σ + 28800 σ − σ + 32000 . This surface is reduced, irreducible, and singular with singular locus given by the conjugacy classesdescribed by( k , k ) ∈ { (0 , / } ∪ { ( − , − } ∪ { (9 / , − } (cid:91) (cid:26)(cid:18) t + 466569 t + 216 t + 1944 t , t + 18 t t + 216 t + 1944 (cid:19) : t (cid:54) = 0 ∈ Q (cid:27)(cid:91) (cid:26)(cid:18) t − − t + 24 , − − t (cid:19) : t ∈ Q \ { } (cid:27)(cid:91) (cid:26)(cid:18) − t − , − t − (cid:19) : t ∈ Q \ { } (cid:27) roof. As usual, we first compute the fixed point multiplier invariants using only the first three: σ = 1 k k − k k + k k − k k + 8 k + 8 k − σ = 1( k k − ( k k − k k − k k + 9 k k + 28 k k + 9 k k − k k − k k + 18 k + 85 k k + 18 k − k − k + 48) σ = 1( k k − ( − k k + 21 k k + 21 k k − k k − k k − k k + 27 k k + 135 k k + 135 k k + 27 k k − k k − k k − k k + 153 k k + 153 k k − k − k k − k + 96 k + 96 k − . We look at the ideal in Q [ σ , σ , σ ] generated by the defining equations of the invariants. Wesaturate by the ideal ( k k −
1) to avoid degeneracy and eliminate the variables k and k . Thisresults in the surface defined by the equation in the statement.Using Magma, we see that the surface is reduced, irreducible, and singular. Since this is ahypersurface, we compute the singular locus as the points on the surface that also vanish on thepartial derivatives of the defining equation. This variety has irreducible components S : (cid:40) σ = σ σ − S : (cid:40) σ + 6 σ + σ −
200 = 36 σ + 12 σ σ + σ − σ + σ S : σ σ + 8 σ − σ σ − σ + 80 σ − σ + 2000 = 90 σ − σ + 9 σ σ + 180 σ − σ − σ + 128 σ σ − σ σ + 1800 σ − σ σ + 1280 σ σ − σ − σ + 30000 σ − σ + 60000 . The general procedure of analyzing the components is similar to the one used in Section 3: westart with the ideal defined by the equations of the component along with the defining equations ofthe fixed point multiplier invariants, eliminate ( σ , σ , σ ), and finally saturate by ( k k − S corresponds to the case k = 0 and k = 4 / S results in a rational curve given by0 = ( k + k + 8)(9 k k − k k + 9 k k + 12 k k + 12 k k + 4 k k − k − k + 64)0 = ( k + 4) (9 k k − k k + 9 k k + 12 k k + 12 k k + 4 k k − k − k + 64) . The common factor gives the rational curve parameterized in the statement. If k = −
4, theneither k = − k = 4 / S results in a variety with the two irreducible components0 = k k + 3 k − k k + 3 k − . These are conjugate by swapping ( k , k ) → ( k , k ) via z (cid:55)→ z . This curve is the final parameteri-zation given in the statement. (cid:3) A ( C ) . We now move on to the family f k ,k ,k ( z ) = z + k z +1 k z + k z .This family has infinitely many conjugacy classes that have the same fixed point multiplierinvariants. Computing the multiplier invariants for points of period two was computationallyinfeasible, but computing the multiplier invariants for the points of formal period two was possible. ecall that points with formal periodic n are the zeros of the n th dynatomic polynomial. Utilizingjust the σ (2) ∗ did result in a finite-to-one map, but there were a number of spurious values appearing.Additionally including σ (2) ∗ resulted in the correct mapping. As the computation is quite time andmemory consuming, we record the two higher multiplier invariants here. σ (2) ∗ = 1( k + k − k k k ) (cid:16) (8 k − k + ( − k k + 40 k − k k + (40 k − k + ( − k k + ( − k + 120) k + ( − k − k ) k + ( − k + 80 k − k + ( − k + ( − k + 24 k ) k + (8 k + 160) k + ( − k + 64 k ) k ) k + ((8 k − k − k k + (16 k − k ) (cid:17) σ (2) ∗ = 1( k + k − k k k ) (cid:16) (24 k − k − k + ( − k k + 320 k − k k + (320 k − k − k k + (168 k + 288)) k + ((24 k − k − k k + (200 k + 1216) k + ( − k − k ) k + ( − k + 2452 k − k + ( − k − k ) k + (360 k − k + 5568)) k + (16 k k + (64 k − k + ( − k − k ) k + (64 k + 664 k + 2712) k + (64 k − k − k ) k + ( − k + 8312 k − k + ( − k − k + 832 k ) k + ( − k + 2656 k − k + 16896)) k + ( − k + 160 k k + (64 k + 32 k − k + ( − k − k ) k + (242 k − k + 6652) k + ( − k − k − k ) k + (40 k − k + 10512 k − k + (32 k − k + 4992 k − k ) k + (16 k − k + 3872 k − k + 16896)) k + ( − k + ( − k + 280 k ) k + (64 k + 312 k − k + (72 k − k + 6008 k ) k + (64 k + 952 k − k + 13152) k + (64 k − k − k + 1920 k ) k + (192 k + 832 k − k − k + (192 k − k + 9728 k − k ) k ) k + (144 k + (128 k − k ) k + (24 k − k + 532 k − k + ( − k − k + 6016 k ) k + (16 k + 152 k − k + 5184) k + ( − k − k + 22528 k ) k + ( − k + 2112 k − k + 13312) k ) k + (( − k + 1224) k + ( − k + 680 k − k ) k + (320 k − k − k + ( − k + 1776 k − k ) k + (736 k + 512 k − k + ( − k + 512 k − k ) k ) k + ((24 k − k + 1554) k + ( − k + 1696 k ) k + (80 k − k + 3392) k + ( − k + 2560 k ) k + (32 k − k + 512) k ) (cid:17) . The first two fixed point multiplier invariants are given by σ (1)1 = 1 k + k − k k k (cid:16) − k k k − k k + k + 4 k k k − k k − k + 4 k + 16 k (cid:17) σ (1)2 = 1( k + k − k k k ) (cid:16) k k k − k k k + 4 k k k + 4 k k − k k − k k k + 16 k k k + 18 k k k − k k − k k k + 28 k k − k + 8 k k k − k k k − k k k + 20 k k + 4 k k k − k k + 70 k + 96 k k k − k k − k k + 16 k k − k − k − k k k − k k + 96 k + 32 k (cid:17) . roposition 5.7. The map ϕ : A \ { k + k = k k k } → M ( k , k , k ) (cid:55)→ [ f k ,k ,k ]is two-to-one. The map τ : ϕ ( A \ { k + k = k k k } ) → A [ f k ] (cid:55)→ ( σ (1)1 , σ (1)2 , σ (2) ∗ , σ (2) ∗ )is injective. Proof.
Trying to compute the degree generically as in the previous families was not feasible withour hardware resources, so we instead specialize to a particular choice of invariants and show thatthe degree is invariant under perturbation.First note that the domain of ϕ is irreducible and the composition τ ◦ ϕ is a morphism, so theimage is also irreducible. From Milne [32, Section 10] the dimension of fibers in this situation canonly go up in a closed set, and the number of points in a specific fiber is at most the degree of themorphism and is equal to the degree for nonsingular fibers. We choose a nonsingular fiber wherethe fiber dimension is zero and has two points in it showing that the degree is two.Choosing ( σ (1)1 , σ (1)2 , σ (2) ∗ , σ (2) ∗ ) = (cid:18) − , − , , (cid:19) generates a system of equations in ( k , k , k ). We take the associated ideal and saturate withrespect to ( − k k k + k + k ) to remove any degenerate solutions from the system. The resultingideal is given by I = ( k − , k − k , k − . This has the two solution (1 , ,
3) and ( − , , − z (cid:55)→ iz . Call X thevariety associated to I . The projective closure of X is dimension 0 and degree 2, so there should betwo points when counted with multiplicity. The two given solutions are both solutions of multiplicity1, so they are all the solutions to the system.In general, we always get the two conjugate solutions ( k , k , k ) and ( − k , k , − k ), so ϕ istwo-to-one. Since these two solutions are conjugate τ is injective. (cid:3) Computing the equation of the dimension three hypersurface for this family via elimination wasbeyond the capabilities of our hardware.6.
Rational Preperiodic Structures in M Having completed the description of the loci A and A and some of their geometric properties,we now turn to arithmetic dynamical properties of the families that make up the automorphismloci. Specifically, we look at the possible structures of Q -rational preperiodic points. The mainmotivation is the far-reaching and open conjecture of Morton and Silverman that the number ofrational preperiodic points should be bounded independently of the particular map chosen. Conjecture 6.1 (Morton-Silverman [34]) . Let f : P N → P N be a morphism of degree d ≥ K of degree D . Then the number of K -rational preperiodic points for f is bounded by a constant C depending only on N , d , and D .The best known results typically make some kind of restriction of f , such as good reduction atcertain primes. We are more interested in results such as Poonen [38] or Manes [28] that restrictto special families. There are a number of results in this area, but we focus primarily on these twoas they are closest in type to the our results for the families in A and A . Poonen completely lassified all possible Q -rational preperiodic graph structures for the family f c ( z ) = z + c assumingthat there are no rational periodic points with minimal period at least 4. This has been generalizedto quadratic fields in [9]. Manes did the same for a family of quadratic rational maps with C automorphisms assuming there were no Q -rational periodic points with minimal period at least5. In particular, her family was the automorphism locus A . We proceed along the same lines asfollows:(1) Provide computational evidence of an upper bound on the minimal period of a Q -rationalpreperiodic point.(2) Analyze all possible rational preperiodic structures assuming that bound on the minimalperiod.In the case of the families with dimension in moduli space at most 1, we are able to completethe classification of rational preperiodic graph structures, with the exception of A ( D ), where theclassification has possibly finitely many exceptional parameters. These results make heavy use oftechniques for finding rational points on curves. For the families of dimension 2 and 3, the difficultyin finding all rational points on surfaces and dimension 3 varieties is the impediment to completingthose classifications. In lieu of a full classification, we examine the existence of periodic points andtake a computational census of the possible rational preperiodic graph structures. As techniquesin these areas improve, it would be good to return to this topic and complete those classifications.We end this introduction with a helpful lemma for points of curves and some helpful references. Lemma 6.2.
Let C be a projective curve defined over Q . Suppose that there is a birationalmap defined over Q between C and a smooth projective hyperelliptic curve X over Q . Then thenonsingular rational points on C is bounded above by | X ( Q ) | . Proof.
Since both C and X are curves, and C is birationally equivalent to X , we know that thereis a Q -rational isomorphism between X and C , the smooth projective model of C [16, Theorem 3in Section 7.5]. In particular, | C ( Q ) | = | X ( Q ) | . Furthermore, we have a surjective map from ¯ C to C , so the number of non-singular rational points on C is bounded above by | C ( Q ) | = | X ( Q ) | . Theonly other rational points on C are the singular points that could come from non-rational pointson X . (cid:3) For references on curve quotienting and blow-ups, see Lorenzini [26] and Liu [27].We recall the definition of dynatomic polynomials and dynatomic curves. For a rational map f ( z ) = F ( z ) G ( z ) define Φ ( f ) = F ( z ) − zG ( z ). Its roots are the points with period 1. Note thatΦ n ( f ) = Φ ( f n ). We define the n th dynatomic polynomial as Φ ∗ n = (cid:81) d | n Φ d ( f ) µ ( n/d ) where µ isthe M¨obius function. Its roots are the points of formal period n and contain among them thepoints of minimal period n . Similarly, for preperiodic points, we define the generalized ( m, n ) -dynatomic polynomial as Φ ∗ m,n ( f ) = Φ ∗ n ( f m )Φ ∗ n − ( f m ) . Its roots are the points of formal period ( m, n ) andcontain among them the points of minimal period ( m, n ). See Silverman [43, § The Single Conjugacy Classes.
The cases A ( C ), A ( D ), and A ( A ) consist of singleconjugacy classes. Using the algorithm from Hutz [20], we compute each such structure. Theorem 6.3.
We have the following rational preperiodic structures. • For A ( C ) = A ( D ), we represent the conjugacy class as f ( z ) = z . This function hasrational periodic structure given by • • ∞ (cid:35) (cid:35) (cid:98) (cid:98) • (cid:121) (cid:121) • − (cid:118) (cid:118) For A ( A ), we represent the conjugacy class as f ( z ) = z − − z . This function has rationalperiodic structure given by • • ∞ (cid:47) (cid:47) (cid:119) (cid:119) Proof.
Direct computation via the algorithm of Hutz [20] as implemented in Sage. (cid:3) A ( C ) . We saw in Proposition 2.8 that the family f a ( z ) = z + aaz with a (cid:54) = 0 gives A ( C ).We first examine an upper bound on the minimal period of a Q -rational periodic point. Wemake use of Lemma 6.4. Lemma 6.4. If F ∈ Q [ x, y ] is irreducible over Q but is reducible over some extension field K of Q , and all the components are defined over K , then every rational point on the curve defined by F = 0 is singular. Proof.
Suppose F factors into F = g g . . . g n over K . Consider the set S = { g σ i : σ i ∈ Gal( K/ Q ) } .Observe first that g σ i has to be another component of F , since F σ i = g σ i g σ i . . . g σ i n = F . Then thepolynomial g = (cid:81) h ∈ S h must be invariant under the Galois action and is, therefore, defined over Q . Since g | F and F is irreducible, we know that g = F (up to scaling). In particular, for every g i and g j , we can find a σ ∈ Gal( K/ Q ) such that g σi = g j . If P is a rational solution to F = 0, itis a rational solution to some g i = 0. But g j ( P ) = g σi ( σ ( P )) = σ ( g i ( P )) = 0, so P is in fact a rootof all the components of F and, thus, must be in the intersection. It then follows that P must bea singular point on the curve defined by F = 0. (cid:3) Proposition 6.5.
Let f a ( z ) = z + aaz .(1) The point ∞ is fixed for all choices of parameter a . There is a second Q -rational fixed pointfor the parameters a = − t for t ∈ Q \ { , } . These are the only occurring Q -rational fixedpoints.(2) There are no rational parameters a where f a ( z ) has a Q -rational periodic point with minimalperiod 2 or 3. Proof.
Clearly ∞ is fixed, so to look for additional fixed points, we examine the first dynatomicpolynomial Φ ∗ ( f a ) = (1 − a ) z + a, and the associated dynatomic curve Φ ∗ ( f a ) = 0. We want values of a where this curve admitsrational points. Linearity in a allows us to quickly solve when z = t ∈ Q , a = z z − − t . Each finite fixed point z ∈ Q determines a unique a , so we can never have more than two fixedpoints.To determine rational 2-cycles, we compute the second dynatomic polynomialΦ ∗ ( f a ) = ( a + 1) z + ( − a + a + 2 a ) z + a . The curve defined by Φ ∗ ( f a ) = 0, called the second dynatomic curve, has genus 3; so by Falting’stheorem, it only has finitely many rational points. It is not hyperelliptic, but observe that bysetting x = z , we can see that it covers a curve of lower genus: X : ( a + 1) x + ( − a + a + 2 a ) x + a = 0 . The curve X has genus 0. In fact, X has a rational parameterization from P given by t (cid:55)→ ( x, a ) = (cid:18) − t − t − t + t , t + 9 t + 5 t + 1 t + t (cid:19) . hus, there are infinitely many rational points on X . However, not every rational point can belifted back to a rational point on the second dynatomic curve. In fact, a rational point ( x, a ) canbe lifted to be a rational point on the dynatomic curve only if x = − t − t − t + t = k for some rationalnumber k . This gives us another curve: Y : − t − t − − k (2 t + t ) = 0 . This curve has genus 2, so it is hyperelliptic with simplified model
Y Y : y = x − x − . Moreover, the rank of its Jacobian is 0 and the torsion subgroup has order 6. A search using heightbound 1000 produces all six points on the Jacobian:(1 , , , (1 , x , , (1 , − x , , ( x + 1 , x + 1 , , ( x + 1 , − x − , , ( x − x + 1 , , . The points with 1 as x -coordinate correspond to points at infinity on the curve Y Y , and the onlyaffine rational points on
Y Y are the ones where x = −
1. Then the only affine rational point of
Y Y is ( − , , x − x − − , (1 : 1 : 0). Thus, there are at most three rational points on the curve Y , and a search using height bound 1000 finds all of them:(0 : 1 : 0) , (1 : 0 : 0) , ( − − , where t is the first coordinate, k the second, and a homogenization variable the third. Observethat we cannot have t = 0 (which will give us a 0 in the denominator of the first coordinate in( x, a )) or t = − x, a )).When t = 1, we have ( x, a ) = ( − / , / Rational Algebraic Curves by J. Rafael,Sendra Franz Winkler, and Sonia P´erez-D´ıaz [39, Theorem 6.22], we can determine that this rationalparameterization is not normal (a rational parameterization is normal if and only if it is surjective),and the only missing point is ( − , Rational Algebraic Curves worksfor curves and rational parameterizations defined over some algebraically closed field. Both thecurve X and the rational parameterization are defined over Q , but we can certainly embed into Q ,and the algorithm says the only missing point is ( − ,
6) when we consider them over Q . This pointis still Q -rational.Now we look at the rational 3-cycles. We want to determine if there are any rational points onthe third dynatomic curve defined by the vanishing ofΦ ∗ ,f a ( a, z ) =( a + a + 1) z + ( a + a + a + 6 a + 7 a + 8 a ) z + ( a a + a + 2 a + 3 a + 4 a + 15 a + 21 a + 28 a ) z + ( a + 2 a + 3 a + 2 a + 5 a + 20 a + 35 a + 56 a ) z + ( a + 2 a + 4 a − a − a + 15 a + 35 a + 70 a ) z + ( a + 3 a − a − a − a + 6 a + 21 a + 56 a ) z + ( a − a − a + a + 7 a + 28 a ) z + ( − a + a + 8 a ) z + a . his curve is reducible over Q , and it has the following components: X : a z + az + z + a z + 2 az + a = 0 X : a z + a z + a z + z + 2 a z + a z + 6 az + 15 a z − a z + 20 a z − a z + 15 a z + 6 a z + a = 0 . Both components are irreducible over Q but reducible over some extension field of Q . For curvessatisfying this condition, we have Lemma 6.4. Thus, any Q -rational points on the curve X mustbe in the intersection of the two components. In particular, any Q -rational points must be singular.The only singular point on X is (0 , X . It turnsout that X is reducible over the extension field K / Q whose generator is defined by x + x + =0, and both components are defined over this extension. Thus, we can use the same argument andcompute the singular points for X . The only singular point is (0 , Q -rationalpoint on the dynatomic curve is (0 , a = 0 since a = 0 gives us a zeroin the denominator for the rational map f a ( z ). Therefore, no member of the family has rational3-cycles. (cid:3) A search for rational preperiodic structures with the parameter up to height 10 ,
000 using thealgorithm from [20] as implemented in Sage yields no parameters where f a has a Q -rational periodicpoint with minimal period at least 4. So we make the following conjecture. Conjecture 6.6.
There is no a ∈ Q so that f a ( z ) = z + aaz has a Q -rational preperiodic point withminimal period at least 4.Assuming this conjecture, we are able to classify all possible Q -rational preperiodic structures.One of the curves appearing in the proof of Theorem 6.8 is a non-hyperelliptic genus 3 withtrivial automorphism group. The standard implementations do not yield a sharp point estimate,so we treat computing its rational points in Lemma 6.7. Lemma 6.7.
Assuming the (weak) Birch-Swinnerton-Dyer conjecture, the curve C ⊆ P definedby C : x y + xy − x − x y − xy + y = 0has exactly the following six points as Q -rational points. { (1 : 1 : 1) , (1 : 1 : − , (0 : 1 : 0) , (0 : 0 : 1) , (1 : 0 : 0) , ( − } . Proof.
We first show that the differences of the six known points form a rank 1 subgroup of theMordell-Weil group of the Jacobian of C , denoted J . This is adapted directly from Michael Stoll’sMagma code for computing Q -rational 6-cycles [45]. We know that prime-to- p torsion in J ( Q )injects into J ( F p ) for primes of good reduction. Magma computes J ( F ) = 3 ·
79 J ( F ) = 7 · . We conclude J ( Q ) has trivial torsion. To show the rank assertion, we use the homomorphismΦ S : (cid:77) i =0 Z P i → Pic C → (cid:89) p ∈ S Pic C/ F p , where S is the set of primes of good reduction. We take S = { , , , , } and compute that thekernel of Φ S is a subgroup of rank 5 in Z = ⊕ Z P i . We apply LLL and find that there are (at least)four relations among the points; this gives an upper bound of the rank as 1. However, looking at Thanks to Michael Stoll for detailed help with the computation and Andrew Sutherland for access to preliminarydata on the analytic rank for Lemma 6.7. he image of Φ S , we see that the degree 0 subgroup of Z surjects onto Z / Z ; and since there is notorsion, the rank must be at least 1. Hence, the rank is exactly 1.This is just the rank of the subgroup supported on the known points, so is not a conclusive rankcalculation. However, Andrew Sutherland in private communication has calculated the analyticrank as 1 assuming that the L -function lies in the (polynomial) Selberg class, which is implied bythe Hasse-Weil conjecture. Assuming BSD (and Hasse-Weil), this is a conclusive rank calculation.The curve C is genus 3 and we assume that rank of J ( Q ) is 1, so we can apply methods ofChabauty to show that these six points are the only Q -rational points on C [2]. Recall that thereis a pairing Ω J ( Q p ) × J ( Q P ) → Q p ( ω, Q ) (cid:55)→ (cid:90) Q ω that induces a perfect Q p -bilinear pairingΩ J ( Q p ) × J ( Q p ) ⊗ Z p Q p → Q p , where J ( Q p ) denotes the kernel of reduction. If G ⊂ J ( Q p ) is a subgroup of rank less thandim( J ) = 3, then there is a nonzero differential ω that kills G under this pairing. We apply thiswith p = 2 and G the subgroup generated by the known rational points. Fix a basis of regulardifferentials w = x ( zdx − xdz ) F y w = y ( zdx − xdz ) F y w = z ( zdx − xdz ) F y , where F ( x, y, z ) is the defining polynomial of C and F y the partial derivative with respect to y . Fora given point P , find a uniformizing parameter of C at P that is also a uniformizer at P modulo2. We find a basis of differentials in terms of the uniformizer that annihilate the known rationalpoints. Looking at the degree of vanishing, we can determine whether one or two rational pointslie above each of the points modulo 2.The points ( − −
2) are in the same residue class. We calculate the basis ofannihilating differentials as1 + t + t + t + t + t + t + t + t + t + t + O ( t )1 + t + t + t + t + t + t + t + t + t + t + O ( t )1 + t + t + t + t + t + t + t + t + O ( t ) . Since there is a non-vanishing constant term, there are most two rational points in this residueclass modulo 2 [44]. Thanks to Michael Stoll for sharing the details of this computation. or the remain four points, we get the following four bases of annihilating differentials. t + t + t + t + t + t + t + t + t + O ( t ) t + t + t + t + t + t + t + t + t + t + t + O ( t )1 + t + t + t + t + t + t + t + t + t + t + t + t + O ( t ) t + t + t + t + t + t + t + t + O ( t )1 + t + t + t + t + t + t + t + O ( t )1 + t + t + t + t + t + t + t + t + t + t + t + t + t + O ( t ) t + t + t + t + t + t + t + t + t + t + O ( t )1 + t + t + t + t + t + t + t + t + t + t + t + O ( t )1 + t + t + t + t + t + t + t + t + t + t + t + t + t + O ( t ) t + t + t + t + t + t + t + t + t + t + t + O ( t ) t + t + t + t + t + t + t + t + t + t + O ( t )1 + t + t + t + t + t + t + t + t + t + t + t + t + t + O ( t ) . For each point there is an element such that the constant term is nonzero and the linear term iszero. This implies that the corresponding logarithm has at most one zero on the residue disk ofthe point, so there is at most one rational point in the disk.Consequently, there are at most six rational points on the curve. (cid:3)
Theorem 6.8.
Let f ∈ A ( C ). If f does not have a Q -rational periodic point of period at least4, then the Q -rational preperiodic structure is one of the following; G := • • ∞ • (cid:47) (cid:47) (cid:119) (cid:119) (cid:121) (cid:121) , a = 11 − t , t ∈ Q \ { , } G := • • ∞ ••• (cid:47) (cid:47) (cid:119) (cid:119) (cid:121) (cid:121) (cid:47) (cid:47) (cid:31) (cid:31) , a = t t − , t = xy − xz for ( x : y : z ) ∈ { n (1 : 0 : 1) : n ≥ y − y = x − } G := • • • ∞ (cid:47) (cid:47) (cid:47) (cid:47) (cid:119) (cid:119) , a = t , t ∈ Q \ { } G := • • ∞ (cid:47) (cid:47) (cid:119) (cid:119) , all other parameters a . Proof.
Proposition 6.5 describes the periodic points, so we need only consider the strictly preperi-odic points.The point at infinity denoted ∞ is fixed for f a for every a . Its only preimage is 0. So we alwayshave at least the preperiodic structure • • ∞ (cid:47) (cid:47) (cid:119) (cid:119) We can also look for preimages of 0. Solutions to z + a = 0 for rational a must look like a = l for some l ∈ Q . Of the three possible preimages of 0, only one can ever be rational. So there areinfinitely many parameter values for either preperiodic structure G or G .To have both a second fixed point and a second preimage of ∞ (i.e., the union of G and G ),we need a = − t and a = l for t, l ∈ Q . Setting these two equal, the problem comes down to nding rational points on the (affine) curve(3) l (1 − t ) − . This curve has genus 1 and a rational point at ( t, l ) = (0 , y − y = x −
27, which has rank0 and whose torsion subgroup is isomorphic to Z / Z . The rational points of the projective closureare [ t : l : r ] = [0 : 1 : 0] , [0 : 1 : 1] , [1 : 0 : 0]. Since we only have three rational points on the ellipticcurve, there are only three on our original curve. Doing a search for points of small height onprojective closure of our curve (3), we find the three projective points [0 : 1 : 0] , [0 : 1 : 1] , [1 : 0 : 0],and only one of these is not at infinity. This point corresponds to a = 1 and gives Φ ∗ ( f )( z ) = 1which is never 0. Thus, we can never have both a second fixed point and a preimage of 0. Notethat this eliminates the union of G and G as well.We can also look at when the preimage of 0 itself has a preimage. We already know a = t forsome t ∈ Q , so we need z + t t z = − t . This determines the curve z + z t + t = 0 , which is a singular genus 3 non-hyperelliptic curve. It has a C automorphism given by ( z, t ) (cid:55)→ ( ωz, ωt ), where ω is a cube root of unity. Quotienting out by this action in Magma gives usall of P , so a different analysis is necessary. We can also perform the change of coordinates( z, t ) (cid:55)→ ( u, v ) = ( zt , z t ) to get v + u + 1 = 0 - this is exactly P as expected.The first birational transformation we perform is sending the coordinates ( z, t ) to ( x, y ) = ( tz , t z ),which gives us y = − x − x . The simple change y (cid:55)→ − y allows us to write the nicer version y = x + x . We see that this is now a superelliptic curve, but in a singular model so still not desirable to workwith. Another birational transformation ( x, y ) (cid:55)→ ( u, v ) = ( x , yx ) gives v = u + u, a nonsingular Picard curve (this is curve p = 2 is a prime of good reduction (define the curve in Magma over F andcheck if it is singular). Code written by Jan Tuitman and Jennifer Balakrishnan [3] verifies that,in fact, there can be at most three rational points on the curve. Thus, there are only three pointson our original curve, which Magma finds to be [0 : 1 : 0] , [0 : 0 : 1] , [1 : 0 : 0]. The only affine pointis (0 , z = 0 , t = 0 is the only solution. However, these values send the expression z + t t z toinfinity, not 0, so it is not a valid solution. Thus, the rational preimage of 0 cannot have a rationalpreimage.We next see if the second fixed point can have rational preimages. To have a second fixed point,we must have a = t t − where t is the affine fixed point. Substituting for a and computing the (1 , z and t , we have(4) ( t − t + 3 t − t ) z + ( − t + 2 t ) z − t z . We also know that f a ( z ) = t, which produces the additional equation(5) 1 t − z ( t −
1) + − t z + t ) = 0 . he variety defined by equations (4) and (5) has two irreducible components: the degenerate case z = t = 0 and the genus 1 curve defined by C : z t − z − zt − t = 0 . Using the point (0 : 1 : 0) as the point at infinity, we get the model(6) y − y = x − . It is rank 1 with generator (1 : 0 : 1) with trivial torsion. We have a mapping ψ : E → Cψ ( x, y, z ) = ( xz, xy − yz, yz − z ) . So every rational point on the elliptic curve corresponds to a rational point on the curve C . Theimage of (1 : 0 : 1) gives t = 1, which is the degenerate case, but the other rational points correspondto parameters a where the additional fixed point has a rational preimage. Furthermore, since thecurve C defining the pair ( z, t ) is degree 2 in z , if there is one rational preimage of the fixed point,then there must be two rational preimages of the fixed point. This is structure G .Now we check if structure G can be extended by the finite fixed point having a second rationalpreimage. We are looking for a rational point on the curve defined by the vanishing of the (2 , ∗ , ( f ) =( t − t + 55 t − t + 330 t − t + 462 t − t + 165 t − t + 11 t − t ) z + (8 t − t + 377 t − t + 1730 t − t + 1693 t − t + 360 t − t + 8 t ) z + (28 t − t + 1080 t − t + 3648 t − t + 2362 t − t + 252 t − t ) z + (56 t − t + 1680 t − t + 4010 t − t + 1568 t − t + 56 t ) z + (70 t − t + 1550 t − t + 2470 t − t + 490 t − t ) z + (56 t − t + 865 t − t + 840 t − t + 56 t ) z + (28 t − t + 282 t − t + 140 t − t ) z + (8 t − t + 48 t − t + 8 t ) z + t − t + 3 t − t and the equation f a ( z ) = t for a = t t − (( t − t + 6 t − t + 1) z + (4 t − t + 9 t − t ) z + (3 t − t + 3 t ) z + t − t )= t (( t − t + t ) z + (2 t − t ) z + t z ) . Taking the ideal generated by the vanishing of these two equations and saturating by the ideal( t, z ) (the degenerate case) leaves the irreducible genus 13 curve defined by z t − z t − z t − z t + 2 z t + 3 z t + z t − z t − z t + t − z + 2 z t − t = 0 . Utilizing Magma, we quotient by the automorphism ( z, t ) (cid:55)→ ( ζ z, ζ t ), where ζ is a primitive thirdroot of unity. This results in the genus 3 curve defined by − x y + y + 5 x − x y + xy − y + 11 x − xy + 136 y + 23 x − y + 517 = 0 . Reducing the (projective closure) equations with the SL ( Z ) element m = − −
40 1 − , we getthe reduced equation x y + xy − x − x y − xy + y = 0 . emma 6.7 calculates the Q -rational points of the projective closure of this curve as { (1 : 1 : 1) , ( − / − / , (0 : 1 : 0) , (0 : 0 : 1) , (1 : 0 : 0) , ( − } . These give the six points on the nonreduced curve { (1 , , , (1 , − / , − / , (1 , − , − , ( − , , , (1 , , , ( − , , } . On the original curve, we find the four points { (0 : 1 : 0) , (0 : 0 : 1) , (0 : 1 : 1) , (1 : 0 : 0) } . Computing the inverse image of the quotient map of the six points on the nonreduced curve, wefind only the four rational points on the original curve already known. So these are the only fourrational points on the original curve. These are all either points at infinity or degenerate cases, sothere are no second rational preimages of the finite fixed point.There are no possible ways to extend the structures G , G , and G rationally, so these are theonly possible structures of Q -rational preperiodic points for f a ( z ) for a ∈ Q . (cid:3) A ( D ) First Component.
There are two families with a D symmetry. We first look at thefamily f a ( z ) = az +1 z + az . Note that a = ± Proposition 6.9.
The following describes Q -rational periodic points for f a ( z ) = az +1 z + az .(1) For every a , f z ( a ) has the rational periodic points • • ∞ • • − (cid:35) (cid:35) (cid:98) (cid:98) (cid:121) (cid:121) (cid:118) (cid:118) (2) The points ± Q -rational fixed points for all a ∈ Q \ {± } .(3) If there are any additional rational periodic points with period 2, then all six points ofperiod 2 are Q -rational.(4) There is no a ∈ Q \ {± } so that f a ( z ) has a Q -rational 3-cycle. Proof.
The first dynatomic polynomial is Φ ∗ ( f a ) = − z + 1. In particular, every member of thefamily has two rational fixed points 1 and − ± i . The seconddynatomic polynomial is given byΦ ∗ ( f a ) = ( a − z + (2 a − a ) z + ( a − z. The associated curve is reducible over Q and the irreducible components are z = 0 a − a + 1 = 0 z + 2 az + 1 = 0 . Note that a = ± ∞ form a 2-cycle. From the lastcomponent, for every z , we can find a = − z − z such that z is periodic with period 2. Furthermore,we know if z is a rational point of period 2, then 1 /z , − z , and − /z are all points of period 2 sincethe maps z (cid:55)→ ± /z and z (cid:55)→ − z are automorphism of f a ( z ). Since f a is a degree 3 map, we knowthat there are at most three rational 2-cycles and six points of period 2. Thus, for every memberof the family, either we have only 1 rational 2-cycle (0 and ∞ ) or all 2-cycles are rational. ow we look at rational 3-cycles. The third dynatomic polynomial is calculated asΦ ∗ ( f a ) = z + ( a + 2 a + 9 a ) z + ( a + 9 a + 14 a + 41 a + 1) z + (7 a + 28 a + 55 a + 118 a + 12 a ) z + (15 a + 54 a + 118 a + 248 a + 59 a + 1) z + (12 a + 73 a + 140 a + 376 a + 180 a + 11 a ) z + (3 a + 52 a + 115 a + 334 a + 361 a + 58 a + 1) z + (12 a + 73 a + 140 a + 376 a + 180 a + 11 a ) z + (15 a + 54 a + 118 a + 248 a + 59 a + 1) z + (7 a + 28 a + 55 a + 118 a + 12 a ) z + ( a + 9 a + 14 a + 41 a + 1) z + ( a + 2 a + 9 a ) z + 1 . The associated dynatomic curve has genus 27 and is difficult to work with directly. First observethat if ( a, z ) is a point on Φ ∗ ( f a ) = 0, then ( a, f a ( z )) and ( a, f a ( z )) are also on the curve. Thus,we can quotient this curve by a C symmetry by setting t = z + f a ( z ) + f a ( z ). We need to findthe curve X obtained through this quotient. We know that finding this curve X is equivalent tofinding the minimal polynomial of t = z + f a ( z ) + f a ( z ) ∈ K , where K is the extension field of Q ( a ) defined by the third dynatomic polynomial. We compute the minimal polynomial as t + ( a + 4 a − a ) t + ( a + 2 a − a + 8 a − t + (3 a + 4 a − a + 20 a ) t + 9 a − a + 16 . The curve X defined by this minimal polynomial is given by X : z +( a +4 a − a ) z +( a +2 a − a +8 a − z +(3 a +4 a − a +20 a ) z +9 a − a +16 = 0 . Observe that we can quotient out by another C action by identifying x = z . This gives us thecurve Y : x +( a +4 a − a ) x +( a +2 a − a +8 a − x +(3 a +4 a − a +20 a ) x +9 a − a +16 = 0 . This curve has genus 3 and is hyperelliptic with simplified model
Y Y : y = x − x + 19 x − x + 9 . Observe, that this curve covers a genus 1 curve by setting x = x , E : y = x − x + 19 x − x + 9 . This curve E has smooth model as an elliptic curve E (cid:48) : y − xy − y = x + 43 x . The curve E (cid:48) has rank 0 and the torsion subgroup has order 6. Therefore, there are at most sixrational points on E . A search using height bound 1000 on the projective closure of E finds fiverational points: (3 : 3 : 1) , (3 : − , (0 : 1 : 0) , (0 : − , (0 : 3 : 1) . This list is complete since (0 : 1 : 0) is a singular point and has multiplicity 2. Observe that affinerational points on E can be lifted to affine rational points on Y Y only if the x -coordinate is asquare. Thus, the only affine rational points on Y Y are (0 , −
3) and (0 , Y Y is of the form y = f ( x ), where f ( x ) has even degree with leading coefficient a square, Y Y hastwo points at infinity, both of which are rational. Thus, on
Y Y , there are only four rational points.Then Y has at most four non-singular rational points. Searching using a height bound 1000 on theprojective closure of Y , we find four rational points:(0 : 1 : 0) , (1 : 0 : 0) , ( − , (1 : − , where the first coordinate is a , the second x , and the third a homogenizing variable. All thesepoints are singular points. When we blow them up, (0 : 1 : 0) blows up to be two rational points, − − Y . We only lift the affine rational points ( − ,
1) and(1 , − a = ±
1. We know when a = ±
1, the dynamical system degenerates.Thus, there is choice of a ∈ Q for which f a ( z ) has a Q -rational 3-cycle. (cid:3) A search for rational preperiodic structures with the parameter up to height 10 ,
000 using thealgorithm from [20] as implemented in Sage yields no parameters when f a ( z ) has a Q -rationalperiodic point with minimal period at least 4. Conjecture 6.10.
There are no a ∈ Q such that f z ( a ) = az +1 z + az has a Q -rational periodic point ofperiod at least 4. Theorem 6.11.
Assuming Conjecture 6.10, the possible rational preperiodic structures for f a ( z ) = az +1 z + az are the following. G : • • − • • ∞ (cid:121) (cid:121) (cid:118) (cid:118) (cid:35) (cid:35) (cid:98) (cid:98) , a not in one of the families G , G , or G G : • •• • − •• • • ∞ (cid:31) (cid:31) (cid:47) (cid:47) (cid:121) (cid:121) (cid:31) (cid:31) (cid:47) (cid:47) (cid:118) (cid:118) (cid:35) (cid:35) (cid:98) (cid:98) , a = t + 31 − t , t ∈ Q \ {± , } G : • • − • • ∞ • • • • (cid:121) (cid:121) (cid:118) (cid:118) (cid:35) (cid:35) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) , a = − t − t , t ∈ Q \ {± , } G : • • − • • ∞ •• •• (cid:121) (cid:121) (cid:118) (cid:118) (cid:35) (cid:35) (cid:98) (cid:98) (cid:39) (cid:39) (cid:55) (cid:55) (cid:103) (cid:103) (cid:119) (cid:119) , a = − t , t ∈ Q \ {± , } . Proof.
Proposition 6.9 combined with Conjecture 6.10 classifies the rational periodic structure. Forpreperiodic structures, we start by looking for rational preimages of the fixed points. Recall fromProposition 6.9 that for every a ∈ Q , f a ( z ) has exactly two Q -rational fixed points ±
1. Observethat if f a ( z ) = 1, then f a ( − z ) = −
1, so it suffices to consider the preimages of 1. To have f a ( z ) = 1,we must have ( az + 1) − ( z + az ) = 0. This defines a reducible curve over Q , and the irreduciblecomponents are z − az − z − z − . We only need to consider the second component. Note that the second equation is quadratic in z ,so we can look at the discriminant to check rationality. The discriminant is ( a − −
4. To havea rational solution, there must be a k ∈ Q so that ( a − − k . This equation defines a genus0 curve with rational parameterization t (cid:55)→ (cid:18) t + 3 − t + 1 , t − t + 1 (cid:19) = ( a, k ) . Thus, for every t ∈ Q \ {± , } , we can find an a such that 1 has a Q -rational preimage under f a ( z ).Since z (cid:55)→ z is an automorphism of f a ( z ), if f a ( z ) = 1, then f a (1 /z ) = 1. Furthermore, since thisirreducible component is quadratic, it can have at most two rational points. Thus, the two rationalpoints must be z and 1 /z .Now we look for a second rational preimage of 1, i.e., a z ∈ Q with f a ( z ) = 1. We need to find(rational) solutions to( a z + a z + z + 3 az + 5 a z + 3 a z + az + a z − ( a z + 3 a z + a z + az + 5 a z + a z + 3 az + 1) = 0 . his defines a reducible curve over Q with irreducible components z − az − z − z − f a ( z ) = 1 points), and a z + a z − a z − az − z − a z − az − a z + az + a z − az − z − az − . We need only consider the third component. It defines a genus 3 curve, but we can quotient by a C symmetry by identifying ( a, z ) with ( a, /z ). If we set t = z + 1 /z , we get the curve after thisquotienting as X : t + ( − a + a ) t + ( a + 2 a − t − a + 3 a − a + 1 = 0 . This is a genus 1 curve and is birationally equivalent to the elliptic curve X (cid:48) : y − xy + 2 y = x − x + 2 x via the map on projective closures x =2 at + 2 a t h − at h − a th + 4 at h − t h + 16 ath + 26 t h − th y =2 at + 4 t h − a h + 2 a th − at h − t h − a h + 36 ath − t h + 58 ah + 90 th − h z = − t + 2 t h. We compute that X (cid:48) has rank 0 and its torsion subgroup has order 6. Thus, there are at most sixnon-singular rational points on X . Searching using height bound 1000 on the projective closureproduces ( a, t, h ) ∈ { ( − − , (3 : 2 : 1) , (0 : 1 : 0) , (1 : 0 : 0) , (1 : 0 : 1) , ( − } , where h is a homogenizing variable. The curve X has singular points (1 : 0 : 1) and ( − − t = z + 1 /z , so we can solve for z . Only t = ± t = ± z = ±
1, which are fixed points. Thus, there is no a such that f a ( z ) = 1 and f a ( z ) (cid:54) = 1 for some z ∈ Q .Now we look at preimages of the 2-cycle comprised of 0 and ∞ . First consider if there is an a such that f a ( z ) = 0. It suffices to look at f a ( z ) = 0 since if f a ( z ) = ∞ , we have f a (1 /z ) = 0 due tothe automorphism z (cid:55)→ /z . Solving az + 1 = 0, we get z = ± (cid:113) − a . Thus, as long as a = − k forsome k ∈ Q , we have f a ( z ) = 0 for two Q -rational z values.Now we consider second preimages: f a ( z ) = 0 for some z such that f a ( z ) (cid:54) = 0. We need to findrational solutions to the equation a z + a z + z + 3 az + 5 a z + 3 a z + az + a z = 0 . This equation defines a reducible curve over Q . The irreducible components are z = 0 z + a = 0 a z + z + 2 az + 3 a z + a = 0 . (7)The first two components correspond to z = 0 and f a ( z ) = 0, so we only need to study the thirdcomponent. It has genus 3, but observe that it covers a genus 1 curve: X : a x + x + 2 ax + 3 a x + a = 0 . his curve is birationally equivalent to the elliptic curve E : y + 8 xy + 2 y = x − x − x via the map (of projective closures) x = 2 a x − a x h + 4 ax h − a x h + 8 ax h + 8 x h − ax h + 4 x h − axh − x h − xh − h y = − a x − ax h + 36 a x h − ax h − x h + 8 ax h − x h + 96 axh + 36 x h + 32 xh + 8 h z = − x h − x h − x h + 2 x h + 3 xh + h , where h and z are the homogenizing variables of X and E , respectively. The curve E has rank 0and its torsion subgroup has order 6. Searching with a height bound 1000 on the projective closure,we find five rational points on X :( a, x, h ) ∈ { (0 : 1 : 0) , (0 : 0 : 1) , (1 : 0 : 0) , ( − , (1 : − } . Among these five points, four of them are singular points:(1 : − , ( − , (0 : 1 : 0) , (1 : 0 : 0) . The first point blows up to be two rational points, the second blows up to be two rational points,and the third blows up to be one rational point. The last one blows up to a point, but that pointis not rational. Thus, we have found all six points on the projective smooth model of X , and thelist must be complete. Recall that we cannot have a = 1 or −
1, and we only look at affine rationalpoints. Thus, the only rational point on the projective closure of X that we care about is (0 : 0 : 1).It lifts to (0 : 0 : 1) on the third irreducible component in equation (7), but z = 0 is a periodicpoint. Therefore, we can conclude that there is no a such that f a ( z ) = 0 and f a ( z ) (cid:54) = 0.Now we look for preimages of one of the other possible rational 2-cycles. In particular, if wecan find an a such that f a ( z ) enters into a 2-cycle that is not the 0 − ∞ cycle. Recall that weneed a = − t − t for t ∈ Q \ {± , } to have extra 2-cycles. Furthermore, we can compute the (1 , ∗ (1 , ( f a ) = ( x + a )( ax + 1)( x + (2 a + 2 a ) x + ( a + 6 a − x + (2 a + 2 a ) x + 1) . When we set a = − t − t , the irreducible components of Φ ∗ (1 , ( f a ) become xt − x ∗ t + 2 t − x = 0 xt + 2 x t − t − x = 02 x t − xt + x − t = 02 x t + xt − x − t = 02 x t − t − x t + x − t = 0 . The first four are quadratic in x and all have discriminant t + 14 t + 1. So rational points occurwhen there are rational points on the curve k = t + 14 t + 1 . This covers the curve y = x + 14 x + 1 , hich is birational to the rank 0 elliptic curve y = x − x + 192 x with torsion subgroupisomorphic to Z / Z × Z / Z . A point search gives the eight points(0 : 1 : 0) , (0 : 0 : 1) , (8 : 16 : 1) , (8 : −
16 : 1) , (12 : 0 : 1) , (16 : 0 : 1) , (24 : 48 : 1) , (24 : −
48 : 1) . The curve y = x + 14 x + 1 has the seven rational points( − , (4 : 1 : 1) , ( − , (4 : − , ( − − , (1 : 0 : 0) , (1 : 0 : 1) , where (1 : 0 : 0) is singular. On the original curve, we find the same seven rational points( − , (4 : 1 : 1) , ( − , (4 : − , ( − − , (1 : 0 : 0) , (1 : 0 : 1) . These all have t ∈ {± , } so correspond to degenerate cases. The last two components are bothbirational to the same rank 0 elliptic curve, y + 2 xy = x − x + 2 x , with torsion subgroupisomorphic to Z / Z × Z / Z , so they have at most four rational points. In both cases, we have therational points ( x, t ) = ( ± , ± t = ± f a ( z ) entering a 2-cycle that is not the 0 − ∞ cycle.Now that we have identified each of the separate possible components of the rational preperiodicstructure, we need to consider which of these components can occur simultaneously. First we askif 0 can have a rational preimage at the same time there are the additional 2-cycles. Recall thatwe need a = − k for some k ∈ Q to have f a ( z ) = 0 and we need a = − z − z to have extra 2-cycles.Thus, we are looking for rational points on the curve X : z + 1 − k z = 0 . This is a genus 1 curve that is birationally equivalent to the elliptic curve E : y + 2 xy = x − x + 2 x. This elliptic curve E has rank 0 with torsion subgroup of order 4. Searching with a height bound1000 on the projective closure, we find five rational points on X :( k, z, h ) ∈ { (1 : 1 : 1) , (1 : 0 : 0) , ( − , (1 : − , ( − − } , where h is a homogenizing variable. Only one of these points is singular, so we know this is acomplete list of rational points. The affine rational points force z = ±
1, which are fixed points.Thus, we cannot have f a ( z ) = 0 and extra 2-cycles at the same time.Now we check if both the fixed points and the 0 − ∞ a = − k for some k ∈ Q to have f a ( z ) = 0 and we need a = t +3 − t +1 for some t (cid:54) = 0 ∈ Q to have f a ( z ) = 1. Thus, we are interested in the following curve X : t + 3 + k ( − t + 1) = 0 . This curve is a genus 1 curve that is birational to the elliptic curve E : y = x + 2 x + 12 x + 24 . This curve E has rank 0 and a torsion subgroup of order 4. Searching using height bound 1000 onthe projective closure, we find two rational points on X :( k, t, h ) ∈ { (0 : 1 : 0) , (1 : 0 : 0) } . Both are singular points and both blow up to be two rational points. Thus, this search producesall the rational points X . They are both points at infinity, so there are no affine rational points on X . Thus, we can conclude that there is no a such that f a ( z ) = 0 and f a ( z ) = 1.Next we check if we can have additional 2-cycles as well as preimages of the fixed points. Recallthat we need a = t +3 − t +1 to have f ( z ) = 1 and we need a = z +12 z to have extra 2-cycles. Thus, weneed to find rational points on the curve X : ( t + 3)(2 z ) = ( z + 1)( − t + 1) . his is a genus 1 curve that is birationally equivalent to the elliptic curve E : y = x − x − x + 24 . This curve E has rank 0 and the torsion subgroup has order 4. Searching using height bound 1000on the projective closure, we find four rational points on the curve X :( t, z, h ) ∈ { (0 : 1 : 0) , ( − , (1 : 0 : 0) , (1 : 0 : 1) } . Among these four points, two of them are singular: (0 : 1 : 0) , (1 : 0 : 0). The point (0 : 1 : 0) blowsup to be two rational points, and the point (1 : 0 : 0) blows up to be a point that is not rational.Thus, this search produces all the rational points on X . Considering only the affine points, we areleft with ( − z = 0 corresponds to periodic points. Thus, there is no a such that we can have extra 2-cycles and f a ( z ) = 1.We have exhausted all possibilities for rational preperiodic structures, leaving only those enu-merated in the statement. (cid:3) A ( D ) Second Component.
Now we look at the family with D symmetry f a ( z ) = az − z − az .We first consider the possible rational periodic points. Proposition 6.12.
Let f a ( z ) = az − z − az for a (cid:54) = ± t ∈ Q \ { } , the value a = t +12 t satisfies f a ( z ) has the four fixed points (cid:8) ± t, ± t (cid:9) .For no other values of a ∈ Q does f a ( z ) have a Q -rational fixed point.(2) For every a ∈ Q \ ±
1, the function f a ( z ) has exactly two Q -rational 2-cycles: swapping 0and ∞ and swapping 1 and − a ∈ Q so that f a ( z ) has a Q -rational 3-cycle. Proof.
The first dynatomic polynomial isΦ ∗ ( f a ) = − z + 2 az − . This polynomial is linear in a , so it has a zero when a = z +12 z . Thus, for every z ∈ Q , we can findan a such that z is a fixed point. Once z is a fixed point, we know from the automorphism groupthat − z , z and − z are all fixed points. Furthermore, for every a , the first dynatomic polynomialhas at most four zeros, so z, − z, z and − z are all the fixed points.Now we look for rational 2-cycles. The second dynatomic polynomial is given byΦ ∗ ( f a ) = ( − a + 1) z + ( a − z = (1 − a ) z ( z − z + 1)( z + 1) . Since a = ± ∗ ( f a ) do not depend on a and everymember of the family f a has exactly two Q -rational 2-cycles: swapping 0 and ∞ and swapping 1and − ∗ ( f a ) = − z + ( a + 4 a + 7 a ) z + ( − a − a − a − a + 1) z + (7 a + 48 a + 91 a + 76 a − a ) z + ( − a − a − a − a + a − z + (6 a + 103 a + 348 a + 304 a + 30 a + a ) z + ( − a − a − a − a − a − a + 1) z + (6 a + 103 a + 348 a + 304 a + 30 a + a ) z + ( − a − a − a − a + a − z + (7 a + 48 a + 91 a + 76 a − a ) z + ( − a − a − a − a + 1) z + ( a + 4 a + 7 a ) z − . he dynatomic curve Φ ∗ ( f a ) = 0 has genus 31. We can quotient by a C symmetry by identifying( a, z ) with ( a, f a ( z )) and ( a, f a ( z )). Setting t = z + f a ( z ) + f a ( z ), this quotienting produces thefollowing curve: X : t + ( − a − a − a ) t + ( a + 14 a + 46 a + 56 a + 1) t + ( − a − a − a − a + 4 a ) t + 4 a + 12 a + 25 a + 24 a + 16 = 0 . This curve has genus 11. We should be able to quotient by another C symmetry by identifying( a, z ) with ( a, z ). However, in this curve we have used t = z + f ( z ) + f ( z ) so we need to identify( a, t ) with ( a, t (cid:48) ), where t (cid:48) = z + f ( z ) + f ( z ) . We need to express t (cid:48) in terms of a and t . If wequotient the original curve by t (cid:48) = 1 /z + 1 /f ( z ) + 1 /f ( z ), we will get the same quotient curve.Equivalently, the minimal polynomial of t is the same as the minimal polynomial of t (cid:48) . Thus, wecan check which root of the minimal polynomial of t corresponds to t (cid:48) . We can express t (cid:48) in termsof a and t as t (cid:48) = 12 a + 13 a + 41 a + 75 a + 85 a + 56 a + 16 (cid:16) ( − a − a − t + ( a + 8 a + 25 a + 35 a + 22 a ) t + ( − a − a − a − a − a − a − t + (4 a + 33 a + 114 a + 210 a + 205 a + 87 a + 4 a ) t (cid:17) . We can find the minimal polynomial of t + t (cid:48) , which defines the curve Y (cid:48) : u + ( − a − a − a − a − a − u + 4 a + 12 a + 25 a + 36 a + 34 a + 24 a + 9 = 0 . This curve has genus 5, but we can identify u = x and get Y : x + ( − a + 2 a − a + 10 a − a + 8) x + 4 a − a + 25 a − a + 34 a − a + 9 = 0 . The new curve Y has genus 2 so is hyperelliptic. It is birational to the curve H : y = 4 x − x + 25 x − x + 25 x − x + 4 . The curve H has genus 2 and its Jacobian has rank 0 with torsion subgroup isomorphic to Z / Z × Z / Z . Using Chabauty’s method for rank 0 Jacobians as implemented in Magma yields the sixrational points on (the projective closure of) H as(1 : − , (1 : 2 : 0) , (1 : 2 : 1) , (1 : − , (0 : 2 : 1) , (0 : − . A point search up to height bound 1000 yields the points on (the projective closure of) Y :( z, a, h ) ∈ { (4 : 1 : 0) , (0 : − , (4 : 1 : 1) , (1 : 0 : 0) , (36 : 1 : 1) } . The point (0 : − Y . The affine points all have a = ±
1, which is the degenerate case for this family. So there are nopoints that corresponds to a rational 3-cycle. (cid:3)
A search for rational preperiodic structures with the parameter up to height 10 ,
000 using thealgorithm from [20] as implemented in Sage yields no parameters where f a has a Q -rational periodicpoint with minimal period at least 4. Conjecture 6.13.
There are no a ∈ Q such that f z ( a ) = az − z − az has a Q -rational periodic point ofperiod at least 4.Assuming Conjecture 6.13, we classify all rational preperiodic structures. heorem 6.14. Assuming Conjecture 6.13, the possible rational preperiodic structures for f a ( z ) = az +1 z + az are the following. G = • • − • • ∞ (cid:36) (cid:36) (cid:98) (cid:98) (cid:35) (cid:35) (cid:98) (cid:98) , for t not in the following cases G = • • − • • ∞ • t • − t • /t • − /t (cid:36) (cid:36) (cid:98) (cid:98) (cid:35) (cid:35) (cid:98) (cid:98) (cid:121) (cid:121) (cid:119) (cid:119) (cid:117) (cid:117) (cid:112) (cid:112) , a = t + 12 t for t ∈ Q \ {± , } G a = • • − • • ∞ •• •• (cid:36) (cid:36) (cid:98) (cid:98) (cid:35) (cid:35) (cid:98) (cid:98) (cid:55) (cid:55) (cid:39) (cid:39) (cid:103) (cid:103) (cid:119) (cid:119) , a = t , for t ∈ Q \ { } G b = • • − • • ∞ •• •• (cid:36) (cid:36) (cid:98) (cid:98) (cid:35) (cid:35) (cid:98) (cid:98) (cid:55) (cid:55) (cid:39) (cid:39) (cid:103) (cid:103) (cid:119) (cid:119) , a = 3 t + 3 t + 32 t + 5 t + 2 , t ∈ Q \ {− , − / } . Proof.
We start with preimages of fixed points. Recall from Proposition 6.12 that we have (four)fixed points when a = t +12 t for some t ∈ Q \ {± , } . We want to know if these points can haverational preimages. We look at the (1 ,
1) generalized dynatomic polynomialΦ ∗ (1 , ( f a ) = x + ( − a − a ) x + ( a + 6 a − x + ( − a − a ) x + 1 . Substituting a = t +12 t this becomes16 t x + ( − t − t − t − t ) x + ( t + 28 t + 38 t + 28 t + 1) x + ( − t − t − t − t ) x + 16 t . Since t is the fixed point and x the (1 ,
1) preperiodic point, the equation f a ( x ) = t must also besatisfied. This equation is − t x + ( t + 1) x + ( t + t ) x − t = 0 . We need both to vanish, so we look at the curve defined by the vanishing of both these equations.The irreducible components are given by t = x = 0 t = x = 1 tx = − t = x, t + 1 = 02 x t + xt − x − t = 0 . The first component is degenerate and the second and third correspond to x is a fixed point. Thefourth does not have Q -rational solutions, so we need only consider the last component. This isa genus 3 curve but is quadratic in x , so we have rational solutions when the discriminant, withrespect to x , is a square. This gives the curve t + 14 t + 1 = k . Replacing u = t , this becomes u + 14 u + 1 = k . Using the point (1 : 0 : 0) as the point at infinity, this is an elliptic curve with model y = x − x + 192 x . This curve has rank 0 and torsion isomorphism to Z / Z × Z / Z . So the original urve has at most eight rational points. A point search up to height 1000 yields the seven points(on the projective closure)( x, t ) ∈ { ( ± ± , ( ± , (1 : 0 : 0) } . The point (1 : 0 : 0) is singular, so we have found all eight Q -rational points. The affine points allhave t values that are degenerate, so there are no a ∈ Q such that f a ( z ) has a rational fixed pointwith a rational preimage.Now we look for rational preimages of the points of period 2. Recall that every member of thefamily has exactly two 2-cycles { , ∞} and { , − } . We want to know if we can have f a ( z ) = 0 or f a ( z ) = 1. It suffices to look for rational preimages of 0 and 1 since preimages of − ∞ arethen obtained from the same automorphisms that produce four rational fixed points when there isone. We first look at f a ( z ) = 0. We need to find rational solutions to the equation az − . This equation defines a genus 0 curve with rational parameterization t (cid:55)→ ( t , /t ) = ( a, z ) . Thus, for every t ∈ Q \ { } , f a (1 /t ) = 0, where a = t .Now we look at f a ( z ) = 1. We need to find rational solutions to the equation ( az − − ( z − az ) = 0 . This equation defines a reducible curve over Q and the irreducible components are z + 1 = 0 az − z + z − . This first component corresponds to a periodic point, so we only look at the second component. Itdefines a genus 0 curve with rational parameterization: t (cid:55)→ (cid:18) t + 3 t + 32 t + 5 t + 2 , t + 22 t + 1 (cid:19) . Thus, for every t ∈ Q , we can find a = t +3 t +32 t +5 t +2 such that 1 has a rational preimage under f a .Now we look for rational second preimages; i.e., determine if we can have f a ( z ) = 0 and f a ( z ) (cid:54) = 0 or ∞ . We need to find rational solutions to the equation a z − a z − z + 3 az − a z + 3 a z + az − a z = 0 . This equation defines a reducible curve and the irreducible components are z = 0 z − a = 0 a z − z + 2 az − a z + a = 0 . The first component corresponds to periodic points. The second component corresponds to z suchthat f ( z ) = ∞ . Thus, we can just look at the third component. It covers an elliptic curve: X : a z − z + 2 az − a z + a = 0 . The elliptic curve X is birationally equivalent to E : y − xy − y = x − x − x. This elliptic curve E has rank 0 and its torsion subgroup has order 6. Searching using height bound1000, we find five rational points on the projective closure of X :( a, z, h ) ∈ { , (0 : 1 : 0) , (0 : 0 : 1) , (1 : 0 : 0) , ( − − } , where h is a homogenizing variable. Among these five points, ( − − − − oint, and point (1 : 0 : 0) blows up to be a point that is not rational. Thus, our search has foundall the rational points on X . The only affine rational point such that a (cid:54) = ± z = 0, which is the periodic point. Thus, there is no a such that f a ( z ) = 0 and f a ( z ) (cid:54) = 0 or ∞ for rational z .Now we look for rational second preimages of 1; i.e., rational z so that f a ( z ) = 1 and f a ( z ) (cid:54) = 1or −
1. We need to find rational solutions to the equation a z − a z − z +3 az − a z +3 a z + az − a z − ( − a z +3 a z − a z + az − a z + a z +3 az −
1) = 0 . This equation defines a reducible curve over Q and the irreducible components are z − az + z + z + 1 = 0 a z − a z − a z + az − z − a z + 2 az − a z − az + a z + 2 az − z + az − . The first component corresponds to z = 1 and the second component to f ( z ) = −
1. Thus, we focuson the third component. It defines a genus 3 curve. We can quotient by a C action by identifying( a, z ) with ( a, /z ). This gives curve X : t + ( − a − a ) t + ( a − a − t + a + 3 a + 3 a + 1 = 0 , where t = z + z . This is a genus 1 curve birational to the elliptic curve E : y − xy + 2 y = x − x + 2 x. The curve E has rank 0 and its torsion subgroup has six elements. A search for rational points onthe projective closure of X using height bound 1000 finds( a, t, h ) ∈ { (1 : − , (1 : 2 : 1) , (0 : 1 : 0) , ( − , (1 : 0 : 0) , ( − } , where h is the homogenizing variable. Among these points, ( − X . Observe that the onlyaffine rational point where a (cid:54) = ± − t = z + 1 /z = 2. The onlysolution is z = 1. Thus, there is no a such that f a ( z ) = 1 and f a ( z ) (cid:54) = 1 or − z .Next we check if both pairs of 2-cycles can have preperiodic tails at the same time; i.e., if we canhave f a ( z ) = 0 and f a ( z ) = 1. Recall that we need a = t to have f a ( z ) = 0 and a = t +3 t +32 t +5 t +2 to have f a ( z ) = 1. Thus, we need to find rational solutions to(3 t + 3 t + 3) − t (2 t + 5 t + 2) = 0 . This defines a genus 1 curve that is birationally equivalent to the elliptic curve E : y + 36 xy − y = x + 180 x + 202176 x − . The curve E has rank 0 and its torsion subgroup has order 4. Searching for rational points on theprojective closure of X using height bound 1000 produces( t , t , h ) ∈ { (0 : 1 : 0) , (1 : 0 : 0) , (1 : 1 : 1) , ( − } , where h is the homogenizing variable. Among these points, (0 : 1 : 0) and (1 : 0 : 0) are singularpoints. The first one blows up to be two rational points, where as the second one blows up tobe only one point that is not rational. Thus, we have found all rational points on X . Checkingwhether these points produce valid members of the family, we see that t = 1 so that a = t = 1,which produces degeneracy. Thus, we cannot have f a ( z ) = 0 and f a ( z ) = 1 at the same time forrational z and z . ow we need to ask if we can have rational fixed points at the same time as preperiodic tails fora 2-cycle. Recall that we have rational fixed points if a = t +12 t and we have we have f a ( z ) = 0 if a = u and we have f a ( z ) = 1 if a = v +3 v +32 v +5 v +2 . We first study the curve X : t + 1 − u t = 0 . This defines a genus 1 curve birational to the elliptic curve E : y + 2 xy = x − x + 2 x. The curve E has rank 0 and 4 rational torsion points. Searching up to height 1000 on the projectiveclosure of X , we find( t, u, h ) ∈ { (1 : 1 : 1) , (0 : 1 : 0) , ( − , (1 : − , ( − − } . The point (1 : 0 : 0) is a singular point and blows up to be two rational points. Thus, we havefound all rational points on X . Note that all the affine points have t ∈ { , ± } so correspond toeither periodic t = 0 or to degenerate t = ±
1. Thus, we cannot have a rational preperiodic tail for0 and rational fixed points at the same time. Now we study the curve X :( t + 1)(2 v + 5 v + 2) − (3 v + 3 v + 3)( t ) = 0corresponding to rational preperiodic tail for 1 and rational fixed points at the same time. Thiscurve covers a genus 1 curve X (cid:48) : 2 v t + 5 vt − v t + 2 t − vt + 2 v − t + 5 v + 2 = 0birational to the elliptic curve E : y − xy + 216 y = x − x + 3240 x − . The curve E has rank 0, and 4 rational torsion points. Searching using height bound 1000 on theprojective closure of X (cid:48) produces:( t, z, h ) ∈ { (0 : 1 : 0) , (0 : − , (0 : − / , (1 : 0 : 0) } , where h is the homogenizing variable. The points (0 : 1 : 0) and (1 : 0 : 0) are singular. The firstpoint blows up to be two rational points and the second point blows up to be a point that is notrational. Thus, we have found all the rational points on X (cid:48) . The only affine rational points have t = 0, but t = 0 are periodic points. Therefore, we cannot have a rational preperiodic tail of 1 andrational fixed points at the same time. (cid:3) C First Component.
We move to the the two -parameter family f a,b ( z ) = az +1 z + bz .We start by examining the fixed points and 2-periodic points. Proposition 6.15.
For f a,b ( z ) = az +1 z + bz , we have the following periodic points.(1) The pairs ( a, b ) for which f a,b has a Q -rational fixed point are parameterized by P withcoordinates ( u : v : w ) as a = u v − vw − u vw + u w b = u w − w − u vw + u w for uw (cid:54) = 0 and v (cid:54) = w . There are exactly two Q -rational fixed points given by x = ± wu .
2) For every (non-degenerate) pair ( a, b ), f a,b has the 2-cycle 0– ∞ . The pairs ( a, b ) for whichthere are additional Q -rational 2-cycles are parameterized by P with coordinates ( u : v : w )as a = u v − vw − u vw + u w b = u w − w − u vw + u w for uw (cid:54) = 0 and v (cid:54) = − w . For each such pair, all three 2-cycles are rational with the twoadditional cycles comprised of (cid:8) ± uw (cid:9) and (cid:8) ± wu (cid:9) . Note that when uw = ±
1, then there areexactly two 2-cycles.
Proof.
We first look at the fixed points. We compute the first dynatomic polynomial asΦ ∗ ( f a,b ) = − z + ( a − b ) z + 1 . We consider the projective closure of the surface Φ ∗ ( f a,b ) = 0, which is a rational surface with pa-rameterization given in the statement. To see that there are exactly two fixed points, we substitutethe parameterizations of a and b into Φ ∗ ( f a,b ) to getΦ ∗ ( f a,b ) = − ( uz − w )( uz + w )( w z + u ) u w . From this we see the two Q -rational points z = ± wu and the two complex points z = ± wiu .Now we consider the second periodic points via the second dynatomic polynomialΦ ∗ ( f a,b ) = z ( ab − z + ( a + b ) z + 1) . We see every choice of parameter gives the cycle 0– ∞ . The component z + ( a + b ) z + 1 = 0 is arational surface parameterized by P as given in the statement. (cid:3) Periodic points with higher periods and preperiodic points are difficult to study for this familymainly because computational tools for rational points on surfaces is much less well developed thanfor curves. Consequently, we content ourselves with a census of Q -rational preperiodic structuresfor parameters a and b with small height. By no means do we think this census is exhaustive;rather, it gives a sense of the diversity of possibilities when there are only C symmetries.(0,0) (0,-1) (0,-2) • • ∞ • • (cid:35) (cid:35) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ •• (cid:35) (cid:35) (cid:98) (cid:98) (cid:119) (cid:119) (cid:103) (cid:103) • • ∞ • • (cid:35) (cid:35) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (-1,-1/4) (-1,-5/2) (-1,11/4) • • ∞ •••• (cid:55) (cid:55) (cid:39) (cid:39) (cid:35) (cid:35) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) • • ∞ •••• (cid:47) (cid:47) (cid:47) (cid:47) (cid:35) (cid:35) (cid:98) (cid:98) (cid:55) (cid:55) (cid:39) (cid:39) • • ∞ • ••• (cid:55) (cid:55) (cid:39) (cid:39) (cid:35) (cid:35) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) (-1/2,-15/4) (2, -7/4) • • ∞ • • • • (cid:35) (cid:35) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ • • • • (cid:35) (cid:35) (cid:98) (cid:98) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) (2,-4) (8/5,-13/10) (7/8, -23/8) • • ∞ •• • • (cid:35) (cid:35) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ •• •• (cid:35) (cid:35) (cid:98) (cid:98) (cid:79) (cid:79) (cid:47) (cid:47) (cid:15) (cid:15) (cid:47) (cid:47) • • ∞ • • •• (cid:35) (cid:35) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) -1, -13/4) (1/2,-5/2) • • ∞ •• • • • • (cid:55) (cid:55) (cid:39) (cid:39) (cid:35) (cid:35) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ •••• • • (cid:55) (cid:55) (cid:39) (cid:39) (cid:35) (cid:35) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (-1/4,-1/4) (-3/2,-3/2) • • ∞ •••• • • (cid:55) (cid:55) (cid:39) (cid:39) (cid:35) (cid:35) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ • •• •• • (cid:35) (cid:35) (cid:98) (cid:98) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (-9/4, -16) (-1,-77/45) • • ∞ •••• •• (cid:55) (cid:55) (cid:39) (cid:39) (cid:35) (cid:35) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) (cid:111) (cid:111) (cid:111) (cid:111) • • ∞ •••• •• (cid:47) (cid:47) (cid:47) (cid:47) (cid:35) (cid:35) (cid:98) (cid:98) (cid:47) (cid:47) (cid:47) (cid:47) (cid:55) (cid:55) (cid:39) (cid:39) (-17/8,-17/8) (3/2,-9/4) • • ∞ • • • • •• (cid:35) (cid:35) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ •• • •• •• • (cid:35) (cid:35) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (-5/2,-7/4) (-13/5, -13/15) • • ∞ •••• • • • • (cid:55) (cid:55) (cid:39) (cid:39) (cid:35) (cid:35) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ • •• ••• •• (cid:35) (cid:35) (cid:98) (cid:98) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) (cid:31) (cid:31) (cid:95) (cid:95) (cid:111) (cid:111) (-9/4, -23/8) (-19/3, -25/9) • • ∞ ••••••• • (cid:35) (cid:35) (cid:98) (cid:98) (cid:53) (cid:53) (cid:41) (cid:41) (cid:79) (cid:79) (cid:63) (cid:63) (cid:47) (cid:47) (cid:47) (cid:47) (cid:31) (cid:31) (cid:15) (cid:15) • • ∞ • ••• •• ••• • (cid:35) (cid:35) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (cid:55) (cid:55) (cid:39) (cid:39) (cid:103) (cid:103) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) C Second Component.
We consider the two -parameter family f a,b ( z ) = z + azbz +1 .We start by examining the fixed points and 2-periodic points. Proposition 6.16.
For f a,b ( z ) = z + azbz +1 , we have the following periodic points.(1) The points 0 and ∞ are always fixed. There are two additional Q -rational fixed points forpairs ( a, b ) parameterized by P with coordinates ( u : v : w ) as a = ( u v − vw u v − w b = u w − w u v − w for u v (cid:54) = w . The two additional Q -two rational fixed points are given by z = ± wu .
2) There are two rational surfaces parameterizing pairs ( a, b ) for which f a,b has a Q -rational2-cycle. We parameterize each surface with a copy pf P with coordinates ( u : v : w ): S : (cid:40) a = u v + uvw − u v − uw b = u w + uw ) − u v − uw S : (cid:40) a = u v + vw − u vw − u w b = u w + w − u vw − u w . For S there is one 2-cycle with points z = ± wu . For S there are two 2-cycles with points z = ± wu and z = ± uw . These two surfaces intersect on the line a + b = − z = ± Proof.
We first look at the fixed points. We compute the first dynatomic polynomial asΦ ∗ ( f a,b ) = − bz + z + az − z. After removing the component z = 0, we consider the projective closure of the surface bz − z − a + 1 = 0. This is a rational surface with parameterization given in the statement. To see thatthere are exactly two fixed points, we substitute the parameterizations of a and b into Φ ∗ ( f a,b ) toget Φ ∗ ( f a,b ) = − zu v − w ( − v + w )( uz − w )( uz + w ) . From this we see the two new Q -rational points are z = ± wu .Now we consider the second periodic points via the second dynatomic polynomialΦ ∗ ( f a,b ) = ( bz + z + a + 1)( z + az + bz + 1) . Each component gives the rational surface parameterized by the given parameterization. Theirintersection contains two curves: a + b = − ab = 1, but the second curve is the degeneratepairs ( a, b ). (cid:3) Similar to the first two-parameter family, periodic points with higher periods and preperiodicpoints are difficult to study for this family. Again, we content ourselves with a census of Q -rationalpreperiodic structures for parameters a and b with small height.(0,1) (0,-1) (0,0) (0,-2) • • (cid:121) (cid:121) (cid:121) (cid:121) •• • • (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:121) (cid:121) • •• • (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) • •• • (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) (0,-3) (0,-5/4) (0,-9/16) • • • •• • (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) • • • •• • (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) •• • • •• (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (1,-9) (-1,-1/4) (-1,-5/2) •• • • • • (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) •• • •• • (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) •• •• • • (cid:47) (cid:47) (cid:47) (cid:47) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:121) (cid:121) -1/2,-15/4) (2,-7/4) (-7/4,-11/8) • • • • •• (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) • • • • •• (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (cid:111) (cid:111) (cid:121) (cid:121) (cid:121) (cid:121) • •• • • • (cid:47) (cid:47) (cid:79) (cid:79) (cid:47) (cid:47) (cid:79) (cid:79) (cid:121) (cid:121) (cid:121) (cid:121) (0,-14/3) (-1,-13/4) (1/2,-7) • • ∞ •••• •• (cid:55) (cid:55) (cid:39) (cid:39) (cid:35) (cid:35) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) (cid:121) (cid:121) (cid:121) (cid:121) •• • • •• • • (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) • • • •• • •• (cid:47) (cid:47) (cid:121) (cid:121) (cid:34) (cid:34) (cid:98) (cid:98) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (-3,-5/4) (-2/3,-11/4) (-3/2,-1/9) • •• • • •• • (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) • •• • •• •• (cid:55) (cid:55) (cid:121) (cid:121) (cid:39) (cid:39) (cid:55) (cid:55) (cid:121) (cid:121) (cid:39) (cid:39) (cid:121) (cid:121) (cid:121) (cid:121) • •• • •• • • (cid:55) (cid:55) (cid:121) (cid:121) (cid:39) (cid:39) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (-1/4,-11/5) (11/4,-16) (-1,-25/9) • •• • •• • • (cid:55) (cid:55) (cid:121) (cid:121) (cid:39) (cid:39) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) •• •• • • •• (cid:47) (cid:47) (cid:47) (cid:47) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) •• •• • •• • (cid:47) (cid:47) (cid:47) (cid:47) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (-16/9,-21/4) (-17/5,-17/20) (-9/25,-25/16) ••••• • •• (cid:121) (cid:121) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) • •• • • •• • (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) •• ••• ••• (cid:60) (cid:60) (cid:124) (cid:124) (cid:121) (cid:121) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:39) (cid:39) (cid:55) (cid:55) (-2,-13/12) (-3,-9/16) • • •• • • •• •• (cid:47) (cid:47) (cid:55) (cid:55) (cid:121) (cid:121) (cid:39) (cid:39) (cid:47) (cid:47) (cid:55) (cid:55) (cid:121) (cid:121) (cid:39) (cid:39) (cid:121) (cid:121) (cid:121) (cid:121) • •• • •• • •• • (cid:55) (cid:55) (cid:121) (cid:121) (cid:39) (cid:39) (cid:55) (cid:55) (cid:121) (cid:121) (cid:39) (cid:39) (cid:55) (cid:55) (cid:121) (cid:121) (cid:39) (cid:39) (cid:121) (cid:121) (-5/2,-7/4) (4/5,-9/4) •• • • •• • • •• (cid:55) (cid:55) (cid:39) (cid:39) (cid:34) (cid:34) (cid:98) (cid:98) (cid:55) (cid:55) (cid:39) (cid:39) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) •• • • •• •• • • (cid:55) (cid:55) (cid:39) (cid:39) (cid:34) (cid:34) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:121) (cid:121) (-5/4,-16/9) (7/8,-17/2) • •• • • •• ••• • • (cid:47) (cid:47) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:47) (cid:47) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:121) (cid:121) • •••• •• • •• (cid:34) (cid:34) (cid:98) (cid:98) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:103) (cid:103) (cid:119) (cid:119) (cid:111) (cid:111) (cid:121) (cid:121) (cid:121) (cid:121) (-19/3,-25/9) • ••• • ••• ••• • (cid:34) (cid:34) (cid:98) (cid:98) (cid:55) (cid:55) (cid:39) (cid:39) (cid:34) (cid:34) (cid:98) (cid:98) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:39) (cid:39) (cid:55) (cid:55) (cid:121) (cid:121) . Rational Preperiodic Structures in M In this section, we examine the Q -rational preperiodic point structures of the families covering A given in Section 4.We start with the dimension 0 family.7.1. The zero dimension loci A ( C ) and A ( D ) . Recall from Proposition 4.2 that A ( D ) = A ( C ) are given by the single conjugacy class f ( z ) = z . Theorem 7.1.
For A ( D ) = A ( C ) the single conjugacy class f ( z ) = z has Q -rational prepe-riodic structure given by • • ∞ • • − (cid:35) (cid:35) (cid:98) (cid:98) (cid:47) (cid:47) (cid:118) (cid:118) Proof.
Direct computation with the algorithm of Hutz [20] as implemented in Sage. (cid:3)
The dimension one family A ( C ) . The automorphism locus of A ( C ) is covered by thefamily f a ( z ) = z +1 az . We start with the periodic points. Proposition 7.2.
We have the following periodic points for f a ( z ) = z +1 az .(1) ∞ is a fixed point for every a . When a = 1 + t for t ∈ Q \ { } , we have two additional Q -rational fixed points, ± t .(2) There are two Q -rational 2-cycles for each of the following two families of parameters a = t + t − for t ∈ Q \ { , ± } a = − ( t + t − ) for t ∈ Q \ { , ± } and one 2-cycle for the family a = − (1 + t ) for t ∈ Q \ { } . None of these pairs of 2-cycles occur at the same time, i.e., there are at most two Q -rational2-cycles.(3) The only Q -rational fixed point that occurs in conjunction with a Q -rational 2-cycle is thefixed point ∞ , which occurs for every parameter value. Proof.
Starting with rational fixed points, we can immediately see that ∞ is always a fixed point.The first dynatomic polynomial is Φ ∗ ( f a ) = (1 − a ) z + 1. Since this is linear in a , there areadditional rational fixed points when a = z + 1 z . In particular, for the four fixed points ± z, ± iz . Only two of these are Q -rational. Substituting z = t , we get a = 1 + t .Next we look for rational 2-cycles. The second dynatomic polynomial isΦ ∗ ( f ) = ( z − az + 1)( z + az + 1)((1 + a ) z + 1) . All three components are linear in a , so we get one-parameter families each of which has two Q -rational 2-cycles.(1) The first factor has a = z +1 z , which for z = ± t or z = ± t becomes a = t + t − . So thereare two Q -rational 2-cycles for t ∈ Q \ { } . When t = ±
1, the two 2-cycles collapse to fixedpoints.(2) The second factor has a = z +1 − z , which for z = ± t or z = ± t becomes a = − ( t + t − ). Sothere are two Q -rational 2-cycles for t ∈ Q \ { } . when t = ±
1, the two 2-cycles collapse tofixed points.
3) The third factor has a = z − z , where if z = ± t , then a = − (1 + t ). So there is one Q -rational 2-cycle for t ∈ Q \ { } .We must also check if any of these three sets of 2-cycles can occur at the same time. We must findpoints on the pairwise intersection of the three curves.(1) For a = t + t − = − ( t + 1 /t ) there are two components: t + t = 0 t t = − a = t + t − and a = − (1 + t ), we get the genus 5 curve t t + t + t + 1 = 0 . This covers the elliptic curve uv + u + u + 1 = 0 . This curve is birational to the elliptic curve y = x − x + x , which is rank 0 with torsionsubgroup isomorphic to Z / Z . The four torsion points are { (0 : 1 : 0) , (1 : 0 : 0) , ( − , ( − − } . Only the two points { (0 : 1 : 0) , (1 : 0 : 0) } correspond to rational pointson the original curve, and these points do not correspond to valid parameters.(3) For a = − ( t + t − ) and a = − (1 + t ), we get the genus 5 curve t t − t + t − . This covers the elliptic curve uv − u + u − . This curve is birational to the elliptic curve y = x − x + x , which is rank 0 with torsionsubgroup isomorphic to Z / Z . The four torsion points are { (0 : 1 : 0) , (1 : 0 : 0) , ( − , ( − − } . Only the two points { (0 : 1 : 0) , (1 : 0 : 0) } correspond to rational pointson the original curve, and these points do not correspond to valid parameters.Now we investigate whether we can get rational fixed points if we have rational 2-cycles. The firstcase is when the third factor of Φ ∗ ( f ) = 0 has a solution, so we want to see if a = − (1 + t ) = 1 + l is possible for some rational t and l . This gives the curve2 + l + t = 0 , which clearly has no solutions over Q . Doing the same thing for the second component of Φ ∗ ( f ) = 0yields t + (1 + l ) t + 1 = 0 . The substitution x = t , y = l reduces this curve to X : x + (1 + y ) x + 1 = 0 , which is a genus 1 curve with the rational point ( x, y ) = ( − , − E : y = x − x + 96 x − . The curve E is a rank 0 elliptic curve with torsion subgroup isomorphic to Z / Z . Thus, the curve X has four rational (projective) points, which we find to be { (0 : 1 : 0) , (1 : 0 : 0) , ( − , ( − − } . The only affine points are ( − , ± Q since we covered by the squaring map. Because there re no rational points on the original curve, there cannot be any rational values of a for which f a has rational fixed points and rational 2-cycles in this case.The first factor of Φ ∗ ( f ) = 0 is similar. The curve we get this time is t − (1 + l ) t + 1 = 0 , which differs from the previous case by a minus sign. We can use the same cover by x and y to getthat this curve is birational to the same elliptic curve E in equation (8). Again there are only fourrational points: { (1 : 1 : 1) , (0 : 1 : 0) , (1 : 0 : 0) , (1 : − } . This time we see that one point has two positive coordinates, so the original curve has rational(affine) solutions { (1 , , (1 , − , ( − , , ( − , − } . They all correspond to the value a = 2 ( t = ± (cid:3) A search for rational preperiodic structures with the parameter up to height 10 ,
000 using thealgorithm from [20] as implemented in Sage yields no parameters where f a has a Q -rational periodicpoint with minimal period at least 3. Conjecture 7.3.
There is no a ∈ Q such that f a ( z ) = z +1 az has a Q -rational periodic point ofperiod at least 3.Assuming Conjecture 7.3, we classify the Q -rational preperiodic structures. Theorem 7.4.
Assuming Conjecture 7.3, the possible rational preperiodic structures for f a ( z ) = z +1 az are the following. G : • • ∞ (cid:47) (cid:47) (cid:119) (cid:119) , all a not in families G , G , or G G : • • ∞ • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:121) (cid:121) (cid:121) (cid:121) , a = 1 + t for t ∈ Q \ { } G : • • ∞ • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) , a = − (1 + t ) for t ∈ Q \ { } G : • ∞ • • • • (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) , a = ± ( t + t − ) for t ∈ Q \ { , ± } . Proof.
Starting with rational fixed points, we can immediately see that ∞ is always a fixed pointand 0 is its only rational (first) preimage for all parameters a . Since the numerator of f a ( z ) is z + 1, there are no rational preimages of 0 in Q . However, we may have rational preimages of theadditional rational fixed points when they do appear. We examine each of the two rational fixedpoints z = ± t .In the first case, we look at z = t . We know that a = 1 + t , so finding the preimages of thispoint amounts to solving the equation z + 1(1 + t ) z = 1 t , which determines the curve (1 − zt )( z t − z + zt + t ) = 0 . The first factor corresponds to the fixed point itself since z = t is fixed. The second factor can bereduced to the form y = x + x + x sing the birational transformation x = zt, y = z . This contains the point (0 ,
0) and is isomorphicto the elliptic curve E : y = x + 4 x + 16 x. The curve E has rank 0 and torsion subgroup isomorphic to Z / Z . Since one of the coordinates ofthe birational transformation was the square map, each point of E has one or two preimages. Wefind the original curve has three rational projective points { (0 : 1 : 0) , (0 : 0 : 1) , (1 : 0 : 0) } . Theonly affine one of these is (0 , (cid:54) = .Now we look at z = − t . In this case, we want to solve the equation z + 1(1 + t ) z = − t , which is the same as finding rational points on(1 + zt )( z t + z − zt + t ) = 0 . The same birational transformation as above can be used to get y = − x + x − x, which is isomorphic to the same elliptic curve E . For the same reason, there are three projectiverational points but only one corresponds to an affine point. In particular, we again see that (0 , z and t , which is invalid. Thus, the extra fixed points never havepreimages.We now investigate the case of points in 2-cycles having rational preperiodic tails. The first caseis when a = − (1 + t ), in which case the points z = t and z = − t map to each other. Findingrational preimages of these points results in the same equations as solving for preimages of fixedpoints, which we know do not exist.The second case is a = t + t − = t +1 t , which has 2-cycles { t, t } and {− t, − t } . The cases of f a ( z ) = ± t reduce to the genus 3 curves( zt ± z t ∓ z ± zt + t ) = 0 . The first component is the other point in the 2-cycle, so we focus on the second component. Theprojective closure has automorphism ( z, t, h ) (cid:55)→ ( − z, − t, h ). The quotient curve by this automor-phism is a genus one curve birational to the elliptic curve E : y = x + x + x. The curve E is rank 0 with torsion subgroup isomorphic to Z / Z . There are at most two rationalpoints on the quotient curve, which we find through a small height search as { (0 : 0 : 1 : 0) , (0 : 1 : 0 : 0) } . The only rational points on the original curve that map to these points can be found by solving forpreimages of the projection map. We get the points { (0 : 0 : 1) , (0 : 1 : 0) , (1 : 0 : 0) } . The affine point has t = 0, which is degenerate.Next we look at f a ( z ) = t ( z +1) z ( t +1) = ± t , which give the curves( z ± t )( z t ± z − zt ± t ) = 0 . The first component is the other point in the 2-cycle, so we focus on the second component.This curve is genus 3 and we can quotient by the order 2 automorphism( z, t, h ) (cid:55)→ ( − z, − t, h ) o obtain a genus 1 curve. A point search of low height gives the two points { ( − /
25 : 0 : 1 : 0) , (17 /
125 : 0 : 1 : 0) } . Using the first point, the curve is birational to the elliptic curve y = x + 83521 / x ,which is rank 0 with torsion subgroup isomorphic to Z / Z . So every rational point on the originalcurve must map to one of these two rational points on the quotient curve. Using the equations ofthe quotient map, we find the rational points in the inverse image of each to get { (0 : 0 : 1) , (0 : 1 : 0) , (1 : 0 : 0) } . The affine point has t = 0, which is the degenerate case. Thus there are no rational preimages ofthe points in any 2-cycle for the given parameterization of a .The final case is when a = − ( t + t − ), but it reduces to finding rational points on the samecurves as the previous case. (cid:3) A ( D ) . The model for this family is f a ( z ) = z + azaz +1 . We start by classifying periodic points. Proposition 7.5.
For the family f a ( z ) = z + azaz +1 , we have the following Q -rational periodic points.(1) The points 0, 1, and ∞ are fixed points for every choice of a ∈ Q \ {± } . There are noother rational fixed points.(2) For a = t + t + 1 + t − + t − with t ∈ Q \ { , ± } , f a ( z ) has a single Q -rational 2-cycle (cid:8) t, t (cid:9) . Proof.
We start with rational fixed points and see that ∞ is always fixed. For additional Q -rationalfixed points, we consider the first dynatomic polynomialΦ ∗ ( f ) = (1 − a ) z ( z − z + z + 1) . Note that a = 1 is degenerate since it would lead us to cancelling a factor of z + 1 in the function,resulting in a function that is not degree 4. The roots of z + z + 1 = 0 are cube roots of unity soare not rational; thus, the fixed points are 0, 1, and ∞ regardless of a .Looking for Q -rational 2-cycles, we consider the second dynatomic polynomialΦ ∗ ( f ) = ( a +1)( z + z +(1 − a ) z + z +1)( z − z + az +( a +1) z +( a − a − z +( a +1) z + az − z +1 . Looking at the curve Φ ∗ ( f ) = 0, we do not consider the degenerate component a = −
1. The degree4 component produces a genus 0 curve since it is linear in a . Solving for a , we get a = t + t + 1 + t − + t − for the 2-cycle consisting of z ∈ (cid:8) t, t (cid:9) . We see this is the only rational 2-cycle by factoring thesecond dynatomic polynomial over the function field Q ( t ). Note that t = − t = 1 has the 2-cycle collapsing to the fixed point z = 1.Now the degree 8 component is irreducible over Q but is reducible over Q ( ω ) where ω is a cuberoot of unity. We can factor it as z − z + az + ( a + 1) z + ( a − a − z + ( a + 1) z + az − z + 1 =( az − ωz + ( ω + 1) z − z − ωz + ω + 1)( az + ( ω + 1) z − ωz − z + ( ω + 1) z − ω ) . Since the curve factors over an extension of Q , rational points on it must be singular (Lemma 6.4).The singular points are just ( a, x ) ∈ { ( − , , (1 , − } but a = ± (cid:3) urning to periodic points of period 3, we look at the vanishing of the third dynatomic polynomialΦ ∗ ( f a ). It has a degree 6 and a degree 54 component. The degree 6 component is given by( a + a + 1) z + ( a + 4 a + 1) z + a + a + 1 = 0 . This is irreducible of Q , but reduced over Q ( ω ), where ω is a cube root of unity. Since the curvefactors over an extension of Q , rational points on it must be singular (Lemma 6.4). The singularpoints are ( a, x ) ∈ { ( − , , (1 , − } but a = ± z with z to have adegree 18 equation in z . This gives a genus 23 curve which is still problematic computationally.The only points of small height on it corresponded to a = ±
1, so are degenerate; however, we arenot able to fully analize this curve.A search for rational preperiodic structures with the parameter up to height 10 ,
000 using thealgorithm from [20] as implemented in Sage yields no parameters where f a has a Q -rational periodicpoint with minimal period at least 3. Conjecture 7.6.
There is no a ∈ Q such that f a ( z ) = z + azaz +1 has a Q -rational periodic point ofperiod at least 3.Assuming Conjecture 7.6, we classify the Q -rational preperiodic structures. Theorem 7.7.
Assuming Conjecture 7.6, the possible rational preperiodic structures for f a ( z ) = z + azaz +1 are the following. G : • − • • • ∞ (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (cid:119) (cid:119) , all a not in G , G , G , or G G : • − • •• • • ∞ (cid:47) (cid:47) (cid:31) (cid:31) (cid:15) (cid:15) (cid:121) (cid:121) (cid:121) (cid:121) (cid:119) (cid:119) , a = t + t − for t ∈ Q \ { , ± } G : • − • • • • • ∞ (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:119) (cid:119) , a = t for t ∈ Q \ { , ± } G : • − • • • ∞ • • (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) , a = t + t + 1 + t − + t − for t ∈ Q \ { , ± } G : •• • − • • • ∞ (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (cid:119) (cid:119) , for a = − − t t + t for t ∈ Q \ , ± . Proof.
We start with rational preimages of the fixed points. We begin with 0, which amounts tosolving z + az = 0, giving us the point z = − t as the preimage of 0 when a = t . Similarly,preimages of ∞ come from solutions to az + 1 = 0, which are parameterized by a = t as well,but the preimage is z = − t . Thus, the parameter a = t gives two additional rational points inthe preperiodic structure. Preimages of 1 come from solutions to f a ( z ) = 1, which gives the curve z − az + az − z − z + 1)( z − az + 1) = 0 . The factor of z − z = − a . Finally, the last factor is genus 0, so when a = t + t − the point z = t maps to 1. Furthermore, this image is invariant when we replace t by t , so in fact there willbe two rational points that map to 1 for these values of a . Note that t = ± ow we check if we can have additional preimages of 1 at the same time as preimages of 0 and ∞ . We need values of a such that a = t + t − = l for t, l ∈ Q . The curve X : t − tl + 1 = 0is a genus 2 hyperelliptic curve with model E : y + x y + 1 = 0 , although the map to this model is not an isomorphism. The curve E has a (isomorphic) simplifiedmodel E (cid:48) : y = x − Z / Z . Magma’s Chabauty methodtells us that there are only two rational points on this hyperelliptic curve. However, since themap from the original curve was not an isomorphism, we still have some work to do: we needto investigate the behavior at the singular points. Fortunately, the only singular point on theprojective model is (1 : 0 : 0), and an initial search on the projective closure of X finds the rationalpoints (0 : 1 : 0) and (1 : 0 : 0), which are all we expect to have. We need to look at the places ofthe divisor associated to this singular point. There is only one; because it is of degree 1, we havenot missed any rational points. Thus, the only rational points are at infinity, so the affine modelhas none and we can never have Q -rational preimages of all three fixed points at the same time.Now we look at second preimages of the fixed point 1. First we consider preimages of −
1. Weare looking for rational points on the curve z + az + az + 1 = 0 . This equation is linear in a , so we can solve as a = − − t t + t for t ∈ Q \ { , ± } . Now we considerpreimages of the additional preimages of 1, which occur for a = t + t − . We are looking for rationalpoints on the (genus 6) curve tz + ( − t − t ) z + ( t + 1) z − t = 0or, equivalently, the curve t z + ( − t − z + ( t + t ) z − t = 0 . These are equivalent problems since the two preimages are t and t and the automorphism z (cid:55)→ z takes rational points on one curve to rational points on the other curve. As these are genus 6curves, they can have at most finitely many rational points, due to Falting’s Theorem, contributingto the finitely many exceptions in the statement.Now we consider preimages of the preimages of −
1. We are looking for rational points on the(genus 1) curve C : ( t + 1) z + 2 tz + t + 1 = 0 . Over Q ( i ) it has the rational point ( − i : 0 : 1), which produces the Weierstrass model E : y = x + 4 x − x − . This curve is rank 0 with torsion subgroup isomorphic to Z / Z × Z / Z . Taking the inverse imageof the eight rational points on E , we get the rational points on the curve C are { (0 : 1 : 0) , ( i : 0 : 1) , (1 : 0 : 0) , (0 : − i : 1) , [(0 : i : 1) , ( − i : 0 : 1) } . The only rational values of t is 0, which is not a valid parameter. So there are no second Q -rationalpreimages of − (cid:8) t, t (cid:9) occurs for a = t + t + 1 + t − + t − with t ∈ Q \ { , ± } . By the automorphism z (cid:55)→ /z , it is equivalent to look t the preimage of either point in the cycle. Looking at preimages of t , we need to find rationalpoints on the genus 6 curve tz − t ( t + t + t + 1) z − ( t + t + t + 1) z + t = 0 . or t z − ( t + t + t + 1) z − t ( t + t + t + 1) z + t ) . As these are genus 6 curves, they can have at most finitely many rational points, due to Falting’sTheorem, contributing to the finitely many exceptions in the statement. (cid:3)
Note that both genus 6 curves that contribute to the finitely many exceptions are expected tohave no rational points contributing to additional graph structures. We, however, are unable tofully analyze the rational points on these curves.7.4.
The locus A ( C ) . This family is given by f k ,k ( z ) = z + k z k z +1 . We first examine periodicpoints. Proposition 7.8.
For the family f k ,k ( z ) = z + k z k z +1 , we have the following Q -rational periodicpoints.(1) The points 0 and ∞ are fixed for all pairs ( k , k ).(2) For ( u, v ) ∈ A ( Q ) and k = u v − u + 1 k = uv,f k ,k ( z ) has one additional rational fixed point z = u . Proof.
We look at the first dynatomic polynomial. The points 0 and ∞ are factors for all choicesof parameters k and k . The other component is given by( k − z − k + 1 = 0 . This forms a rational surface that has the given parameterization. Substituting the parameteriza-tion back into Φ ∗ ( f k ,k ), we see there is one additional fixed point. (cid:3) As with the degree 3 families with multiple parameters, periodic points with higher periodsand preperiodic points were difficult to study. We content ourselves with a census of Q -rationalpreperiodic structures for parameters k and k with small height.(0,1) (0,0) (0,-1/2) (1,-1) •• • ∞ (cid:121) (cid:121) (cid:119) (cid:119) (cid:47) (cid:47) • •• • ∞ (cid:121) (cid:121) (cid:119) (cid:119) (cid:47) (cid:47) (cid:121) (cid:121) •• • • ∞ (cid:121) (cid:121) (cid:119) (cid:119) (cid:47) (cid:47) (cid:47) (cid:47) • • • • ∞ (cid:121) (cid:121) (cid:119) (cid:119) (cid:47) (cid:47) (cid:47) (cid:47) (1,-1/3) (1,5/2) (1/2,1/2) • • • • • ∞ (cid:121) (cid:121) (cid:119) (cid:119) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) • • • •• ∞ (cid:121) (cid:121) (cid:119) (cid:119) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) • • • • •• ∞ (cid:121) (cid:121) (cid:119) (cid:119) (cid:47) (cid:47) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) -2,16/3) (-2,19/12) (-3,15/4) • • •• • • ∞ (cid:121) (cid:121) (cid:47) (cid:47) (cid:119) (cid:119) (cid:39) (cid:39) (cid:55) (cid:55) (cid:47) (cid:47) • • • • ∞ • • (cid:121) (cid:121) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) • • • • ∞ •• (cid:121) (cid:121) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (-4/3,5/6) (-4,-5/2) (-1/2,5/2) • • • • • ∞ • (cid:121) (cid:121) (cid:47) (cid:47) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:121) (cid:121) • • • • ∞ •• (cid:121) (cid:121) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) • • • •• •• ∞ (cid:121) (cid:121) (cid:119) (cid:119) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:47) (cid:47) (4/3,-13/6) (-8/3,-11/6) (3/2,3/2) • • • • • ∞ • • (cid:121) (cid:121) (cid:47) (cid:47) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) • • • • • ∞ • • (cid:121) (cid:121) (cid:47) (cid:47) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:47) (cid:47) (cid:121) (cid:121) • • • • ∞ • •• • (cid:121) (cid:121) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) (5/2,5/2) (-1/6,-1/6) • • • • ∞ •• • • (cid:121) (cid:121) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:121) (cid:121) • • • • • • ∞ • • (cid:121) (cid:121) (cid:47) (cid:47) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:121) (cid:121) (11/6,11/6) (17/10,17/10) • • • • ∞ • •• • (cid:121) (cid:121) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:47) (cid:47) (cid:121) (cid:121) (cid:34) (cid:34) (cid:98) (cid:98) • • • • ∞ •• • • (cid:121) (cid:121) (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:39) (cid:39) (cid:55) (cid:55) (cid:47) (cid:47) (cid:121) (cid:121) The locus A ( C ) . This family is given by f k ,k ,k ( z ) = z + k z +1 k z + k z . The point at infinityis always a fixed point with preimage 0. Additional fixed points are given by the first dynatomicpolynomial whose vanishing defines a rational hypersurfaceΦ ∗ ( f ) = ( k − z − k z + k z − . For this family, we again content ourselves with a census of Q -rational preperiodic structures forparameters k , k , and k with small height.(0,0,1) (0,0,2) (0,0,-2) (0,1,-1) (-2,0,1) • • ∞ (cid:47) (cid:47) (cid:119) (cid:119) • • ∞ • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) • • ∞ •• (cid:47) (cid:47) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (0,1/3,-4/3) (0,3,-5) (0,-3,5) (0,2/3,-8/3) • • ∞ •••• (cid:79) (cid:79) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:47) (cid:47) • • ∞ • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ •• •• (cid:47) (cid:47) (cid:119) (cid:119) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ • • • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ • • •• (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) • • ∞ • • • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (cid:111) (cid:111) (-2,0,9/2) (-2,1,2) (-2,1,-1/4) • • ∞ •••• (cid:47) (cid:47) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:47) (cid:47) • • ∞ •• •• (cid:47) (cid:47) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ •• •• (cid:47) (cid:47) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:15) (cid:15) (cid:79) (cid:79) (-2,1,-5/2) (1/4,9,-9/2) (1/4,9/4,-9/2) • • ∞ •• • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ •• •• (cid:47) (cid:47) (cid:119) (cid:119) (cid:79) (cid:79) (cid:47) (cid:47) (cid:47) (cid:47) (cid:79) (cid:79) • • ∞ •• • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) (1,7,-7) (1,-7,7) (1,7/2,-7/2) • • ∞ • •• • • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (cid:111) (cid:111) • • ∞ • • • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) • • ∞ • ••• •• (cid:15) (cid:15) (cid:85) (cid:85) (cid:47) (cid:47) (cid:111) (cid:111) (cid:55) (cid:55) (cid:39) (cid:39) (cid:103) (cid:103) (cid:119) (cid:119) (-1/2,3,-9/2) (-1/2,-3,9/2) (-1/2,3/2,3/2) • • ∞ • •• • • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (cid:111) (cid:111) • • ∞ • •• • • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ • • •• • • (cid:79) (cid:79) (cid:10) (cid:10) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (-1/2,3/2,-9/4) (-1/2,-3/2,9/4) (-2,0,9/4) • • ∞ • • •• • • (cid:79) (cid:79) (cid:10) (cid:10) (cid:47) (cid:47) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:47) (cid:47) (cid:121) (cid:121) • • ∞ • ••• • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:32) (cid:32) (cid:96) (cid:96) (cid:47) (cid:47) (cid:47) (cid:47) (cid:111) (cid:111) (cid:111) (cid:111) • • ∞ •• • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (-2,0,-9/4) (-2,1/4,7/2) (-2,4/5,-5) • • ∞ •• • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ • ••• •• (cid:79) (cid:79) (cid:10) (cid:10) (cid:47) (cid:47) (cid:111) (cid:111) (cid:55) (cid:55) (cid:39) (cid:39) (cid:103) (cid:103) (cid:119) (cid:119) • • ∞ • •• •• • (cid:15) (cid:15) (cid:85) (cid:85) (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) (cid:111) (cid:111) -2,6,-3/8) (-2,7,1/2) (-2,-7,-1/2) • • ∞ • •• •• • (cid:15) (cid:15) (cid:85) (cid:85) (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) (cid:111) (cid:111) • • ∞ •• • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) • • ∞ •• •• •• (cid:47) (cid:47) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:47) (cid:47) (cid:66) (cid:66) (cid:2) (cid:2) (1/3,8/3,-3/2) (-3,1,1) (-3,1,-3/2) • • ∞ • • • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ •• • •• • (cid:79) (cid:79) (cid:10) (cid:10) (cid:47) (cid:47) (cid:31) (cid:31) (cid:121) (cid:121) (cid:47) (cid:47) (cid:31) (cid:31) (cid:121) (cid:121) • • ∞ • • • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (-3,-1,-1) (-3,-1,3/2) (-3,1/3,-4/3) • • ∞ •• • • •• (cid:79) (cid:79) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:34) (cid:34) (cid:98) (cid:98) (cid:103) (cid:103) (cid:119) (cid:119) • • ∞ • • • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ • • • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) (-3,2/3,-1/6) (-3,3/2,-1) (-3,-3/2,1) • • ∞ • • • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:47) (cid:47) (cid:79) (cid:79) (cid:111) (cid:111) (cid:15) (cid:15) • • ∞ • •• • •• (cid:79) (cid:79) (cid:10) (cid:10) (cid:47) (cid:47) (cid:87) (cid:87) (cid:7) (cid:7) (cid:47) (cid:47) (cid:87) (cid:87) (cid:7) (cid:7) • • ∞ • • •••• (cid:79) (cid:79) (cid:10) (cid:10) (cid:31) (cid:31) (cid:63) (cid:63) (cid:79) (cid:79) (cid:95) (cid:95) (cid:127) (cid:127) (cid:15) (cid:15) (-3,4/3,-1/3) (-3,-4/3,1/3) (-3,1/6,-2/3) • • ∞ •••• •• (cid:15) (cid:15) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:47) (cid:47) (cid:66) (cid:66) (cid:2) (cid:2) • • ∞ •••• •• (cid:15) (cid:15) (cid:119) (cid:119) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:47) (cid:47) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ • •• •• • (cid:15) (cid:15) (cid:85) (cid:85) (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) (cid:111) (cid:111) (-4/3,8/9,-2/9) (-5/4,-5/4,2) (-3/4,1/6,-8/3) • • ∞ • • • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ • • • •• • (cid:47) (cid:47) (cid:119) (cid:119) (cid:34) (cid:34) (cid:98) (cid:98) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) (cid:121) • • ∞ • • • • •• • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (-5/4,5/3,-8/3 (-5/4,8/3,-5/3) (-3/4,7/3,-7/3) • • ∞ • •• • • •• • (cid:79) (cid:79) (cid:10) (cid:10) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) (cid:34) (cid:34) (cid:98) (cid:98) • • ∞ •• •• •• •• (cid:79) (cid:79) (cid:10) (cid:10) (cid:79) (cid:79) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:79) (cid:79) (cid:47) (cid:47) (cid:79) (cid:79) (cid:47) (cid:47) • • ∞ • ••••• •••• (cid:15) (cid:15) (cid:85) (cid:85) (cid:47) (cid:47) (cid:111) (cid:111) (cid:59) (cid:59) (cid:51) (cid:51) (cid:43) (cid:43) (cid:35) (cid:35) (cid:99) (cid:99) (cid:107) (cid:107) (cid:115) (cid:115) (cid:123) (cid:123) (-3/4,7/2,-7/2) (-3/4,-7/2,7/2) • • ∞ • • ••• • • ••• (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:55) (cid:55) (cid:39) (cid:39) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (cid:103) (cid:103) (cid:111) (cid:111) (cid:119) (cid:119) • • ∞ • • •• •• •• •• (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:47) (cid:47) (cid:31) (cid:31) (cid:15) (cid:15) (cid:121) (cid:121) (cid:47) (cid:47) (cid:31) (cid:31) (cid:15) (cid:15) (cid:121) (cid:121) • • ∞ • • • • • •• • • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (cid:111) (cid:111) (cid:47) (cid:47) (cid:34) (cid:34) (cid:98) (cid:98) (cid:111) (cid:111) • • ∞ • • • • • •• • • • (cid:47) (cid:47) (cid:119) (cid:119) (cid:31) (cid:31) (cid:15) (cid:15) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) (cid:47) (cid:47) (cid:121) (cid:121) Notice that there are Q -rational periodic points of (minimal) period { , , , , } . Are there Q -rational points of (minimal) period 5?8. Cycle Statistics for A d In finite fields, every rational point is preperiodic, so discussing the number of elements in thedirected graph of rational preperiodic points is no longer appropriate. However, there are numerousquestions one may ask about these graphs, such as what is the overall structure, the number ofperiodic versus non-periodic points, or the number of connected components. If one assumes that f : S → S is a random mapping on a finite set S , where we define random mapping as the imageof any given z ∈ S is equally likely to be any element of S , then the statistics have been wellstudied because of their connections with computational number theory and cryptography; see, forexample, the survey [35]. Hence, the question of which functions behave as random maps is also wellstudied in certain instances. For quadratic polynomials f c ( z ) = z + c , Pollard, in his ρ -factoringalgorithm [36], advised not to use z or z − z or z − f ) defined over the base field places restrictions on the graph structures thatare possible. For example, since the set of periodic points of a given period is invariant under theaction of the automorphism group, if there is an order two automorphism defined over the basefield and a rational cycle of odd length, then there must, in fact, be two such cycles. We proposethe following general philosophy. Without an underlying group structure, the big-O asymptoticsremain the same, but the implied constants are affected. • The existence of rational automorphisms means if there is one rational cycle of length n ,then there (typically) are several. Since this is true for every n , the overall average lengthremains unchanged. However, the number of cycles and number of periodic points shouldincrease. • Similarly, for preperiodic, if there is 1 rational preperiodic point, then there (typically)should be several. Since every point is preperiodic causing the total number to be un-changed, the “tree” of preperiodic points must be shorter causing a decrease in average taillength.Proving statistical distributions associated to this philosophy is beyond the scope of this article, butit would be an interesting future project. The graphs in Figures 1, 2, 3 provide empirical evidencefor this philosophy. Note that the field of definition of the automorphism plays a role; for example,the C data points in black split into two components depending on the residue class of p modulo3, i.e., whether or not third roots of unity exist in F p . The data for the red line representing thebehavior of a theoretical random mapping is taken from Flajolet and Odlyzko [13]. The yellow datapoints are randomly generated maps with trivial automorphism group. In all families the statistics Thanks to Institute for Computational and Experimental Research in Mathematics and the Brown UniversityCenter for Computation and Visualization for providing the computer time to compute this data.
00 1000 1500 2000 2500 3000 3500 p avg tail length FamiliesthIdC2C3D2
Figure 1.
Average Tail Lengthby Family p num cycles FamiliesthIdC2C3D2
Figure 2.
Total Number of Pe-riodic Cycles by Family
500 1000 1500 2000 2500 3000 3500 p num periodic FamiliesthIdC2C3D2
Figure 3.
Total Number of Periodic Points by Familyare averaged over 100 randomly generated maps of each type. The data for families in M show asimilar pattern of behavior, so we omit those graphs. eferences [1] Andrew Bridy. The artin-mazur zeta function of a dynamically affine rational map in positive characteristic. Journal de Theorie des Nombres de Bordeaux , 28(2):301–324, 2016.[2] N. Bruin, B. Poonen, and M. Stoll. Generalized explicit descent and its application to curves of genus 3.
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Department of Mathematics, Stonybrook University, Stony Brook NY, 11794
E-mail address : [email protected] Department of Mathematics and Statistics, Saint Louis University, St. Louis, MO 63103
E-mail address : [email protected] Department of Mathematics, Florida State University, Tallahassee, FL 32306
E-mail address : [email protected] Department of Mathematics, Brown University, Providence, RI 02912
E-mail address : srinjoy [email protected] Department of Mathematics and Statistics, Colby College, Waterville, Maine 04901
E-mail address : [email protected]@colby.edu