Doubly isogenous genus-2 curves with D_4-action
Vishal Arul, Jeremy Booher, Steven R. Groen, Everett W. Howe, Wanlin Li, Vlad Matei, Rachel Pries, Caleb Springer
DDOUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION VISHAL ARUL, JEREMY BOOHER, STEVEN R. GROEN, EVERETT W. HOWE, WANLIN LI,VLAD MATEI, RACHEL PRIES, AND CALEB SPRINGERA
BSTRACT . We study the zeta functions of curves over finite fields. Suppose C and C (cid:48) arecurves over a finite field K , with K -rational base points P and P (cid:48) , and let D and D (cid:48) be thepullbacks (via the Abel–Jacobi map) of the multiplication-by-2 maps on their Jacobians. Wesay that ( C , P ) and ( C (cid:48) , P (cid:48) ) are doubly isogenous if Jac ( C ) and Jac ( C (cid:48) ) are isogenous over K and Jac ( D ) and Jac ( D (cid:48) ) are isogenous over K . For curves of genus 2 whose automorphismgroups contain the dihedral group of order eight, we show that the number of pairs of doublyisogenous curves is larger than na¨ıve heuristics predict, and we provide an explanation forthis phenomenon.
1. I
NTRODUCTION
Suppose C and C (cid:48) are smooth projective geometrically connected curves of genus g overa field K . The Jacobians Jac ( C ) and Jac ( C (cid:48) ) may be isogenous over K even when C and C (cid:48) are not isomorphic to one another. Over non-prime finite fields K , it is easy to createsuch examples by taking C (cid:48) to be a Galois conjugate of C . Smith [18] showed that evenin characteristic 0 there exist non-isomorphic curves C and C (cid:48) of arbitrarily large genuswith Jac ( C ) isogenous to Jac ( C (cid:48) ) . Later, Mestre [12, 13] proved that such pairs of curvesexist for every genus in characteristic 0; in particular, for every g ≥ g + g with a 2-power isogeny betweentheir Jacobians. Thus the isogeny class of the Jacobian is not an invariant that can alwaysdistinguish two curves from one another.Motivated by this observation, one can ask whether we can distinguish curves usingthe isogeny classes of the Jacobians of certain covers, which we now define. Let ( C , P ) and ( C (cid:48) , P (cid:48) ) be pointed curves over K , that is, curves provided with a K -rational point.Let D → C be the pullback of the multiplication-by-2 map on Jac ( C ) via the embedding C → Jac ( C ) that sends P to the identity, and let D (cid:48) → C (cid:48) be the analogously defined coverof C (cid:48) . We say that ( C , P ) and ( C (cid:48) , P (cid:48) ) are doubly isogenous if Jac ( C ) and Jac ( C (cid:48) ) are isogenousover K and Jac ( D ) and Jac ( D (cid:48) ) are isogenous over K .If K is a finite field, then C and C (cid:48) are isogenous if and only if they have the same zetafunction over K . One can use arithmetic statistics to develop heuristics for the number ofpairs of pointed curves ( C , P ) and ( C (cid:48) , P (cid:48) ) that are isogenous or doubly isogenous. Thenone can collect data to test these heuristics.In as-yet-unpublished work, Howe, Sutherland, and Voloch studied genus-2 curves hav-ing an automorphism of order 3; the full automorphism group of these curves contains thedihedral group of order 12. They gave a heuristic for the number of such curves over finite Date : 22 February 2021.2020
Mathematics Subject Classification.
Primary 11G20, 11M38, 14H40, 14K02, 14Q05; Secondary 11G10,11Y40, 14H25, 14H30, 14Q25.
Key words and phrases.
Curve, Jacobian, finite field, zeta function, isogeny, unramified cover, arithmeticstatistics. a r X i v : . [ m a t h . N T ] F e b ARUL, BOOHER, GROEN, HOWE, LI, MATEI, PRIES, AND SPRINGER fields of characteristic not 2 or 3 that are doubly isogenous. Their data showed that thenumber of such pairs was larger than expected. This over-abundance was explained by theexistence of a pair of doubly isogenous pointed curves over the number field Q ( √ ) ; forevery prime p of this number field, the reductions of these curves modulo p gives a pair ofdoubly isogenous pointed curves having an automorphism of order 3.We explore another instance of this problem, working over a field K of characteristic not2 containing a primitive 4th root of unity. We consider curves of genus 2 with an automor-phism ρ of order 4. The automorphism groups of these curves contain the dihedral group oforder 8. We study their elementary abelian 2-group covers, in some cases restricting to thesituation where the Weierstrass points of the genus-2 curves are K -rational. An imprecisesummary of our main results is that, taking K = F q for a prime q ≡ F q whose Jacobians are isogenous over F q grows as expected; see Theorem 4.6 and Table 2.(2) The number of pairs of such curves that are “ [ − ρ ∗ ] -isogenous” over F q grows asexpected; see Example 5.5 and Table 3(B). (The terminology is explained in Exam-ple 5.5, but roughly speaking, the definition of being [ − ρ ∗ ] -isogenous is the sameas that of being doubly isogenous, except the multiplication-by-2 map on Jac ( C ) isreplaced by a degree-4 endomorphism of Jac ( C ) .)(3) The number of pairs of such curves that are doubly isogenous over F q is larger thanexpected; see Lemma 5.3 and Table 3(A). We explain this discrepancy in Section 6 byfinding unexpected relationships between the Prym varieties of certain covers.We remark that this family of curves is Moonen’s fourth special family [16]. It would beinteresting to study isogenies between curves in the other special families of Moonen. Conventions. A curve over a field K is a smooth projective geometrically connected 1-dimensional variety over K . If C is a curve over a field K , then Aut ( C ) is the group of K -rational automorphisms of C ; if L is an extension field of K , then Aut L ( C ) is the group of L -rational automorphisms of C .2. T HE FAMILY OF GENUS -2 CURVES WITH D - ACTION
Let K be a field of characteristic not 2. In this section, we find an equation that describesevery genus-2 curve over K whose automorphism group contains the dihedral group D of order 8. We also show that the Jacobian of a genus-2 curve with D contained in itsautomorphism group is isogenous to the square of an elliptic curve.We fix a presentation of the dihedral group of order 8: D = (cid:104) a , b | a = b = ( ab ) = (cid:105) .We let ξ denote the automorphism of D that interchanges a and b . Notation 2.1. A curve with D -action is a curve Z together with an embedding (cid:101) : D (cid:44) → Aut ( Z ) . Two curves with D -action ( Z , (cid:101) ) and ( Z (cid:48) , (cid:101) (cid:48) ) are isomorphic if there is a K -rationalisomorphism ϕ : Z → Z (cid:48) such that the following diagram commutes: D (cid:101) (cid:123) (cid:123) (cid:101) (cid:48) (cid:36) (cid:36) Aut ( Z ) δ (cid:55)→ ϕ ◦ δ ◦ ϕ − (cid:47) (cid:47) Aut ( Z (cid:48) ) OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 3 Let Z denote the set of K -isomorphism classes of genus-2 curves with D -action over K .Using Igusa’s classification [8, §8] of the automorphism groups of genus-2 curves, we checkthat if ( Z , (cid:101) ) is a genus-2 curve with D -action, then (cid:101) (( ab ) ) is the hyperelliptic involution. Remark 2.2. If ( Z , (cid:101) ) is a curve with D -action and α is an automorphism of Z , we let (cid:101) α denote the inclusion D (cid:44) → Aut ( Z ) that sends x to α(cid:101) ( x ) α − . Note that α : Z → Z then givesa morphism of pairs ( Z , (cid:101) ) → ( Z , (cid:101) α ) , which shows that conjugating (cid:101) by an automorphismof Z does not change the isomorphism class of the pair ( Z , (cid:101) ) .2.1. A family of genus- curves with D -action. Let c and s be elements of K with c (cid:54) = s (cid:54) = ±
2, and let Z be the genus-2 curve(2.1) Z : y = c ( x + )( x + sx + ) .The hyperelliptic involution κ of Z is given by ( x , y ) (cid:55)→ ( x , − y ) . The curve Z also has other K -rational involutions, including σ : ( x , y ) (cid:55)→ ( − x , y ) and τ : ( x , y ) (cid:55)→ ( x , y / x ) .Let ρ = στ , so that ρ takes ( x , y ) to ( − x , y / x ) . We note that ρ = κ . The group G generated by σ and τ is isomorphic to D . More precisely, we specify an inclusion (cid:101) : D (cid:44) → Aut ( Z ) (2.2) a (cid:55)→ σ b (cid:55)→ τ .Thus, Equations (2.1) and (2.2) give us a family of genus-2 curves with D -action. Remark 2.3.
When s ∈ {− −
1, 14 } , the curve (2.1) has geometric automorphism groupstrictly larger than D ; using Igusa’s classification [8, §8], we can show that all other valuesof s give curves with geometric automorphism group isomorphic to D .2.2. Classifying genus- curves with D -action up to isomorphism.Lemma 2.4. Let ( Z (cid:48) , (cid:101) (cid:48) ) be a genus- curve over K with D -action. Then there are values c , s ∈ Ksuch that the curve with D -action ( Z , (cid:101) ) given by (2.1) and (2.2) is isomorphic to ( Z (cid:48) , (cid:101) (cid:48) ) . Thevalue of s is unique, and the value of c is unique up to multiplication by elements of K × .Proof. Let α = (cid:101) (cid:48) ( a ) , let β = (cid:101) (cid:48) ( b ) , and let ι be the hyperelliptic involution on Z (cid:48) . Thequotient of Z (cid:48) by α has genus 1 because α (cid:54) = ι . The Riemann–Hurwitz formula shows that α has two geometric fixed points. If P is one of these fixed points, then ι P = ια P = αι P , so ι P is fixed by α as well.We claim that P (cid:54) = ι P . To see this, consider the V -subgroup H = (cid:104) α , ι (cid:105) . The stabilizerof any point under H is the decomposition group of the corresponding place in the geo-metric cover Z (cid:48) → Z (cid:48) / H . This decomposition group is cyclic since the extension is tamelyramified. Hence, no fixed point of α is fixed by ι .Consider the quotient P = Z (cid:48) / (cid:104) ι (cid:105) . The two fixed points of α are P and ι P . These twopoints map to the same point Q in P and Q must be K -rational. Since α and β both com-mute with ι , they descend to automorphisms α and β of P . The point Q is one of the fixedpoints of the involution α , so both fixed points of α must be K -rational. By choosing thecoordinate x on P appropriately, we may assume that the fixed points of α are x = x = ∞ . This means that an equation for Z (cid:48) is y = a x + a x + a x + a , ARUL, BOOHER, GROEN, HOWE, LI, MATEI, PRIES, AND SPRINGER for some constants a , a , a , a ∈ K , and in this model α sends ( x , y ) to ( − x , y ) .Since ( αβ ) = ι and ι induces the trivial automorphism on P , we see that αβ is aninvolution of P , implying that α and β commute. This means that β must be a linearfractional transformation of the form x (cid:55)→ d / x for some d ∈ K × . Since the fixed points of β are also K -rational, d is a square in K × . By scaling x by √ d , we may assume that d = a = a and a = a , so that an equation for Z (cid:48) is y = a x + a x + a x + a ,and so that β sends ( x , y ) to ( x , y / x ) . Let c = a and s = a / a , and let ( Z , (cid:101) ) bethe genus-2 curve with D -action given by (2.1) and (2.2). Our model for Z (cid:48) gives us anisomorphism ( Z (cid:48) , (cid:101) (cid:48) ) → ( Z , (cid:101) ) .Demanding that the fixed points of α be 0 and ∞ and that the fixed points of β be 1and − Z (cid:48) , up to scaling y by aconstant. These scalings modify c by multiplication by a square in K × . This proves the finalstatement of the lemma. (cid:3) Lemma 2.5.
The two curvesZ : y = c ( x + )( x + sx + ) and Z (cid:48) : y = c (cid:48) ( x + )( x + s (cid:48) x + ) are isomorphic to one another if and only if either c (cid:48) = c ( in K × / K × ) and s (cid:48) = s, or c (cid:48) = c ( s + ) ( in K × / K × ) and ( s (cid:48) + )( s + ) = . Note that the lemma says that every curve of the form given by Equation (2.1) has exactlyone other model of the same form, unless s = − − Proof of Lemma . If either of the given relations among c , c (cid:48) and s , s (cid:48) hold, it is easy tocheck that the two curves are isomorphic to each other. The isomorphism in the secondcase is given by ( x , y ) (cid:55)→ (cid:0) ( x + ) / ( x − ) , y / ( x − ) (cid:1) .On the other hand, suppose we have a curve Z as in the lemma. We would like to see howmany other models it has that are also of the form given by Equation (2.1). Notation 2.1,Remark 2.2, and Lemma 2.4 show that these models correspond to the embeddings of D into Aut ( Z ) , up to conjugation by Aut ( Z ) , so we just need to count the number of suchembeddings up to conjugacy.If s (cid:54)∈ {− −
1, 14 } then Aut ( Z ) ∼ = D by Remark 2.3. The outer automorphism groupof D has two elements, so there are two embeddings of D into Aut ( Z ) up to conjugationand hence two models of the form (2.1). These are accounted for by the two models in thelemma.If s ∈ {−
1, 14 } , then by computing Igusa invariants and consulting [8, §8] we findthat Aut K ( Z ) is a certain group of order 24, so Aut ( Z ) is a subgroup of this group thatcontains D . By enumeration, we find that for each such subgroup G there are two conju-gacy classes of embedding D (cid:44) → G . Once again, these are accounted for by the two modelsin the lemma.When s = −
6, we find from Igusa that Aut K ( Z ) is either a certain group G of order 48(if K has characteristic not 5) or a certain group G of order 240 (if K has characteristic 5).Both of these groups contain a unique subgroup G of order 16. For every subgroup G of G or G that contains D , we find that the number of conjugacy classes of embeddings D (cid:44) → G is equal to 2 if G does not contain G , and is equal to 1 if G does contain G . OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 5 In terms of the model y = c ( x + )( x − x + ) for Z , the group G is generated bythe involutions σ and τ together with the automorphism υ of order 8 given by ( x , y ) (cid:55)→ (( x − ) / ( x + ) , 2 √− y / ( x + ) ) . We see that G is contained in Aut ( Z ) if and only if − K . Combined with the results of the preceding paragraph, we find twomodels for Z when − (cid:3) An invariant of the curve Z . Lemma 2.5 shows that two curves Z and Z (cid:48) of the formgiven by Equation (2.1) are geometrically isomorphic to one another if and only if either s (cid:48) = s or s (cid:48) = ( − s + ) / ( s + ) . The function(2.3) I ( s ) : = − ( s − ) ( s + ) = − s + s (cid:48) s ↔ s (cid:48) and is rational of degree 2, so it gives a geometricinvariant for the curve Z .2.4. Structure of the Jacobian of the curve Z . In this section, we consider the quotients of Z by the non-central involutions of D .Let E be the elliptic curve defined by(2.4) E : y = c ( x + )( x + sx + ) . Lemma 2.6.
The quotient of Z by each of the involutions ( x , y ) (cid:55)→ ( − x , ± y ) is isomorphic to E,and Jac ( Z ) is isogenous to E .Proof. The quotient of Z by the involution ( x , y ) (cid:55)→ ( − x , y ) is clearly E .To find the quotient by ( x , y ) (cid:55)→ ( − x , − y ) , it helps to rewrite the equation for Z as x y = cx ( x + )( x + sx + ) .Since xy and x are both fixed by the involution, the quotient is given by the equation y = cx ( x + )( x + sx + ) .If we replace ( x , y ) with ( x , y / x ) , we obtain (2.4) and hence the second quotient is iso-morphic to E .These two involutions generate a subgroup of Aut ( Z ) isomorphic to the Klein four-group. The product of these involutions is the hyperelliptic involution κ ; the quotient of Z by κ is the projective line. By [10, Theorem C], Jac ( Z ) is isogenous to the product of theJacobians of the three quotients, thus Jac ( Z ) ∼ Jac ( E ) ∼ = E . (cid:3) Let s (cid:48) = ( − s + ) / ( s + ) and c (cid:48) = c ( s + ) , and let E (cid:48) be the elliptic curve E (cid:48) : y = c (cid:48) ( x + )( x + s (cid:48) x + ) .Note that there is a 2-isogeny E → E (cid:48) given by ( x , y ) (cid:55)→ (cid:32) s + ( x + s )( x − )( x + ) , 4 s + ( x + x + s − )( x + ) y (cid:33) ,whose kernel contains the 2-torsion point P = ( −
1, 0 ) of E . The kernel of the dual isogeny E (cid:48) → E contains the 2-torsion point P (cid:48) = ( −
1, 0 ) of E (cid:48) . Lemma 2.7.
The quotient of Z by each of the involutions ( x , y ) (cid:55)→ ( x , ± y / x ) is isomorphicto E (cid:48) . ARUL, BOOHER, GROEN, HOWE, LI, MATEI, PRIES, AND SPRINGER
Proof.
Replacing x with ( x + ) / ( x − ) and y with y / ( x − ) in the equation for Z , wefind that Z can also be written as y = c (cid:48) ( x + )( x + s (cid:48) x + ) .The two involutions ( x , y ) (cid:55)→ ( x , ± y / x ) in the original model become the involutions ( x , y ) (cid:55)→ ( − x , ∓ y ) in the new model. The result follows from Lemma 2.6. (cid:3) Lemma 2.6 says that there is an isogeny E → Jac ( Z ) , but we can be much more precise. Proposition 2.8.
Let E be as above, let P = ( −
1, 0 ) ∈ E [ ] , and let Q and R be the other twogeometric points of order on E. Let ψ : E [ ] → E [ ] be the automorphism that fixes P and swapsQ and R. Then there is an isogeny ϕ : E × E → Jac ( Z ) whose kernel is the graph of ψ , and thepullback via ϕ of the principal polarization on Jac ( Z ) is twice the product polarization on E × E.Proof.
Because there is a degree-2 map Z → E , the general theory set out in [11, §2] showsthat there is an elliptic curve F , an isomorphism ψ : E [ ] → F [ ] , and an isogeny E × F → Jac ( Z ) satisfying the conclusion of the proposition. The explicit construction carried out in[5, §3] shows that F ∼ = E and that ψ is the isomorphism specified in the statement. (cid:3) In fact, almost every pair ( E , P ) consisting of an elliptic curve E over K and a K -rational2-torsion point arises in this way. Proposition 2.9.
Let E be an elliptic curve over K with a rational point P of order . Then there isa genus- curve Z over K with D -action that gives rise to this ( E , P ) as above if and only if E does not have a geometric automorphism α (cid:54) = ± that fixes P.Proof. Given an E and a P as in the statement of the proposition, we may choose a model y = x ( x + ax + b ) for E so that P is the point (
0, 0 ) . Let ψ : E [ ] → E [ ] be the auto-morphism that fixes P and swaps the other two points of order 2. If there is no geometricautomorphism of E that restricts to ψ on E [ ] , then the construction of [5, Proposition 4,p. 324] produces a genus-2 curve Z of the form (2.1), and we check that the E and P pro-duced by this curve as above are the E and P we started with.If there is a geometric automorphism α of E that restricts to ψ , then the geometric isogeny ϕ : E × E → E × E that takes ( U , V ) to ( U + α − ( V ) , V − α ( U )) has kernel equal to the graphof ψ , and the pullback via ϕ of the product polarization is twice the product polarization. Ifthere were a curve Z that gave rise to ( E , P ) , then by Proposition 2.8 the polarized Jacobianof Z would be geometrically isomorphic to E × E with the product polarization, which isimpossible. To complete the proof, we just need to observe that over fields of characteristicnot 2, every automorphism α (cid:54) = ± (cid:3) Related families of genus- curves with D -action. If K is algebraically closed, Car-dona and Quer [2, Proposition 2.1] show that every genus-2 curve Y with Aut ( Y ) ∼ = D isa member of the family Y v : y = x + x + vx ,where v ∈ K \ {
0, 1/4, 9/100 } .The advantage of this family is that every geometric isomorphism class of a curve withautomorphism group D corresponds to exactly one value of v , as opposed to the family Z in (2.1). The disadvantage is that the automorphisms of this curve are not necessarilydefined over the field generated by the parameter v . OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 7 Moonen [16] studied cyclic covers of P given by monodromy data. One of the twentyfamilies of curves that appear in his work is the cyclic degree-4 cover of P given by X T : z = x ( x − ) ( x − T ) .The curve X T has genus 2 and Aut K ( X T ) ∼ = D for a generic choice of T . This model makesthe order-4 automorphism very apparent, but the hyperelliptic structure is not as clearlyvisible. 3. T HE TORSION AND UNRAMIFIED ELEMENTARY ABELIAN COVERS
In this section, we study unramified elementary abelian 2-covers of the curves Z definedby (2.1). Throughout this section, we assume all Weierstrass points of Z are defined over K .Let ζ be a primitive fourth root of unity in K . Then Z can be given by Z : y = c ( x − ζ )( x + ζ )( x − t )( x + t )( x − t )( x + t ) , where(3.1) s = − ( t + ) / t .(3.2)3.1. Unramified elementary abelian 2-covers.
Let P ∈ Z ( K ) be the Weierstrass point ( ζ , 0 ) .Since there is a K -rational automorphism of Z taking P to ( − ζ , 0 ) , the choice of ζ does notaffect the K -isomorphism class of the 2-covers we construct. Note that P and ( − ζ , 0 ) aredistinguished from the other Weierstrass points of Z by the fact that they form an orbit ofsize 2 under the action of D ; the other Weierstrass points form an orbit of size 4. Definition 3.1.
Let ι P : Z (cid:44) → Jac ( Z ) be the Abel-Jacobi embedding that sends Q ∈ Z ( K ) tothe divisor class [ Q − P ] . Let π : (cid:101) Z → Z be the pullback of the multiplication-by-2 map onJac ( Z ) by ι P : Z (cid:44) → Jac ( Z ) .Our assumption that the Weierstrass points of Z are K -rational implies that the coverJac ( Z ) → Jac ( Z ) given by the multiplication-by-2 map is Galois, with Galois group isomor-phic to Jac ( Z )[ ] ∼ = ( Z /2 Z ) ; the group Jac ( Z )[ ] acts on the cover by translation. Since π is defined as a pullback of this cover, π is also Galois, with Galois group canonicallyisomorphic to Jac ( Z )[ ] . In fact, geometric class field theory shows that we can recognize π as the maximal unramified abelian extension of Z with Galois group of exponent 2 inwhich the base point P = ( ζ , 0 ) splits completely. Definition 3.2.
For a subgroup H of Jac ( Z )[ ] , let (cid:101) Z H be the quotient of (cid:101) Z by H . Let π H : (cid:101) Z H → Z be the quotient cover.For example, (cid:101) Z = (cid:101) Z and (cid:101) Z Jac ( Z )[ ] = Z . More generally, the degree of π H equals theindex of H in Jac ( Z )[ ] . Since Jac ( Z )[ ] is abelian, π H is Galois with Galois group iso-morphic to Jac ( Z )[ ] / H . Furthermore, the genus of (cid:101) Z H equals [ Jac ( Z )[ ] : H ] + Remark 3.3.
If we pick a different basepoint P (cid:48) instead of P and keep track of the basepointdependence by labeling the 2-covers as (cid:101) Z P and (cid:101) Z P (cid:48) , and if we let Q ∈ Jac ( Z )( K ) be a pointwith 2 Q = P − P (cid:48) , then translation by Q on Jac ( Z )( K ) yields a geometric isomorphismfrom (cid:101) Z P to (cid:101) Z P (cid:48) . We will prove in the following paragraph that there exists an elementaryabelian 2-extension L of K of degree at most 2 such that Q ∈ Jac ( Z )( L ) , so this translationisomorphism will be defined over L . In particular, when K is a finite field, then L is at worst ARUL, BOOHER, GROEN, HOWE, LI, MATEI, PRIES, AND SPRINGER a quadratic extension of K , and the curve (cid:101) Z P (cid:48) is a (possibly trivial) quadratic twist of (cid:101) Z P .From now on, we fix the base point to be P = ( ζ , 0 ) for all π H .The following argument was provided by Bjorn Poonen. The obstruction to dividing apoint of Jac ( Z )( K ) by 2 lies in H ( K , Jac ( Z )[ ]) . Since all the Weierstrass points are definedover K , we know that Jac ( Z )[ ] ∼ = µ as a Galois module. Hence, the obstruction to divid-ing P (cid:48) − P by 2 lies in H ( K , Jac ( Z )[ ]) ∼ = ( H ( K , µ )) = ( K × / K × ) , so there exists anelementary abelian 2-extension L / K of degree at most 2 such that the image of this classin H ( L , Jac ( Z )[ ]) = ( L × / L × ) is trivial. When K is a finite field of characteristic not 2,over its unique elementary abelian 2-extension L , this obstruction class vanishes. Thus (cid:101) Z P (cid:48) is a quadratic twist of (cid:101) Z P .3.2. Decomposition of the Jacobian.
In this section, we determine the isogeny decompo-sition of Jac ( Z H ) over K . Definition 3.4.
Given a cover π : V → Z , let Prym π denote the Prym variety of π , thatis, the identity component of the kernel of the induced norm homomorphism Jac ( V ) → Jac ( Z ) . There is a K -isogeny Jac ( V ) ∼ Jac ( Z ) × Prym π . Definition 3.5.
For a subgroup H of Jac ( Z )[ ] , we set Prym H : = Prym π H , where π H is thecover defined in Definition 3.2. Proposition 3.6.
Let Z be a genus- curve with D -action whose Weierstrass points are definedover K. For every H ⊆ Jac ( Z )[ ] , there is an isogeny (3.3) Jac ( (cid:101) Z H ) ∼ E × ∏ H ⊆ H (cid:48)⊆ Jac ( Z )[ ][ Jac ( Z )[ ] : H (cid:48) ]= Prym H (cid:48) . Proof.
Let G = Jac ( Z )[ ] and r = [ G : H ] , and enumerate the index-2 subgroups of G containing H by H (cid:48) , · · · , H (cid:48) r − . We apply [10, Theorem C] to the { H (cid:48) i } together with H and G . More precisely, we define H i : = H (cid:48) i for i =
1, . . . r − H for i = r , G for i = r +
1, and n i : = − i =
1, . . . r − i = r , r − i = r + g ij be the genus of Z H i H j . The group H i H j must be one of H (cid:48) , · · · , H (cid:48) r − , H , G . We knowfrom Riemann–Hurwitz that the genus of (cid:101) Z H is r +
1, the genus of each (cid:101) Z H (cid:48) i is 3, and thegenus of (cid:101) Z G is 2. Using this information and some casework, we see that g ij = i (cid:54) = j and i , j ≤ r − i = j ≤ r − i ≤ r − j = r , or i = r and j ≤ r − r + i = j = r ,2 if i or j is r + H i H j = H j H i is satisfied because G is abelian; and second, ∑ i n i g ij = j ∈ {
1, . . . , r + } by our computations above. OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 9 Therefore the conclusion of [10, Theorem C] holds, namely, there exists a K -isogeny ∏ n i > Jac ( (cid:101) Z H i ) n i ∼ ∏ n j < Jac ( (cid:101) Z H j ) − n j ,which for us becomes Jac ( (cid:101) Z H ) × ( Jac ( Z )) r − ∼ ∏ H ⊆ H (cid:48)⊆ Jac ( Z )[ ][ Jac ( Z )[ ] : H (cid:48) ]= Jac ( (cid:101) Z H (cid:48) ) .Now we substitute Jac ( (cid:101) Z H (cid:48) ) ∼ Jac ( Z ) × Prym H (cid:48) , cancel ( Jac ( Z )) r − from both sides, andsubstitute Jac ( Z ) ∼ E (from Lemma 2.6) to finish. (cid:3) The Weil pairing.Definition 3.7.
Let R = { ζ , − ζ , t , − t , 1/ t , − t } . For r ∈ R , let W r denote the Weierstrasspoint ( r , 0 ) of Z . Lemma 3.8.
If D ∈ Jac ( Z )[ ] , then there exist u , v ∈ R such that D = [ W u − W v ] .Proof. We know Jac ( Z )[ ] is generated by [ W r − W ζ ] for r ∈ R \ { ζ } with the single relation ∑ r [ W r − W ζ ] =
0. The conclusion follows from a straightforward computation. (cid:3)
Definition 3.9.
Let e ( · , · ) denote the Weil pairing on Jac ( Z )[ ] , which takes values in {± } ⊂ K × . For every subgroup H of Jac ( Z )[ ] , define H ⊥ by H ⊥ : = { S ∈ Jac ( Z )[ ] : e ( Q , S ) = Q ∈ H } .For later use, we give an explicit description of the Weil pairing on 2-torsion points. Lemma 3.10.
For nonzero elements [ W u − W v ] and [ W u − W v ] of Jac ( Z )[ ]( K ) , we havee ([ W u − W v ] , [ W u − W v ]) = − if and only if ( { u , v } ∩ { u , v } ) = .Proof. This is a direct calculation using a well-known formula for the Weil pairing (see [4,Theorem 1]). (cid:3)
Proposition 3.11.
Let H (cid:48) be an index- subgroup of Jac ( Z )[ ] and let U = [ W u − W v ] be thegenerator of ( H (cid:48) ) ⊥ . Define a ∈ K × / K × bya : = (cid:40) c ( ζ − u )( ζ − v ) if ζ (cid:54)∈ { u , v } ∏ r ∈ R \{ u , v } ( ζ − r ) if ζ ∈ { u , v } . Let E (cid:48) be the genus- curve given by the equation y = a ∏ r ∈ R \{ u , v } ( x − r ) . Then there is a K-isogeny
Prym H (cid:48) ∼ Jac ( E (cid:48) ) . Proof.
Let f = ac ( x − u )( x − v ) and f = a ∏ r ∈ R \{ u , v } ( x − r ) , and consider the V -diagramof function fields K ( x , (cid:112) f , (cid:112) f ) K ( x , (cid:112) f ) K ( x , (cid:112) f f ) K ( x , (cid:112) f ) K ( x ) . If we let C be the genus-0 curve y = f , the diagram above gives us a V -diagram of curves(3.4) Y (cid:120) (cid:120) (cid:38) (cid:38) (cid:15) (cid:15) C (cid:37) (cid:37) Z (cid:15) (cid:15) E (cid:48) (cid:121) (cid:121) P ,where Y is the curve with function field K ( x , (cid:112) f , (cid:112) f ) . The value of a was chosen so thatthe point x = ζ of P splits in one of the extensions C → P and E (cid:48) → P and ramifies inthe other, and it follows that the Weierstrass point P = ( ζ , 0 ) of Z splits in the quadraticextension Y → Z . Since Y → Z is a Galois extension with group of exponent 2 in which P splits completely, it must be a subextension of (cid:101) Z → Z , which as we noted earlier is themaximal such extension. This tells us that the element f of K ( Z ) × is a square in K ( (cid:101) Z ) × .In fact, we see that the map U (cid:55)→ f defines an injective homomorphism γ : Jac ( Z )[ ] → ( K ( Z ) × ∩ K ( (cid:101) Z ) × ) / K ( Z ) × . If we let G be the latter group, then Kummer theory says thatthere is a perfect pairing Gal ( (cid:101) Z / Z ) × G → {± } ⊂ K × .In particular, G = ( (cid:101) Z / Z ) = ( Z )[ ] , so the injective homorphism γ is an isomor-phism. This isomorphism, together with the canonical isomorphism Jac ( Z )[ ] ∼ = Gal ( (cid:101) Z / Z ) ,turns the Kummer pairing into a perfect pairingJac ( Z )[ ] × Jac ( Z )[ ] → {± } .As is observed in [4, §2], this pairing is in fact the Weil pairing; this can be seen by usingthe explicit formula for the natural pairing of the m -torsion of an abelian variety with thatof its dual [14, §16] and the fact that the pullback of the Abel–Jacobi map Z → Jac ( Z ) isequal to − λ − : (cid:92) Jac ( Z ) → Jac ( Z ) , where λ : Jac ( Z ) → (cid:92) Jac ( Z ) is the canonical polarizationon Jac ( Z ) [15, Lemma 6.9 and Remark 6.10(c)].From this we conclude the cover Y → Z is π H (cid:48) . Furthermore, we see from Diagram (3.4)and [10, Theorem C] that Jac ( Y ) ∼ Jac ( Z ) × Jac ( E (cid:48) ) , so Prym H (cid:48) ∼ Jac ( E (cid:48) ) . (cid:3) The D -action on the factors of Jac ( (cid:101) Z ) . Applying Proposition 3.11 to the fifteen index-2 subgroups of Jac ( Z )[ ] yields fifteen elliptic curves. Our notation for these curves unfor-tunately depends on the value of t ∈ K used in the defining equation (3.1) for Z ; in the nextsubsection we will see what happens when we choose a different value of t that defines acurve isomorphic to Z . Definition 3.12.
Given a nonzero U ∈ Jac ( Z )[ ] , let E U be the elliptic curve obtained byapplying Proposition 3.11 to the index-2 subgroup (cid:104) U (cid:105) ⊥ . If u and v are the unique elementsof R such that U is equal to the divisor class [ W u − W v ] = [ W v − W u ] , we also write E { u , v } for E U . Corollary 3.13.
There is a K-isogeny
Jac ( (cid:101) Z ) ∼ E × ∏ U Jac ( E U ) , where the product is over nonzero U ∈ Jac ( Z )[ ] . Proof.
Combine Proposition 3.6 with H = and Proposition 3.11. (cid:3) OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 11 Proposition 3.14.
The set of nonzero elements of
Jac ( Z )[ ] breaks up into six orbits under theaction of D , as listed in Table . For each U in an orbit, the table presents a value a ∈ K × as inProposition , and values of λ and d such that E U is isomorphic to y = dx ( x − )( x − λ ) . Orbit Pointlabel label a λ d { ζ , − ζ } t / ( t + ) { t , − t } c ( t + ) ζ t / ( t + ζ ) c ( t + ) { t , − t } c ( t + ) {− t , − t } ζ ct ( t + ) ( t − ) / ( t + ζ ) c ( t + ) { t , 1/ t } ζ ct ( t + ) { t , − t } ζ ct − c ( t + ) {− t , 1/ t } ζ ct { ζ , 1/ t } ζ t ( t + ζ ) − ζ t / ( t − ζ ) ζ { ζ , − t } ( t + ζ ) {− ζ , − t } ζ ct ( t − ζ ) − ζ t / ( t − ζ ) ζ c ( t + ) {− ζ , t } c ( t − ζ ) { ζ , t } ( t − ζ ) ζ t / ( t + ζ ) ζ { ζ , − t } ζ t ( t − ζ ) {− ζ , − t } c ( t + ζ ) ζ t / ( t + ζ ) ζ c ( t + ) {− ζ , 1/ t } ζ ct ( t + ζ ) T ABLE
1. The fifteen elliptic curves E U for nonzero U ∈ Jac ( Z )[ ] , labeled asin Definition 3.12, grouped in their orbits under the action of D . The valueof a is as in Proposition 3.11, and the values of λ and d are such that E U is alsoisomorphic to y = dx ( x − )( x − λ ) . Recall that each E U can be recovered asthe Prym variety of the double cover of Z associated, as in Proposition 3.11,to the subgroup H (cid:48) ⊆ Jac ( Z )[ ] that pairs trivially with U under the Weilpairing. Proof.
The generators σ and τ of the D subgroup of Aut ( Z ) act on the curve labels via σ ( { u , v } ) = (cid:8) − u , − v (cid:9) and τ ( { u , v } ) = (cid:8) u , 1/ v (cid:9) ,so the grouping into orbits is clear. The value of a is determined via Proposition 3.11, andthe associated λ and d are computed by applying a linear fraction transformation to put thecurve E (cid:48) from Proposition 3.11 into Legendre form. (cid:3) Remark 3.15.
Suppose an element α ∈ Aut ( Z ) takes U ∈ Jac ( Z )[ ] to V . If α does notfix the base point ( ζ , 0 ) by which we embedded Z into Jac ( Z ) , then α does not necessarilyprovide a K -rational isomorphism between E U and E V , because the base point determines the appropriate twist of the elliptic curve associated to a 2-torsion point. We see this, forexample, in Orbits 4A and 4B: each of these orbits has two different values of d . Remark 3.16.
The order-4 automorphism of Z induces an order-4 automorphism ζ of Jac ( Z ) ,such that multiplication by 2 factors as ( − ζ )( + ζ ) . A natural object of study is thedegree-4 cover of Z whose Jacobian contains orbits 1 and 2C; it arises as (cid:101) Z H when H : = Ker ( − ζ ) . See Sections 5 and 6 for more details.4. H EURISTICS FOR ISOGENOUS CURVES
Let q be a power of an odd prime p and let K ∼ = F q be a finite field of order q . In thissection, we consider genus-2 curves Z over K having the property that D ⊆ Aut ( Z ) . Westudy unordered pairs of non-isomorphic curves of this type whose Jacobians are isogenousto one another. The main result of this section is Theorem 4.6, which gives upper and lowerbounds for the number of these unordered pairs in terms of q .4.1. The moduli space of genus- curves with D -action. Recall from Notation 2.1 that Z is the set of K -isomorphism classes of objects ( Z , (cid:101) ) , where Z is a genus-2 curve over K andwhere (cid:101) : D (cid:44) → Aut ( Z ) is an embedding. Let Z denote the set of K -isomorphism classes ofgenus-2 curves over K such that D ⊆ Aut ( Z ) , and let ν : Z → Z be the forgetful morphismtaking the object ( Z , (cid:101) ) to the curve Z . At the the beginning of Section 2 we defined ξ to bethe involution of D that swaps the generators a and b . We can define an involution on Z as well, by sending ( Z , (cid:101) ) to ( Z , (cid:101)ξ ) . Notation 4.1.
Let X be the set of isomorphism classes of objects ( E , P ) , where E is an ellipticcurve over K and P is a K -rational point of order 2 on E . Two such objects ( E , P ) and ( E , P ) are isomorphic if there is a K -rational isomorphism E → E taking P to P .Let χ be the involution on X that sends a pair ( E , P ) to the pair ( E (cid:48) , P (cid:48) ) , where E (cid:48) = E / (cid:104) P (cid:105) and where P (cid:48) is the generator of the kernel of the dual isogeny E (cid:48) → E . Let X − ⊂ X be thesubset consisting of those objects ( E , P ) such that E has a K-rational endomorphism β with β = − β ( P ) =
0. Let X − ⊂ X be the subset consisting of those ( E , P ) suchthat E has a geometric automorphism α satisfying α = − α ( P ) = P . Finally, let X (cid:48) = X \ X − . The involution χ on X restricts to an involution on X (cid:48) .In Section 2.4, we associated to every genus-2 curve with D -action ( Z , (cid:101) ) an elliptic curve E and a 2-torsion point P on E . Thus there is a map µ : Z → X that sends the isomorphismclass of ( Z , (cid:101) ) to that of ( E , P ) . Proposition 4.2.
The map µ is injective and has image X (cid:48) . It takes the involution ( Z , (cid:101) ) (cid:55)→ ( Z , (cid:101)ξ ) of Z to the involution χ on X (cid:48) . The map ν : Z → Z that sends ( Z , (cid:101) ) to Z is -to- , unless ( Z , (cid:101) ) is fixed by ξ or, equivalently, unless µ ( Z , (cid:101) ) ∈ X − .Proof. Let ( E , P ) ∈ X . By Proposition 2.9, there exists ( Z , (cid:101) ) ∈ Z such that µ ( Z , (cid:101) ) = ( E , P ) if and only if there is no automorphism α (cid:54) = ± E that fixes P . Combining this with theobservation that an automorphism α (cid:54) = ± ( E , P ) is in the image of µ if and onlyif it lies in X (cid:48) .Next we show that a genus-2 curve with D -action ( Z , (cid:101) ) can be recovered from its image ( E , P ) under µ . To see this, we first write down a short Weierstrass model for E such that P is the point (
0, 0 ) . Such a model is of the form y = x ( x + dx + e ) , and the model is unique OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 13 up to scaling x and y . The coefficient e is nonzero because the model is nonsingular, and d is also nonzero, because otherwise the map ( x , y ) (cid:55)→ ( ζ x , − y ) would be an automorphismof order 4 that fixes P , and E has no such automorphisms because ( E , P ) lies in X (cid:48) . Thereis a unique way to scale x so that the model becomes y = cx ( x + f x − f ) for c , f ∈ K × with f (cid:54) = −
4; the value of f is unique, and the value of c is unique up to squares. Replacing x with x + y = cx ( x + sx + ) , where s = f + (cid:54) = ± P = ( −
1, 0 ) . We have shown that ( E , P ) determines unique values of s ∈ K and c ∈ K × / K × , and these values determine a unique genus-2 curve with D -action, viaEquations (2.1) and (2.2), so µ is injective.Lemmas 2.6 and 2.7 show that if µ ( Z , (cid:101) ) = ( E , P ) then µ ( Z , (cid:101)ξ ) = ( E (cid:48) , P (cid:48) ) , so µ takes theinvolution ( Z , (cid:101) ) (cid:55)→ ( Z , (cid:101)ξ ) to χ .The fact that ν is 2-to-1 except for the objects ( Z , (cid:101) ) that are isomorphic to ( Z , (cid:101)ξ ) followsfrom Lemma 2.5 and its proof. By the preceding statements, ( Z , (cid:101) ) and ( Z , (cid:101)ξ ) are isomor-phic if and only if ( E , P ) is fixed by χ , meaning that ( E , P ) and ( E (cid:48) , P (cid:48) ) are isomorphic.This is true if and only if E has an endomorphism β , whose kernel is generated by P , suchthat β = u for a unit u in End ( E ) . The only possibilities are that: (i) β = ± ± ζ where ζ ∈ Aut ( E ) with ζ = −
1, in which case ( E , P ) (cid:54)∈ X (cid:48) , or (ii) β satisfies β = − (cid:3) Remark 4.3.
As the preceding proof shows, the curves Z ∈ Z that have only one preimagein Z correspond to the exceptional curves in Lemma 2.5, that is, the curves with s = − − s = − E has j -invariant 8000, which is the unique root of the Hilbert class polynomial for Z [ √− ] . Theendomorphisms β ∈ End ( E ) with β = − P = ( −
1, 0 ) , and theseendomorphisms are K -rational if and only if − Remark 4.4.
We can view Z as the coarse moduli space for objects ( Z , (cid:101) ) as in Notation 2.1.Similarly, we can view X as the coarse moduli space Y ( ) ∼ = Y ( ) , namely the modularcurve X ( ) with its cusp removed. Then, under the embedding Z (cid:44) → Y ( ) the involutionassociated to Z → Z is the Fricke involution on Y ( ) . The language of moduli spaces isnot useful to us here since these are not fine moduli spaces and we need to keep track ofthe field of definition of the objects.4.2. Counting isogenous pairs of curves with D -action.Definition 4.5. Let P ( q ) denote the number of unordered pairs { Z , Z } , where Z , Z ∈ Z are not isomorphic to one another and Jac ( Z ) and Jac ( Z ) are isogenous.The following theorem determines the rate of growth of P ( q ) up to logarithmic factors. Theorem 4.6.
For all odd prime powers q > , there are constants d , d > such thatd q ≤ P ( q ) ≤ d q ( log q ) ( log log q ) . If the generalized Riemann hypothesis holds, there is a constant d > such thatP ( q ) ≤ d q ( log log q ) . Remark 4.7.
Direct calculation shows that P ( q ) = q =
3, 5, and 7, so the hypothesisthat q > Proposition 2.8 shows that if Z is a genus-2 curve over K with D -action, then Jac ( Z ) isisogenous to E , where E is an elliptic curve with a rational 2-torsion point. Since E has a2-torsion point, E ( K ) is even, and since q is odd, the trace of Frobenius for E must also beeven. This shows that the Weil polynomial of Jac ( Z ) is of the form ( x − tx + q ) , for aneven integer t with t ≤ q . Definition 4.8.
For each even integer t with t ≤ q , let M ( q , t ) denote the number of Z ∈ Z whose Weil polynomial is ( x − tx + q ) , and let N ( q , t ) denote the number of elliptic curvesover K with trace t . Lemma 4.9.
For all odd prime powers q and even integers t with t ≤ q we have M ( q , t ) ≤ N ( q , t ) . Proof.
For each curve Z ∈ Z with Weil polynomial ( x − tx + q ) , choose an embedding (cid:101) : D (cid:44) → Aut ( Z ) . Proposition 4.2 shows that ( Z , (cid:101) ) gives rise to a unique pair ( E , P ) withtrace ( E ) = t , so M ( q , t ) is at most the number of such pairs. Since an elliptic curve has atmost three rational 2-torsion points, M ( q , t ) ≤ N ( q , t ) . (cid:3) Lemma 4.10.
There is a constant d such that for all odd prime powers q and even integers t witht ≤ q, we have N ( q , t ) < d √ q ( log q )( log log q ) . If the generalized Riemann hypothesis holds, there is a constant d such for all odd prime powers qand even integers t with t ≤ q, we haveN ( q , t ) < d √ q ( log log q ) . Proof.
This follows from the formulas for N ( q , t ) found in [17, Theorem 4.6, pp. 194–196]),combined with the bounds on Kronecker class numbers found in [1, Lemma 4.4, p. 49]. (cid:3) Proof of Theorem . Clearly, P ( q ) = ∑ t ≤ qt even (cid:18) M ( q , t ) (cid:19) = ∑ t ≤ qt even M ( q , t ) /2 − ∑ t ≤ qt even M ( q , t ) /2.The number of even t with t ≤ q is at most 2 √ q +
1. By the Cauchy–Schwarz inequality, ∑ M ( q , t ) ≥ ( ∑ M ( q , t )) ( √ q + ) ,where each sum is over the set of even t with t ≤ q . By Lemma 2.5, the sum of the M ( q , t ) is either q − q −
3, so P ( q ) ≥ ( q − ) ( √ q + ) − q −
22 .From this we can show that P ( q ) ≥ q /23 for q ≥
17. By direct computation we find that P ( ) = P ( ) =
3, and P ( ) =
6, so we have P ( q ) ≥ q /23 for all q > OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 15 To prove the upper bounds on P ( q ) , we use Lemmas 4.9 and 4.10 to see that P ( q ) = ∑ t ≤ qt even (cid:18) M ( q , t ) (cid:19) < ∑ t ≤ qt even N ( q , t ) ≤ ( √ q + ) (cid:40) d q ( log q ) ( log log q ) in general; d q ( log log q ) if GRH holds.The upper bounds in the theorem follow. (cid:3) Gathering data.
Proposition 4.2 and the ideas in Section 4.2 allow us to quickly com-pute the exact value of P ( q ) when q is not too large. The subtleties in the computationinclude computing the objects ( E , P ) that lie in X − and in X − , and, for each even trace t ,determining the number of elliptic curves with trace t that have exactly one point of order2 and the number that have exactly three points of order 2. This latter question is answeredby noting that an elliptic curve with Frobenius endomorphism π has three rational pointsof order 2 if and only if ( π − ) /2 lies in its endomorphism ring, and by noting that thenumber of curves with trace t and with a given endomorphism ring can be computed froma class number; see [17, Theorem 4.5, p. 194].Further details of our method of computing P ( q ) can be found in the comments of theMagma code we used to do so, which can be found on the fourth author’s web page.For 15 ≤ n ≤
24, we computed the values of P ( q ) / q for the 1024 odd prime powers q closest to 2 n . For each of these sets of 1024 prime powers we also computed the standarddeviation and the minimum and maximum values of P ( q ) / q . These values are presentedin Table 2. From Theorem 4.6 and this data, it seems reasonable to model P ( q ) as growinglike a constant times q . n Mean S.d. Max Min15 0.42025 0.01958 0.45974 0.3847316 0.42188 0.01949 0.45828 0.3873217 0.42270 0.01953 0.45792 0.3884518 0.42394 0.01939 0.45914 0.3899619 0.42406 0.01955 0.45865 0.3907820 0.42464 0.01917 0.45862 0.3910521 0.42514 0.01942 0.45851 0.3920322 0.42577 0.01922 0.45830 0.3924823 0.42527 0.01937 0.45843 0.3926224 0.42557 0.01938 0.45853 0.39276T
ABLE
2. Data for isogenous curves. For each n , we give the mean, standarddeviation, and extremal values of P ( q ) / q , where q ranges over the 1024odd prime powers closest to 2 n . Remark 4.11.
The quantity P ( q ) was defined so that it counts the number of unorderedpairs { Z , Z } of non-isomorphic curves with D -action and with isogenous Jacobians, be-cause clearly Z and Z will have isogenous Jacobians if in fact they are the same curve. There’s another “easy” way that two curves can have isogenous Jacobians: If Z and Z are curves over a proper extension F q of F p that are Galois conjugates of one another, theirJacobians will be isogenous to one another via some power of the Frobenius isogeny. If q = p e , these Galois conjugate pairs account for Θ ( e q ) of all the isogenous pairs over F q ,which is an increasingly small fraction of the value of P ( q ) as q → ∞ . However, whenwe consider the more uncommon doubly isogenous pairs in later sections, we will want tospecifically exclude Galois conjugate pairs from our counts.5. I NITIAL HEURISTICS AND DATA FOR DOUBLY ISOGENOUS CURVES
By Theorem 4.6, the number of unordered pairs { Z , Z } of genus-2 curves over K = F q with D -action and with isogenous Jacobians is proportional to q , up to logarithmic fac-tors. The frequency naturally decreases if, in addition, we require that Z and Z be doublyisogenous. In this section, we present our initial heuristic about the expected number ofdoubly isogenous curves over F q and some data that we use to test the heuristic.5.1. An initial heuristic for doubly isogenous curves.
As we noted in Remark 4.11, twocurves Z and Z over a finite field K are trivially doubly isogenous if they are Galoisconjugates of one another. This observation influences the following definition. Definition 5.1.
Let δ ( q ) be the number of unordered pairs { Z , Z } of doubly isogenouscurves over F q , where Z and Z are genus-2 curves with D -action and all Weierstrasspoints rational, and where Z and Z are not Galois conjugates of one another.We formulate a heuristic to estimate δ ( q ) that we label as “na¨ıve” because it turns outnot to match the data we gathered. Later in the paper, we explain this discrepancy andimprove the heuristic. Na¨ıve Heuristic 5.2.
For a fixed odd prime power q, we model the double-isogeny class of Z as asix-tuple of independent random elliptic curves over F q .Justification. By Corollary 3.13, if all of the Weierstrass points of Z are rational then Jac ( (cid:101) Z ) decomposes into a sum of 17 elliptic curves, two of which are E . The remaining 15 ellipticcurves fall into six orbits under the action of D , as in Table 1. The elliptic curve in Orbit 2 C does not depend on s .Suppose Z and Z are genus-2 curves with D -action, lying over elliptic curves E and E as in Lemma 2.6. For Z and Z to be doubly isogenous over K , there must be six geo-metric isogenies of elliptic curves, one between E and E and an additional five for thenon-constant orbits. (These five isogenies may be between orbits with different labels; forexample, orbit 2A for one curve may be isogenous to orbit 2B for the other. This only affectsthe probability that that the five isogenies exist by a constant factor.) With positive proba-bility, a geometric isogeny Jac ( (cid:101) Z ) ∼ Jac ( (cid:101) Z ) comes from a K -rational isogeny, because allthe elliptic curves have a bounded number of twists. Thus it is reasonable to model thedouble-isogeny class of Z as six random elliptic curves. (cid:3) Let n be the number of isomorphism classes of genus-2 curves over K with D -action andrational Weierstrass points. It follows from the parametrization in terms of the variable t given in (3.1) that n (cid:16) q ; the exact count is irrelevant for our purposes. Choose n six-tuplesof random elliptic curves over F q , and denote them by ( E i ,1 , . . . E i ,6 ) for i ∈ {
1, . . . , n } . OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 17 Define the set S q : = { ( i , j ) : i (cid:54) = j and E i , a ∼ E j , a for a =
1, 2, . . . 6 } ⊆ {
1, . . . , n } . Lemma 5.3.
The expected value of S q ( which is the prediction of Na¨ıve Heuristic for the numberof pairs of doubly isogenous curves ) satisfies E ( S q ) (cid:16) q . Proof.
There are Θ ( q ) pairs of ( i , j ) , and in Section 4 we showed the probability of tworandom elliptic curves over F q being isogenous is Θ ( q − ) . (cid:3) Thus Na¨ıve Heuristic 5.2 predicts that the “expected value” of δ ( q ) is (cid:16) q . As we willsee, this does not match the data we gathered; the assumption that the 6 elliptic curves areindependent does not turn out to be completely accurate.5.2. Data that does not support the na¨ıve heuristic.
In order to calculate δ ( q ) for specificvalues of q , we first want to enumerate all isomorphism classes of genus-2 curves Z over K ∼ = F q with D -action and with all Weierstrass points rational. To do so, we vary t in (3.1).Letting s = − ( t + ) / t and taking into account the involution s ↔ s (cid:48) , we see that thefollowing values of t give isomorphic curves:(5.1) ± t , ± (cid:18) t (cid:19) , ± (cid:18) t − t + (cid:19) , ± (cid:18) t + t − (cid:19) .To enumerate isomorphism classes, we fix an ordering of the elements of K and only con-sider values of t for which s (cid:54) = ± Z is non-singular) and for which t is the smallest ofthe values in (5.1). We then include the curve Z from (3.1) and its standard quadratic twistin our enumeration. If n is a fixed quadratic non-residue of K , that means we look at y = ( x + )( x + sx + ) and y = n ( x + )( x + sx + ) .(Recall from Lemma 2.5 that if s = − − K , the curve Z is isomorphicto its standard quadratic twist. However, when s = − t = ± ± √
2, so when theWeierstrass points of Z are rational, the exceptional case in Lemma 2.5 does not occur.)In Table 3(A), we present some data that we collected by enumerating doubly isogenouspairs. For n ranging from 15 to 23, we considered the 1024 primes q ≡ n and computed the sum of δ ( q ) over these values of q . According to Na¨ıve Heuristic 5.2 andLemma 5.3, we would expect this sum to have rate of growth of the form c /2 n for someconstant c . In particular, we would expect the sum to approximately halve as we increase n by 1. This is not what we observe. We explain this discrepancy in the next section byfinding several families of coincidences that cause doubly isogenous pairs to occur moreoften than predicted.We can similarly develop heuristics for covers corresponding to subgroups of Jac ( Z )[ ] . Definition 5.4.
Let H be a subgroup of Jac ( Z )[ ] . We say that Z and Z are H-isogenous ifJac ( Z ) and Jac ( Z ) are isogenous and Jac ( (cid:101) Z H ) and Jac ( (cid:101) Z H ) are isogenous.We can similarly develop a heuristic for H -isogenous curves. Proposition 3.6 gives adecomposition of Jac ( (cid:101) Z H ) , and Table 1 lets us identify the elliptic curves appearing in thisdecomposition. In particular, if m is the number of different non-constant orbits of ellipticcurves corresponding to the 2-torsion points in H ⊥ , then we expect(5.2) { H -isogenous pairs / F q } (cid:16) q ( − m ) /2 . n Examples15 82016 58017 40718 28219 21820 13821 10022 5823 42A. Data for doubly isogenouscurves n Examples15 1169016016 2383799417 4844368818 9760827619 19634321220 39458413021 79383983622 158828277623 3172154548B. Data for [ − ρ ∗ ] -isogenouscurves (see Example 5.5)T ABLE
3. The total number of unordered pairs of doubly isogenous curvesand [ − ρ ∗ ] -isogenous curves over F q for the 1024 primes q ≡ n , restricting to curves with all Weierstrass points rational. Example 5.5.
For example, take H : = Ker ( − ρ ∗ ) , where ρ is the automorphism of order 4defined in Section 2.1. The cover (cid:101) Z H has degree four, and since ( − ρ ∗ ) = − ρ ∗ , it isisomorphic to the pullback of the endomorphism 1 − ρ ∗ on Jac ( Z ) via the embedding Z → Jac ( Z ) . When Z and Z are H -isogenous for this H , we say that Z and Z are [ − ρ ∗ ] -isogenous . The Jacobian Jac ( (cid:101) Z H ) contains orbits 1 and 2C (in addition to E ). Since Orbit 2Cis constant, m = ( (cid:101) Z H ) and Jac ( (cid:101) Z H ) to beisogenous. Thus we expect(5.3) { [ − ρ ∗ ] -isogenous pairs / F q } (cid:16) q .We expect the total number of pairs of [ − ρ ∗ ] -isogenous curves for the 1024 primes q ≡ n to have rate of growth c · n for some constant c , and this is supportedby the data in Table 3(B), as the number of pairs roughly doubles as we increase n by 1.6. F AMILIES WITH UNEXPECTED COINCIDENCES
Na¨ıve Heuristic 5.2 predicts that the “expected value” of δ ( q ) is on the order of 1/ q ,where δ ( q ) is the number of non-conjugate pairs { Z , Z } of doubly isogenous curves overa finite field F q with Z and Z genus-2 curves with D -action and all Weierstrass pointsrational. As seen in Table 3(A), the data we collected does not seem to reflect this rateof growth. In this section, we find a number of families of coincidences that explain thisdiscrepancy and we formulate a more sophisticated heuristic for the number of such pairs,which will be supported by the data in Section 7.6.1. j -invariants for orbits. We begin by computing the j -invariants of the elliptic curvesappearing in Table 1; the middle column of Table 4 gives these j -invariants in terms of theparameter t . For our computations, it will be convenient to note that we can also expressthese j -invariants in terms of the quantity u : = ( )( t − t ) , OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 19 as is shown in the third column of Table 4. This new paramatrization simplifies our compu-tations in Section 6.2, because I = ( u + u ) is a quartic function of u instead of a degree-8function of t . We omit the proofs of the following two facts. j -invariant, in terms of. . .Orbit . . . the variable t . . . the variable u ( t + t + ) ( t − t ) ( u + u + ) u ( u + ) − ( t − t + ) t ( t + ) − ( u − ) ( u + )
2B 2 ( t − t + ) ( t − ) ( t + ) ( u − ) ( u + u )
2C 1728 17284A − ( t − ζ t − t + ζ t + ) ( t − t ) ( t − ζ ) − ( u − ζ u − ) ( u − ζ u ) − ( t + ζ t − t − ζ t + ) ( t − t ) ( t + ζ ) − ( u + ζ u − ) ( u + ζ u ) T ABLE j -invariants for the elliptic curves in Table 1 in terms of t and u = ( t − t ) . Lemma 6.1.
Replacing u with − u, u, or − u does not change the value of I. (cid:3) Lemma 6.2.
The elliptic curve E defined by Equation (2.4) has j-invariant ( t − t + ) t ( t − ) = ( u + ) u , and is isomorphic to y = dx ( x − )( x − λ ) , where λ = t and d = c ( t + ) . (cid:3) Finding generic geometric isogenies.
In this subsection, we work over an algebraicallyclosed field K of characteristic not 2. Our goal is to find families of ordered pairs ( Z , Z ) of genus-2 curves over K with D -action that have a higher-than-expected chance of beingdoubly isogenous. For example, we might search for families where the elliptic curves insome of the orbits listed in Table 1 for Z are automatically isogenous to those in some of theorbits for Z . We carry out this search by using classical modular polynomials Φ n ∈ Z [ x , y ] .(Recall that the polynomial Φ n has the property that there is a geometric cyclic n -isogenybetween two elliptic curves over an arbitrary field K if and only if the j -invariants j and j of the two curves satisfy Φ n ( j , j ) =
0; see [7], [9].)Let Z and Z be two genus-2 curves over K with D -action, and let I and I be theirrespective invariants; see Section 2.3. We can write each Z i in the form (3.1); that is, there are c i , t i ∈ K such that Z i is given by Z i : y = c i ( x − ζ )( x + ζ )( x − t i )( x + t i )( x − t i )( x + t i ) (6.1) = c i ( x + )( x + s i x + ) ,where s i = − ( t i + ) / t i . Since K is algebraically closed, we may take c = c = Z and Z are determined by the valuesof u i = ( )( t i − t i ) , which satisfy I i = ( u i + u i ) . It follows that u and u arereasonable parameters to use for the families we construct.To find families of ordered pairs ( Z , Z ) with a cyclic isogeny of degree n between speci-fied orbits, we work over the algebraic closure K of the 2-variable function field Q ( ζ )( u , u ) ,and consider the curves Z and Z with parameters t , t ∈ K such that u i = ( )( t i − t i ) . Given an orbit for Z and an orbit for Z , we can plug the appropriate formulas forthe j -invariants of the orbits into Φ n in order to obtain an expression in u and u which iszero if and only if there is a cyclic n -isogeny between the curves in the given orbits. Example 6.3.
Let us calculate conditions under which the Orbit 1 elliptic curve for Z isgeometrically isomorphic to the Orbit 1 elliptic curve for Z . This calculation is simplerthan most, because the j -invariants of the Orbit 1 elliptic curves can in fact be expresseddirectly in terms of the invariants of Z and Z ; namely, the Orbit 1 j -invariant for eachcurve is 256 · ( I i − ) / I i . We see that the two j -invariants are equal if and only if ( I − ) I − ( I − ) I = I I + I I − I I + I − I .We compute that the condition I I + I I − I I + = ( u + u + )( u u + u + )( u u + u + )( u u + u + u ) = Example 6.4.
Let us consider the relation between u and u that is satisfied exactly whenthe elliptic curves in Orbit 2A of Z are 2-isogenous to the elliptic curves in Orbit 2B of Z .We will not write down the full polynomial relation in u and u because it involves 94terms. We do observe that it factors over Q ( ζ ) into nine irreducible polynomials; one ofthese irreducible factors is u u + u +
1, which also appears in Example 6.3! Thus, if thesingle relation u u + u + = Z is isomorphic to the Orbit 1curve for Z , and the Orbit 2A curve for Z is 2-isogenous to the Orbit 2B curve for Z . Thisis an unexpected coincidence!In the following subsection we report on what we found by systematically searching forsuch coincidences. The ideal but computationally intensive calculation would be to workover K and consider every pair ( F , F ) of elliptic curves, where each F i is either the quotientof Z i given by (2.4) or one of the curves in an orbit for Z i . For each positive integer n insome a predetermined set of values (see Remark 6.5 for our choice), we would compute anexpression in u and u that equals zero if and only if there is a cyclic n -isogeny between thecurves in the two orbits. Then, for every pair of such expressions, we would compute theirgreatest common divisor. Whenever this greatest common divisor was not 1, we wouldfind a family of pairs ( Z , Z ) of curves associated to a pair of parameters ( u , u ) wherethere are multiple isogenies between the elliptic factors of Jac ( (cid:101) Z ) and those of Jac ( (cid:101) Z ) . OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 21 It is computationally difficult to implement the above strategy because when we sub-stitute the rational functions for the j -invariants into all but the smallest modular poly-nomials, the expressions become quite large. To reduce the size of the coefficients in theexpressions, and to reduce the number of monomials involved, we instead work moduloa prime p ≡ u to a value in F p ( ζ ) . This yields rational functionsin F p ( ζ )( u ) which fit much more easily in memory. We are therefore looking for cyclic n -isogenies between fibers (over u ) of the family Z and a fixed fiber of Z , after reducingmodulo p .There are two risks associated to making these reductions. The first is that we may findspurious relations, nonzero greatest common divisors that occur only modulo p . As ithappens, none of the relations we found involved modular polynomials of high degree, sowe were subsequently able to verify the relations we found over the full ring Q ( ζ )( u , u ) .The second risk is that we might miss a family. This could happen, for example, if thereis a relation that involves a polynomial in Q ( ζ )( u , u ) that reduces modulo p to a constant,or to a polynomial like u u whose solutions require one of the u i to be equal to one of theforbidden values 0 or ζ . It could also happen if we specialize to a value of u that makes thepolynomial constant. Without knowing more about the geometry of the possible familiesin characteristic zero, we are not sure how to rule out these possibilities. We did, however,run our computation several times, with different choices for the prime p and differentchoices for the values of u , and the results did not vary. Thus, we believe we found all ofthe families of coincidences involving isogenies of the degrees we considered. As we willsee in Section 7, we have found enough families to formulate an improved heuristic that issupported by our data. Remark 6.5.
We are left to specify the degrees n of the cyclic isogenies we will consider. Wechoose to look for cyclic n -isogenies for all values of n for which the modular curve X ( n ) has genus 0 (namely, n =
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, and 25) or genus 1 (namely, n =
11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, and 49). We chose these values so that we wouldfind all families of coincidences given by a relation between u and u that defines a curveof geometric genus at most 1. To see that these values of n will lead to all such families,note that every family we find gives us a varying pair of elliptic curves connected by acyclic n -isogeny, and so comes provided with a nonconstant map to X ( n ) . Since no familyof genus 0 or 1 can map to a modular curve with genus larger than 1, for our goal of findingall families defined by genus-0 and genus-1 relations between u and u , it will suffice forus to consider the values of n specified above.In the end, we found sixteen families of coincidences in terms of u and u . However, ifwe count two families as being equivalent if they produce the same pairs ( Z , Z ) — that is,if one family can be obtained from the other by applying transformations from Remark 6.1to u and u — then we have only four equivalence classes of families. We describe thesefour classes of families in the following section, where we keep close track of fields ofdefinition of isogenies. Remark 6.6.
A family given by a relation between u and u can be made more explicitby replacing each u i with ( )( t i − t i ) , and then looking at an irreducible factor of theresulting expression. For example, the relation between t and t obtained from the relation u + u + = t − ζ t . See Section 6.3.1. Description of the families.
The first family.
Let K be a field of characteristic not 2 that contains a primitive 4throot of unity ζ . For i =
1, 2, let c i and t i be elements of K × with t i (cid:54) =
1, let Z i be givenby (6.1), and let E i be the quotient curve(6.2) E i : y = c i ( x + )( x − t i )( x − t i ) . Lemma 6.7.
Suppose t = ζ t , and suppose ( t + )( t + ) and c c are squares in K. Then overK the following statements hold :(1) the elliptic curve in Orbit 1 for Z is isomorphic to the elliptic curve in Orbit 1 for Z ;(2) the elliptic curves in Orbit 2B for Z and the elliptic curves in Orbit 2B for Z are related bya degree- isogeny ;(3) the elliptic curves in Orbit 2C for Z are isomorphic to those in Orbit 2C for Z ;(4) the elliptic curve E ( resp. E ) and the elliptic curves in Orbit 2A for Z ( resp. Z ) are relatedby a degree- isogeny.Furthermore, if K is the algebraic closure of the function field Q ( t ) and t = t, then there are noother isogenies among the orbits associated to Z and Z . The Lemma could also be rephrased in terms of Prym varieties using Proposition 3.11.
Proof of Lemma 6.7.
To avoid a proliferation of subscripts and to aid in visual comprehen-sion of various formulas, in this proof we will write t and c for t and c , and we will write T and C for t and c . More generally, we will use lower case letters for variables associatedwith Z , and upper case letters for variables associated with Z .Since the isomorphism classes of the curves Z i and the elliptic curves E i only depend onthe values of c and C up to squares, and since cC is a square by hypothesis, we may assumethat C = c .By Table 1, the Orbit 1 curves for Z and Z can be written as y = x ( x − )( x − λ ) and Y = X ( X − )( X − Λ ) ,respectively, where λ = t / ( t + ) and Λ = T / ( T + ) = − t / ( t − ) . Then anisomorphism between the curves is given by X = ( x − λ ) / ( − λ ) and Y = y · ( t + ) / ( t − ) .This proves the first statement.The Orbit 2B curves for Z and Z can be written as y = c ( t + ) · x ( x − )( x − λ ) and Y = c ( T + ) · X ( X − )( X − Λ ) ,respectively, where λ = ( t − )( t + ζ ) and Λ = ( T − )( T + ζ ) = ( t + )( t + ) .Then a degree-2 isogeny from the first curve to the second is given by X = ζ ( t + ) · (( t + ζ ) x − ( + ζ )( t + )) x − Y = y · ( t + ζ )( ζ − ) ( t + ) · ( t + ζ ) x − ( t + ζ ) x + ( t − )( x − ) .This proves the second statement. OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 23 The Orbit 2C curves for Z and Z are both twists of y = x − x , by c ( t + ) and by c ( T + ) , respectively. By hypothesis, ( t + )( T + ) is a square in K , so these two twistsare isomorphic to one another. This proves the third statement.By Remark 6.2 and Table 1, we can write E and the Orbit 2A curve for Z as y = c ( t + ) · x ( x − )( x − λ ) and Y = c ( T + ) · X ( X − )( X − Λ ) ,respectively, where λ = t and Λ = ζ T / ( T + ζ ) = t / ( t + ) . Then a degree-2 isogenyfrom the first curve to the second is given by X = ( t + ) · ( x + t ) x and Y = y · ( t + ) · x − t x .This proves one case of the fourth statement. The proof of the other case is similar.Finally, we check that no other pairs of elliptic curves associated to Z and Z are isoge-nous to one another when K is the algebraic closure of Q ( t ) and t = t . As we noted earlier,two elliptic curves over a field are connected by a cyclic n -isogeny over the algebraic closureif and only if their j -invariants satisfy the n -th classical modular polynomial Φ n ∈ Z [ x , y ] .If this relation holds in K , then it will also hold when we reduce modulo p and specialize t and c to specific values for which the resulting curves are nonsingular. We take p = ζ = ∈ F p , t = ∈ F p , and c = ∈ F p . Computing traces shows the elliptic curvesin question are all ordinary. It follows that any geometric isogeny between them is definedalready over F p ; see [6, §5, p. 251]. Thus we simply compute the traces of these ellipticcurves over F p and observe that there are no further matches. (cid:3) We noted in Section 5.2 that replacing t with any of eight linear fractional expressionsin t will result in a curve isomorphic to Z , but possibly with the labels on Orbits 2A and2B swapped, and similarly for Orbits 4A and 4B. If we take the relation t = ζ t and applyone of these transformations to t and a possibly different one to t , we will get anotherfamily of curves satisfying the conclusions of Lemma 6.7, possibly with the roles of variousorbits swapped. There are 64 ways of applying these eight linear fractional transformationsseparately to t and t , but some of these will produce equivalent relations; for example,replacing t with − t and t with − t fixes the relation t = ζ t . In fact, we obtain only 16different families in this way. When we multiply the 16 corresponding relations together,we get a relation that can be expressed in terms of I and I , namely:(6.3) I I + I I − I I + = Definition 6.8.
We say that two curves Z and Z with D -action are in the first family if theirinvariants satisfy (6.3). Remark 6.9.
Equation (6.3) defines a genus-0 curve, which can be parametrized as I = − ( + z ) ( − z ) , I = − ( − z ) ( + z ) .Under this parametrization, the involution swapping I and I corresponds to z ↔ − z .Proposition 3.11 shows that each of the elliptic curves that appears in Table 1 as anisogeny factor of (cid:101) Z i can also be viewed as a Prym variety Prym H for a double cover of Z i specified by an index-2 subgroup H of Jac ( Z i )[ ] , and each such subgroup H is determinedas the set of elements of Jac ( Z i )[ ] that pair trivially with a nonzero element U ∈ Jac ( Z i )[ ] .Lemma 6.7 could therefore be restated in terms of these Pryms. The labeling of these Pryms via the elements U has the same problem as the labeling of the orbits associated to the Z i :The labels depend on which of the eight possible values of t i we used to write down anequation for Z i . However, using Prym varieties we can state a variant of Lemma 6.7 whosehypotheses and conclusions depends only on the isomorphism classes of the curves Z i andnot on the choices we made to write them down. Notation 6.10.
Let Z be a genus-2 curve with D -action over a field K of characteristic not 2.As we see from Table 1, there is a unique nonzero point U of Jac ( Z )[ ]( K ) that is fixed bythe action of D . Let H be the order-2 subgroup of Jac ( Z )[ ] generated by U . In the notationof Definition 3.2, let (cid:98) Z = (cid:101) Z H , so that there is a degree-8 cover (cid:98) Z → Z and (cid:98) Z has genus 9. Proposition 6.11.
Let Z and Z be curves with D -action over a field K of characteristic not ,and suppose Z and Z lie in the first family. If Jac ( Z ) and Jac ( Z ) are geometrically isogenous toone another, then Jac ( (cid:98) Z ) and Jac ( (cid:98) Z ) are geometrically isogenous to one another.Proof. We may assume that K is algebraically closed. Let ζ be a primitive fourth root ofunity in K . Since Z and Z are in the first family and since K is algebraically closed, we canchoose values of t and t with t = ζ t such that each Z i has a model as in (3.1) with t = t i and with c =
1. The hypotheses of Lemma 6.7 are then satisfied.Since Jac ( Z ) and Jac ( Z ) are isogenous to one another and since Jac ( Z i ) ∼ E i for each i ,the elliptic curves E and E are isogenous to one another. Lemma 6.7 then shows that theOrbit 1 curves for Z and Z are isogenous to one another, as are the Orbit 2A curves, theOrbit 2B curves, and the Orbit 2C curves.Proposition 3.6 shows that Jac ( (cid:98) Z i ) decomposes up to isogeny as the product of E i withthe product of the Prym H (cid:48) , where H (cid:48) ranges over the index-2 subgroups of Jac ( Z i )[ ] thatcontain the subgroup H = (cid:104) U (cid:105) , where U = [ W ζ − W − ζ ] . Taking duals with respect to theWeil pairing, we see that the H (cid:48) are the index-2 subgroups such that ( H (cid:48) ) ⊥ ⊆ H ⊥ . Thismeans that the ( H (cid:48) ) ⊥ are precisely the subgroups (cid:104) U (cid:48) (cid:105) , where U (cid:48) is a nontrivial 2-torsionpoint that pairs trivially with U . We see from Lemma 3.10 and Table 1 that these U (cid:48) are thelabels of Orbits 1, 2A, 2B, and 2C, so each Jac ( (cid:98) Z i ) is isogenous to E i times the product ofthe elliptic curves in Orbits 1, 2A, 2B, and 2C. Therefore, the Jac ( (cid:98) Z i ) are isogenous to oneanother. (cid:3) Proposition 6.12.
Let Z and Z be curves with D -action given as in (6.1) by values of t i and c i such that t = ζ t and such that c c and ( t + )( t + ) are both squares. For i =
1, 2 , let E i be the elliptic curve given by (6.2) . Suppose that E and E are isogenous to one another, and thateither of the following two conditions holds :(1) The curve y = ζ x ( x − )( x + ζ t / ( t − ζ ) ) is isogenous to eithery = ζ x ( x − )( x + ζ t / ( t − ζ ) ) ory = ζ c ( t + ) · x ( x − )( x + ζ t / ( t − ζ ) ) , and the curve y = ζ x ( x − )( x − ζ t / ( t + ζ ) ) is isogenous to eithery = ζ x ( x − )( x − ζ t / ( t + ζ ) ) ory = ζ c ( t + ) · x ( x − )( x − ζ t / ( t + ζ ) ) . OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 25 (2) The curve y = ζ x ( x − )( x + ζ t / ( t − ζ ) ) is isogenous to eithery = ζ x ( x − )( x − ζ t / ( t + ζ ) ) ory = ζ c ( t + ) · x ( x − )( x − ζ t / ( t + ζ ) ) , and the curve y = ζ x ( x − )( x − ζ t / ( t + ζ ) ) is isogenous to eithery = ζ x ( x − )( x + ζ t / ( t − ζ ) ) ory = ζ c ( t + ) · x ( x − )( x + ζ t / ( t − ζ ) ) . Then Z and Z are doubly isogenous.Proof. Since Jac ( Z i ) ∼ E i , the assumption that E ∼ E implies that Jac ( Z ) ∼ Jac ( Z ) . ByLemma 6.7, the elliptic curves in Orbit 1 for Z are isogenous to those in Orbit 1 for Z , andsimilarly for Orbits 2A and 2B. The Orbit 2C curves for Z and Z are isomorphic, becauseeach curve is the twist of y = x − x determined by c i ( t i + ) , and the product of thesetwo factors is a square. Therefore, in order for Z and Z to be doubly isogenous, it sufficesthat the elliptic curves in Orbits 4A and 4B for Z are, in some order, isogenous to the Orbit4A and 4B curves for Z . This is equivalent to conditions (1) and (2). (cid:3) The second family.
In Table 1, the four elliptic curves in Orbit 4A are not necessarilyisomorphic to one another over the base field, because there are two possible values for thefactor d that determines the twist of the curve. We will refer to the first two curves (with d = ζ ) as the “first pair” of Orbit 4A, and the last two curves (with d = ζ c ( t + ) ) as the“second pair.” Likewise, we refer to the d = ζ curves in Orbit 4B as the first pair of thatorbit, and the d = ζ c ( t + ) curves as the second pair of that orbit.As in the preceding subsection, for i = i = Z i and E i be the curves givenby (6.1) and (6.2). We will consider the following relation between t and t :(6.4) ( t − ζ ) ( t − ζ ) = − t t . Lemma 6.13.
Suppose (6.4) holds. Then :(1) the elliptic curve in Orbit 1 for Z and the elliptic curve in Orbit 1 for Z are related by adegree- isogeny over K ;(2) if c c ( t + )( t + ) ∈ K × , then the elliptic curves in Orbit 2C for Z are isomorphicover K to the elliptic curves in Orbit 2C for Z ;(3) if c t ( t + ) ∈ K × , then the elliptic curves in Orbit 2A for Z are isomorphic over K tothe first pair of elliptic curve in Orbit 4A of Z , and the elliptic curves in Orbit 2B for Z are isomorphic over K to the first pair of elliptic curve in Orbit 4B of Z ;(4) if c c t ( t + )( t + ) ∈ K × , then the elliptic curves in Orbit 2A for Z are isomorphicover K to the second pair of elliptic curve in Orbit 4A of Z , and the elliptic curves in Orbit2B for Z are isomorphic over K to the second pair of elliptic curve in Orbit 4B of Z ;(5) the statements obtained from (3) and (4) by interchanging the roles of Z and Z also hold.Furthermore, if K is the algebraic closure of the function field Q ( t ) and t = t, then there are noother isogenies among the orbits associated to Z and Z .Proof. We leave the proof to the reader, because it is essentially the same as the proof ofLemma 6.7 and is mostly straightforward. The only non-obvious details involved in theproof of the numbered statements are that if t , t ∈ K satisfy (6.4), then ζ t t and t ( t − ) and t ( t − ) are all squares in K . The first of these is a square because of (6.4) and the factthat − ζ = ( − ζ ) ; the second is a square because (6.4) can be rewritten as t ( t − ) = ( + ζ ) t ( t + ζ ) / ( t − ζ ) ;and the third is a square by symmetry.The final statement can be proven by taking p = ζ = ∈ F p , t = ∈ F p , t = ∈ F p , and c = c = ∈ F p , and comparing traces over F p as in the end of theproof of Lemma 6.7 (cid:3) Proposition 6.14.
Let Z and Z be curves with D -action given as in (6.1) by values of t i thatsatisfy (6.4) and with c i = ζ ( t i + ) , and such that t and ζ t are squares in K × . For i =
1, 2 , letE i be the elliptic curve given by (6.2) . Suppose that :(1) E and E are isogenous to one another ;(2) the first pair of curves in Orbit 4A of Z are isogenous to the second pair of curves in Orbit4A for Z ; and (3) the first pair of curves in Orbit 4B of Z are isogenous to the second pair of curves in Orbit4A for Z .Then Z and Z are doubly isogenous. Remark 6.15.
This proposition follows from making choices that allow us to apply certainstatements from Lemma 6.13; in particular, we make choices that imply that c t ( t + ) and c c t ( t + )( t + ) are squares. Other choices would lead to variations of Proposi-tion 6.14. Proof of Proposition . From Lemma 6.13(1), the curves in Orbit 1 for Z and Z are isoge-nous to one another. From Lemma 6.13(3), the curves in Orbits 2A and 2B for Z are isoge-nous to the first pairs of Orbits 4A and 4B for Z , respectively, and from Lemma 6.13(5)applied to (4), the curves in Orbits 2A and 2B for Z are isogenous to the second pairs ofOrbits 4A and 4B for Z , respectively.Since ζ c ( t + ) and ζ c ( t + ) are both squares, c ( t + ) and c ( t + ) are in the samesquare class in K × . Since the first pair and second pairs of Orbits 4A for Z i are twists of oneanother by c i ( t i + ) , the hypothesis (2) of the proposition implies that the second pair ofcurves in Orbit 4A of Z are isogenous to the first pair of curves in Orbit 4A of Z . Thus,by the preceding paragraph, the curves in Orbit 2A for Z are isogenous to the curves inOrbit 2A for Z . Similarly, the first pair and second pairs of Orbits 4B for Z i are twists ofone another by c i ( t i + ) and the same argument shows that the curves in Orbit 2B for Z are isogenous to the curves in Orbit 2B for Z .Again as c ( t + ) and c ( t + ) are in the same square class in K × , Lemma 6.13(2)implies that the curves in Orbit 2C for Z are isomorphic to the curves in Orbit 2C for Z .As E ∼ E , we conclude that Z and Z are doubly isogenous. (cid:3) As in the preceding subsection, we can apply any of eight linear fractional transforma-tions to t and to t in the relation given by (6.4) to get another family that satisfies a lemmasimilar to Lemma 6.13. Only four of these families are distinct. Multiplying the polynomi-als defining these four families together, we find a relation that can be expressed in termsof the invariants I and I of Z and Z :(6.5) I I = OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 27 Definition 6.16.
We say that two curves Z and Z with D -action are in the second family iftheir invariants satisfy (6.5). Remark 6.17.
Equation (6.5) defines a genus-0 curve, which can be parametrized as I = ( − z ) + z , I = ( + z ) − z .Under this parametrization, the involution swapping I and I corresponds to z ↔ − z . Remark 6.18.
Proposition 6.11 gives an interpretation of the first family that can be statedin terms of the isomorphism classes of the curves, without reference to the choices of t and t that we make to write down the curves; this is possible because the orbits involved inLemma 6.7 are exactly the orbits contained in the Prym variety of a cover that can be definedindependently of the choices of t i . There is no straightforward analog of Proposition 6.11for the second family.6.3.3. The third and fourth families.
We found two further families of pairs of curves wherethere are more isogenies between the associated elliptic curves than expected. They pro-duce fewer doubly isogenous curves than the preceding two families, so here we just sum-marize the results. We also simplify the exposition by assuming in this section that the basefield K is algebraically closed, so that we do not have to worry about twists.Let Z and Z be genus-2 curves with D -action and with invariants I and I . We saythat Z and Z are in the third family if we have(6.6) I I − · I I + I + I = Z and Z are in the third family, then • the curves in Orbit 1 for Z and Z are 4-isogenous to one another; • the curves in either Orbit 4A or Orbit 4B for Z are 2-isogenous to the curves in eitherOrbit 4A or Orbit 4B for Z .Note that (6.6) defines a curve of genus 0, which can be parametrized by I = − z ( z + ) / ( z − ) , I = − z ( z − ) / ( z + ) .Under this parametrization, the involution swapping I and I corresponds to z ↔ − z .We say that Z and Z are in the fourth family if their invariants satisfy(6.7) ( I + I I − · I I + I )( I + I I − · I I + I ) = I and I satisfy the first factor in this expression. Then • the curves in Orbit 1 for Z and Z are 2-isogenous to one another; • one of the following holds: – the curve E is isomorphic to the Orbit 4A curves for Z and the Orbit 2A curvesfor Z are 2-isogenous to the Orbit 4B curves for Z ; – the curve E is isomorphic to the Orbit 4B curves for Z and the Orbit 2A curvesfor Z are 2-isogenous to the Orbit 4A curves for Z ; – the curve E is 2-isogenous to the Orbit 4A curves for Z and the Orbit 2B curvesfor Z are 2-isogenous to the Orbit 4B curves for Z ; – the curve E is 2-isogenous to the Orbit 4B curves for Z and the Orbit 2B curvesfor Z are 2-isogenous to the Orbit 4A curves for Z .If I and I satisfy the second factor in (6.7), then the roles of Z and Z in the above list arereversed. Each factor in (6.7) defines a curve of genus 0. Intersections of families.
Under mild restrictions on the field K , we will producegenus-2 curves Z and Z with D -action that are very close to being doubly isogenous.The invariants I and I of these curves are the two roots of x − ( ) x +
16. Since I I =
16, this pair of curves lies in the second family; since I I + I I − I I + = ( I + I ) − = I I − · I I + I + I = − · · + ( ) = Proposition 6.19.
Let K be a field in which − , , and − are nonzero squares, and let ζ ∈ Ksatisfy ζ = − . In K, lett = ( + ζ )( + ζ √− ) and t = ( − + ζ )( + ζ √− ) For i =
1, 2 , let c i = , and let Z i and E i be defined by (6.1) and (6.2) . Then the invariants of Z and Z are the two roots of x − ( ) x + , and if E is isogenous to E , the curves Z and Z are doubly isogenous.Proof. An easy computation verifies the statement about the invariants of Z and Z .We check that t = ζ t and that ( t + )( t + ) is equal to the square of √ ( + ζ √− ) .Since c c = c t ( t + ) = √ − ( + ζ − √− + ζ √− ) and c t ( t + ) = √ − ( + ζ − √− + ζ √− ) .As we already noted, ( t + )( t + ) is a square, so the hypotheses of statements (1), (2),and (3) of Lemma 6.13 hold, as does the hypothesis of the variation of the lemma’s state-ment (3) obtained by interchanging the roles of Z and Z .Finally, we note that the first pair and the second pair of Orbit 4A for Z are twists of oneanother by c ( t + ) , while the first pair and the second pair of Orbit 4A for Z are twistsof one another by c ( t + ) ; these two twisting factors lie in the same square class in K × .The analogous statement holds for the first and second pairs of Orbit 4B for Z and for Z .Combining the conclusions of Lemma 6.7 and Lemma 6.13 with this last observation, itis straightforward to verify that Z and Z are doubly isogenous. (cid:3) Remark 6.20.
It is easy to check that the only pair ( I , I ) of nonzero elements of K thatsatisfies (6.3) and (6.5) is the pair from Proposition 6.19. This pair is also the only pair tosatisfy both (6.5) and (6.6). There are other pairs that satisfy the defining equations of morethan one of the four families, but in characteristic 0 the curves with those invariants do nothave as many isogeny factors in common as the curves in Proposition 6.19. Remark 6.21.
As we noted in Remark 4.11, if two curves over a finite field are Galois conju-gates of one another, they are necessarily doubly isogenous. We check that the values of s i (see Equation (3.2)) for the two curves in Proposition 6.19 are s = √− s = − √− K is finite and − OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 29 Example 6.22. In K = F , take ζ =
15 and √− =
28, and apply Proposition 6.19. Wefind that t =
107 and t =
23, that s =
55 and s =
58, and that the elliptic curves E and E both have trace 6, so that E ∼ E . This gives an example where the curves inProposition 6.19 are not Galois conjugates of one another, but are doubly isogenous. Remark 6.23.
Let E be the elliptic curve y = ( x + )( x + √− x + ) over the field K = Q ( √− ) and let E be its conjugate over Q . If p is a prime of K such that the reductionsof E and E modulo p are isogenous, then over an extension of the residue field of p , thetwo curves y = ( x + )( x ± √− x + ) will be doubly isogenous. By [3, Theorem 1.1],there are infinitely many such primes. However, the two curves we obtain from such aprime will be conjugate to one another — and hence, will give an uninteresting example —exactly when p is a prime of degree 2.This leads to the following question: Are there infinitely many degree-1 primes p of Q ( √− ) such that the reductions of E and E modulo p are isogenous? We suspect thatthe answer is yes, because if p lies over p then heuristically there is roughly one chance outof √ p that they will be isogenous by chance, and the sum of 1/ √ p diverges. Unfortunately,we do not know how to prove that there are infinitely many such primes.7. H EURISTICS FOR THE FAMILIES OF COINCIDENCES
In the preceding section, we identified four families of pairs ( Z , Z ) of genus-2 curveswith D -action where there are unexpected isogenies among a number of the elliptic curvesthat appear in the decomposition (up to isogeny) of the Jacobians Jac ( (cid:101) Z ) and Jac ( (cid:101) Z ) . Inthis section, we formulate heuristics for the expected number of doubly isogenous pairsover a finite field that occur in these families.7.1. Counting doubly isogenous pairs in families.
First, we introduce notation to keeptrack of the number of doubly isogenous pairs in each family; see also Definition 5.1.
Definition 7.1.
We consider unordered pairs { Z , Z } of doubly isogenous curves over F q ,where Z and Z are genus-2 curves with D -action and all Weierstrass points rational, andwhere Z and Z are not Galois conjugates of one another. For n =
1, 2, 3, 4, let δ n ( q ) bethe number of these pairs { Z , Z } that lie in the n th family. Let δ ( q ) be the number ofthese pairs { Z , Z } that do not lie in any of these four families. Let δ ( q ) be the numberof these pairs { Z , Z } that are simultaneously in families 1, 2, and 3 — that is, the pairswhose invariants are the two roots of x − ( ) x + Remark 7.2.
For n >
0, each δ n ( q ) counts doubly isogenous pairs whose invariants satisfyone of the four equations (6.3), (6.5), (6.6), or (6.7). We do not demand that the doublyisogenous curves come from values of t that are related to one another as in, for example,Lemma 6.7 or Lemma 6.13.We saw in Section 5 that Na¨ıve Heuristic 5.2 did not seem to reflect the data that we hadcollected. Here we present another heuristic which better reflects the data. In each family,let m be the number of pairs of elliptic curves required to be isogenous to ensure doubleisogeny of two curves in that family; for example, Proposition 6.12 shows that m = m -tupleof independent elliptic curves. Given this, as in Section 5.1, one can compute the expectednumber of doubly isogenous pairs in the given families under this heuristic. By abuse of notation, we will denote this by E ( δ i ( q )) , even though for fixed q , the integer δ i ( q ) is a fixedvalue and not a random variable. Heuristic 7.3.
The following are reasonable estimates for the “expected value” of δ n ( q ) for primepowers q ≡ E ( δ ( q )) (cid:16) q , E ( δ ( q )) (cid:16) √ q , E ( δ ( q )) (cid:16) √ q , E ( δ ( q )) (cid:16) √ q , E ( δ ( q )) (cid:16) q , and E ( δ ( q )) (cid:16) √ q . Combining these values, we expect that E ( δ ( q )) (cid:16) √ q.Justification. First we consider δ ( q ) . If we assume that there are no families of unexpectedcoincidences other than the four presented in Section 6.3, then the justification of Na¨ıveHeuristic 5.2 applies to the pairs { Z , Z } that are not in these four families; this suggeststhat the expected value of δ ( q ) is Θ ( q ) .Next we consider δ ( q ) , and in particular we look at the pairs of curves ( Z , Z ) whichcan be written with t = ζ t and c = c . There are roughly q such ordered pairs. ByProposition 6.12, in order for such a pair to be doubly isogenous, it is sufficient for threecoincidences to hold: E and E should be isogenous, and one of the two pairs of isogeniesin items (1) and (2) of Proposition 6.12 should hold. We model each of these coincidences asasking that two random elliptic curves lie in the same isogeny class, which happens withprobability Θ ( √ q ) . Thus, we expect to find on the order of q / ( √ q ) = √ q doublyisogenous pairs with t = ζ t and c = c .There are other relations between t and t that lead to Equation (6.3) holding, and againwe expect Θ ( √ q ) doubly isogenous pairs that satisfy the relation. Thus, in total, weexpect Θ ( √ q ) pairs of doubly isogenous pairs in the first family.For δ ( q ) the argument is similar. By Proposition 6.14, if each of three pairs of ellip-tic curves are isogenous to one another, the curves Z and Z in the proposition are dou-bly isogenous. Again modeling these isogeny class collisions as occuring with probability Θ ( √ q ) , we find that we expect Θ ( √ q ) pairs of curves ( Z , Z ) coming from pairs ofvalues ( t , t ) satisfying (6.4).As for the first family, there are other relations between t and t that lead to Equa-tion (6.5) holding. For each such relation we again expect Θ ( √ q ) doubly isogenous pairsthat satisfy the relation. Again, in total, we expect Θ ( √ q ) pairs of doubly isogenouspairs in the second family.We skip over the third family for the moment, for reasons that will become apparent.For pairs { Z , Z } in the fourth family, we need five coincidental isogenies in order forthe curves to be doubly isogenous. This suggests that the expected number of such curvesis Θ ( q / q ) = Θ ( q ) .Proposition 6.19 suggests that when − F q there is one chance out of √ q that the two values of t in the proposition give rise to a doubly isogenous pair over F q that lies in the first, second, and third families. When − δ ( q ) . For other pairs of curves whose invariants are the roots of x − ( ) x + δ ( q ) is Θ ( √ q ) .For pairs { Z , Z } in the third family, if we argue as above we see that we need fourcoincidental isogenies in order for the curves to be doubly isogenous. This suggests that OUBLY ISOGENOUS GENUS-2 CURVES WITH D -ACTION 31 the expected number of such curves is Θ ( q / q ) = Θ ( q ) . But once again our na¨ıveanalysis needs revision, because clearly δ ( q ) ≥ δ ( q ) . Thus, we take the expected valueof δ ( q ) to be Θ ( √ q ) . (cid:3) Comparison with data.
For n =
15, . . . , 23, we considered the 1024 primes q with q ≡ n . For each such q , we found all unordered pairs { Z , Z } of non-conjugate curves over F q with D -action and with all Weierstrass points rational for which Z and Z are doubly isogenous. A given pair may appear in more than one family. Table 5show which of these pairs are explained by one (or more) of the families.In a Not in a n Total Family Family F1 F2 F3 F4 (F1 ∩ F2 ∩ F3)15 820 586 234 366 222 62 20 3816 580 494 86 286 198 34 0 1217 407 318 89 192 138 24 0 1818 282 238 44 148 96 14 0 1019 218 196 22 116 90 10 2 1020 138 132 6 78 58 4 0 421 100 90 10 54 40 4 0 422 58 58 0 40 16 2 0 023 42 40 2 20 20 0 0 0T
ABLE
5. Data for doubly isogenous curves. For each n , column 2 containsthe total number of (unordered) pairs of doubly isogenous curves over F q forthe 1024 primes q ≡ n . The 3rd (resp. 4th) column con-tains the number of these in (resp. not in) at least one family. The remainingcolumns contain the number for each family.As predicted by Heuristic 7.3, increasing n by two appears to roughly halve the totalnumber of pairs, as well as the pairs in family 1 or family 2 (and possibly in family 3 and inthe intersection of the first three families, although it is harder to tell because the numbersare smaller). In contrast, increasing n by one appears to roughly halve the number of pairscoming from no family. This is as expected as Θ ( q − ) = Θ ( − n /2 ) and Θ ( q − ) = Θ ( − n ) for primes q near 2 n . The numbers for the fourth family drop off too rapidly to easilydetermine the rate of decline, but our heuristics do at least predict that the fourth familywill decrease the fastest. Thanks : This work was supported by a grant from the Simons Foundation (546235) forthe collaboration ‘Arithmetic Geometry, Number Theory, and Computation’, through aworkshop held at ICERM. Booher was partially supported by the Marsden Fund Coun-cil administered by the Royal Society of New Zealand. Li was partially funded by theSimons collaboration on ‘Arithmetic Geometry, Number Theory, and Computation’. Prieswas partially supported by NSF grant DMS-19-01819. Springer was partially supported byNational Science Foundation Awards CNS-2001470 and CNS-1617802.We thank Bjorn Poonen and Felipe Voloch for helpful conversations. R EFERENCES [1] Jeffrey D. Achter and Everett W. Howe,
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