Asymptotics for the number of Simple (4a+1) -Knots of Genus 1
aa r X i v : . [ m a t h . N T ] M a y ASYMPTOTICS FOR THE NUMBER OF SIMPLE (4 a + 1) -KNOTS OF GENUS 1 ALISON BETH MILLER
Abstract.
We investigate the asymptotics of the total number of simple 4 a + 1-knots with Alexanderpolynomial of the form mt +(1 − m ) t + m for some m ∈ [ − X, X ]. Using Kearton and Levine’s classificationof simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. Inparticular, this count is the same as the total number of Z [1 /m ]-equivalence classes of binary quadraticforms of discriminant 1 − m , for m running through the same range. Our heuristics, based on the Cohen-Lenstra heuristics, suggest that this total is asymptotic to X / / log X , and the largest contribution comesfrom the values of m that are positive primes. Using sieve methods, we prove that the contribution to thetotal coming from m prime is bounded above by O ( X / / log X ), and that the total itself is o ( X / ). Introduction
In this paper we will count simple 4 a + 1-knots by way of invariants with arithmetic structure. Informally,an n -knot is a knotted copy of S n in S n +2 : for a formal definition see Section 2.1. For n ≥
5, it is impossibleto classify all n -knots, but there is a restricted family, the simple n -knots, which have been completelyclassified by classical algebraic invariants. In this paper we will look at the case n ≡ n > n ≡ a + 1-knots. For a ≥ g simple knot is precisely 2 g .1.1. Questions and Heuristics.
We are interested in the general question: for a given positive integer g ,what are the asymptotics of the number of distinct 4 a + 1 knots with squarefree Alexander polynomial ofdegree 2 g and coefficients of “size” bounded by X ? It is known by results of Bayer and Michel [1] that thisnumber is always finite. In this paper we address the case of g = 1. In this case, the only possible Alexanderpolynomials are of the form ∆ m = mt + (1 − m ) t + m for m ∈ Z and the question becomes: Question (Counting question, knot version) . Fix a ≥
1. Asymptotically, what is the total number ofequivalence classes of simple 4 a + 1-knots of genus 1 with Alexander polynomial of the form ∆ m for | m | ≤ X ?Perhaps surprisingly, the answer to this question does not depend on the value of a , but is uniform forall a >
1. This follows from the classification of simple knots in terms of their Alexander module andBlanchfield pairing [8]. Using this classification, we can transform our counting question into an equivalententirely algebraic question:
Question (Counting question, equivalent Alexander/Blanchfield version) . Fix a ≥
0. Asymptotically, whatis the total number of isomorphism classes are there of Alexander modules with Blanchfield pairing comingfrom simple 4 a + 1-knots with Alexander polynomial of the form ∆ m for | m | ≤ X ?Note that this question has a uniform answer for all a ≥ Question (Counting question, equivalent quadratic form version) . Asymptotically, what is the total, overall m with | m | ≤ X , of the number of SL ( Z [ m ])-equivalence classes of binary quadratic forms over Z [ m ]having discriminant 1 − m ?Asymptotics of binary quadratic forms are a very-well studied question in number theory, going back toGauss. The difficulty in Question 1 comes from inverting the varying integer m . We observe that there isa clear split in behavior between the cases where the constant term of the Alexander polynomial is prime nd the cases where it is composite. Using the theory of binary quadratic forms, we propose the followingheuristics. In Section 2.4 we justify these heuristics using the Cohen-Lenstra-Hooley heuristics for binaryquadratic forms plus some additional reasonable assumptions. Heuristic 1.
The total number of distinct Alexander modules (with pairing) having Alexander polynomialequal to ∆ p for some prime p in the range [1 , X ] is asymptotic to a constant times X / / log X . Heuristic 2.
The total number of distinct Alexander modules (with pairing) having Alexander polynomialequal to ∆ − p for some prime p in the range [1 , X ] is asymptotic to a constant times X log X . Heuristic 3.
The total number of distinct Alexander modules (with pairing) having Alexander polynomialequal to mt + (1 − m ) t + m for m running over all integers in the range [ − X, X ] with | m | not prime isasymptotic to a constant times X log X . Results.
The difficulty in proving these heuristics is that we are in general counting quadratic formsover rings with infinite unit group. However, the total contribution from m prime and positive can bebounded above using Rosser’s sieve. Theorem 1.1.
The total number of isotopy classes of simple a + 1 -knots having Alexander polynomial equalto pt + (1 − p ) t + p for p running over all primes in the range [1 , X ] is (unconditionally) bounded above by O ( X / / log X ) . Sieve methods are not powerful enough to give a lower bound, so instead we apply the Brauer-Siegeltheorem to obtain
Theorem 1.2.
The total number of isotopy classes of simple a + 1 -knots having Alexander polynomial equalto ∆ p for ≤ p ≤ X is ≫ ( X / − ǫ ) for all ǫ > . Like the Brauer-Siegel theorem, this result is ineffective.
Remark.
Both of these results should generalize to g >
1. To prove the upper bound, we will need to replaceGauss’s asymptotics for binary quadratic forms with asymptotics for Sp g -orbits on 2 g -ary quadratic forms.In a forthcoming paper [14], the author obtains such bounds on Sp g -orbits, and she hopes to apply thesebounds to knot theory in future work.For the lower bound, it should be possible to replace the Brauer-Siegel with a relative Brauer-Siegeltheorem for CM extensions. However there may be technical issues regarding the cases of non-maximalorders.For the other cases, it is much harder to obtain sharp results. For instance, in the case of m = − p Theorem 1.1 remains true with exactly the same proof, but Heuristic 2 suggests a strictly lower order ofgrowth.However, we will show the following weak upper bound on the total:
Theorem 1.3.
The total number of distinct Alexander modules (with pairing) having Alexander polynomialequal to mt + (1 − m ) t + m for m running over all integers in the range [ − X, X ] is bounded above by o ( X / ) . Acknowledgments.
The author would like to thank Manjul Bhargava for leading me to explore thisproblem, and for his guidance and support throughout the process. She would also like to thank BarryMazur for helpful advice on the organization of this paper. She would also like to thank Arul Shankar andLenny Ng for comments on previous versions of this paper. Much of this paper was originally part of theauthor’s Ph.D. thesis at Princeton, where she was supported by an NSF Graduate Fellowship and an NDSEGFellowship. 2.
Proofs
Preliminaries on knots.
There are multiple different ways to formalize the notion of an n -knot. Thefollowing two definitions are both commonly used: Definition. (i) An n -knot K is a PL embedded copy of S n in S n +2 that is locally flat (locally homeo-morphic to R n ⊂ R n +2 ). Equivalence is given by ambient isotopy. ii) An n -knot is a smoothly embedded submanifold K of S n +2 that is homeomorphic to S n (but not nec-essarily diffeomorphic; K might be an exotic sphere). Equivalence is induced by orientation-preservingdiffeomorphism of the S n +2 .In both cases, we will consider both S n and S n +2 as oriented manifolds, so that reversing the orientationof either or both may give an inequivalent knot.Although it is far from obvious, classification results we use will give the same answer regardless of whichof the two definitions above is used. I will talk about knots and equivalence with the understanding that allstatements hold using either formulation. Definition.
The knot K is called simple if π i ( S n +2 − K ) = π i ( S ) for all i ≤ ( n − / Definition. A Seifert hypersurface in S n +2 is a compact oriented ( n + 1)-manifold V with boundary suchthat K = ∂V is homeomorphic to S n . We say that V is a Seifert hypersurface for the knot K .(We again have the choice of working in either the PL category or the smooth category, and again allresults we use hold uniformly in both cases.) Definition. A simple Seifert hypersurface in S n +2 is said to be simple if V is n − -connected.It is known that the simple knots are exactly those with simple Seifert hypersurfaces. Theorem 2.1. If V is a simple Seifert hypersurface in S n +2 , then ∂V is a simple ( n − -knot. Conversely,any simple n -knot K has a (non-unique) simple Seifert hypersurface.Proof. Farber states this as Theorem 0.5 in [5], where he deduces it from results of Levine [10, 13] andTrotter [16]. (cid:3)
We now specialize to n = 4 a +1. If V is a simple Seifert hypersurface in S a +3 it follows from the Hurewicztheorem and Poincare duality that H i ( V, Z ) is trivial for all i except i = 2 a + 1, and H a +1 ( V, Z ) is a free Z -module of even rank. Definition. If V is a simple Seifert surface in S a +3 , we define the genus of V to be half the rank of H a +1 ( V, Z ).The genus of a 4 a + 1-knot K is the minimum genus of any Seifert surface for K .In the classical case a = 0, the genus is a subtle geometric invariant of knots. However for a >
1, thegenus of a simple knot K is easily computable from the Alexander polynomial of K , defined. Theorem 2.2. [11] If V is any simple Seifert surface for K , and P is a Seifert matrix for V , then ∆ K = t − dim ker P (det tP − P t ) is the normalized Alexander polynomial of K , and in particularl is independent of V . Corollary 2.3. If K is any simple knot, then the genus of K is equal to deg ∆ K .Proof. This follows from Theorem 2.2 and the result that any simple knot has a Seifert surface with nonde-generate Seifert pairing (itself a consequence of Theorem 3 and Proposition 1 in [12]). (cid:3)
Relationship between knots, ideal classes, and quadratic forms.
We now use the classificationof simple knots to show that the various forms of Question 1 stated in the introduction are equivalent.
Theorem 2.4 (Classification of simple knots) . [12, 13, 17, 9, 5] The following are in bijection with eachother:(i) equivalence classes of simple a + 1 knots of genus ≤ g (ii) S -equivalence classes of g × g Seifert matrices P [12] (iii) Alexander modules of genus ≤ g equipped with Blanchfield pairing. [13, 17] (iv) R -equivalence classes of Z [ z ] -modules with isometric structures [5] For each of these objects, there is a natural way of defining an Alexander polynomial, and these bijectionspreserve the Alexander polynomial. or g = 1, we can add two more items to the list. First define R m = Z (cid:20) t, t − ] / ∆( t ) ∼ = Z [ 1 m , √ − m (cid:21) . (Note that this ring is called Ø m in the author’s Ph.D thesis [15].)Inspired by [2] we define Definition. An oriented ideal class of R m is a homothety class of fractional ideals I of R m equipped withan isomorphism φ : V I ∼ = Z [ m ] of Z [ m ]-modules.The set of oriented ideal classes of R m is often called the narrow class group of R m , though that term isused as well for other distinct but related groups.Any such φ can be written in the form φ ( α, β ) = tr (cid:18) αβκ √ − m (cid:19) for a unique κ ∈ Z [ m ]. Such a φ maps V I isomorphically to Z [ m ] if and only if κ is a generator of thefractional ideal N I of Z [ m ].Hence we can also describe oriented ideal classes of R m as equivalence [ I, κ ] of pairs where I is a fractionalideal of R m and κ ∈ Z [ m ] is a generator of N I , modulo the equivalence relation (
I, κ ) ∼ ( αI, N ακ ) for every α in Frac( R m ) = Q ( √ − m ).In this paper we will write our narrow ideal classes as [ I, κ ]. We’ll use the same notation for imaginaryquadratic rings.
Remark.
In my Ph.D. thesis [15] I defined conjugate self-balanced modules/ideal classes, which generalizethe definition of oriented ideal above to the case of g >
1. They can be thought of as relative ideal classesfor a quadratic extension of rings.
Corollary 2.5.
When g = 1 we can add the following two items to the list in Theorem 2.4:(v) pairs ( m, [ I, κ ]) where m is an integer and [ I, κ ] is an oriented ideal class of the ring R m = Z [ m , √ − m ] .(vi) pairs ( m, [ Q ]) where m is an integer and [ Q ] is an SL [ Z [ m ]] -equivalence class of binary quadratic formsover Z [ m ] with Disc Q = 1 − m .In both cases, the corresponding Alexander polynomial is ∆ m = mt + (1 − m ) t + m. Proof.
First of all, the bijection between (v) and (vi) is a generalization of the standard bijection betweenbinary quadratic forms and ideal classes in quadratic rings. Given an oriented ideal class (
I, φ ) of R m , chooseany Z [ m ] basis u , u of I with φ ( u ∧ u ) = 1. The function φ (( √ − m ) a ∧ b ) is a symmetric bilinear formon the rank 2 Z [ m ]-module I . If we write φ out in the basis u , u we obtain a binary quadratic form Q ofdiscriminant √ − m .To finish, it’s easiest to either biject Alexander modules with oriented ideal classes, or Seifert matriceswith binary quadratic forms. The former bijection is easier to prove, the latter easier to describe. We provethe former:If M is an Alexander module with Alexander polynomial ∆ m , we can view M as a module over the quotientring R m = Z [ t, t − ] / ∆ m . Because ∆ m is squarefree, we have M ⊗ Z Q ∼ = R m ⊗ Z Q as R m ⊗ Z Q = Q [ t, t − ] / ∆( t )-modules. Hence M is isomorphic as R m -module to some fractional ideal I of R m . Choose such an I and anisomorphism φ : I → M .To put an orientation on I , we use the Blanchfield pairing. The isomorphism φ lets us transfer theBlanchfield pairing on M to a R m -hermitian perfect pairing h , i : I × I → t ) Z [ t, t − ] / Z [ t, t − ] . To obtain an orientation, we compose with the map T : t ) Z [ t, t − ] / Z [ t, t − ] → Q sending [ f ] f ′ (0). Themap T is a special case of “Trotter’s trace function” in knot theory. It’s determined Z [ m ]-linearly from thevalues T ( t ) ) = 0, T ( t ∆( t ) ) = m . y standard arguments, the pairing T ( h a, b i ) : I × I → Z [ m ] is skew-symmetric with determinant a unitin Z [ m ]. Moreover we can recover the pairing h , i from the pairing T ( h a, b i ). (cid:3) We can now apply the formulas for the Alexander module and Blanchfield pairing in terms of the Seifertmatrix original given by Levine in [13] and reproved by Friedl and Powell in [6]. When one works out thedetails, it turns out that the composite map from Seifert matrices to binary quadratic forms sends a Seifertmatrix P to the quadratic form with matrix P + P T . (This is not an integer matrix, but still gives an integerquadratic form.)We then obtain the following corollary. Corollary 2.6.
Two × Seifert matrices P and P with the same Alexander polynomial ∆ m (equivalently,with the same determinant m ) are S -equivalent if and only if there exists X ∈ SL ( Z [ m ]) with P = XP X T . This is a special case of 4.15 in Trotter[16]. For larger Seifert matrices Trotter also shows that S -equivalenceimplies Sp g ( Z [ m ])-equivalence, but the converse is not generally the case.Another corollary (which can also be proved directly): Corollary 2.7.
Any binary quadratic form over Z [ m ] of discriminant − m is SL ( Z [ m ]) -equivalent to aform defined over Z . The map from oriented ideal classes of Z [ γ m ]) to oriented ideal classes of Cl + ( R m ) . Corol-lary 2.7 can also be interpreted as a statement about narrow class groups. Let γ m = √ − m , so that Z [ γ m ]is the ring of integers in Q ( √ − m ).The inclusion ι : Z [ γ m ] ֒ → R m induces a map ι ∗ : Cl + ( Z [ γ m ]) → Cl + ( R m ). More generally, if ( I, κ ) is a(possibly non-invertible) ideal class of Z [ γ m ], then we can map it to the ideal class ( IR m , κ ) of R m . Corollary 2.8.
The map ι ∗ : Cl + ( Z [ γ m ]) → Cl + ( R m ) is surjective. The kernel of ι ∗ can also be described explicitly: Proposition 2.9.
The kernel ker ι ∗ is generated by the classes [( p, γ m ) , p ][( p, − γ m ) , p ] − = [( p, γ m ) , p ] where p runs through the set of all prime factors of m .Proof. First, observe that R m = Z [ γ m , m ], and that m factors in R m as m = Y p | m ( p, γ m )( p, − γ m ) . Let [
I, κ ] be an arbitrary element of ker ι : since we are in the kernel we can rescale so that I · R m = R m and κ = 1. Then I is a fractional ideal of Z [ γ m ] which becomes trivial when we invert the element m , so wecan factor I as I = Y p | m ( p, γ m ) a p ( p, − γ m ) b p . In order to have
N I = (1) we must have a p = − b p for all p . The result follows. (cid:3) Note that not all oriented ideal classes of Z [ γ m ] are invertible. However: Proposition 2.10.
Two (possibly non-invertible) oriented ideal classes [ I, κ ] and [ I ′ , κ ′ ] of Z [ γ m ] becomeequivalent in R m if and only if there exists [ J, λ ] ∈ ker ι ∗ with [ I, κ ] = [
J, λ ][ I ′ , κ ′ ] . Proof.
As before, can reduce to the case κ = κ ′ . For each p let a p = v p ( I ) − v p ( I ′ )where p is the ideal ( p, γ m ) and let J = Y p [( p, γ m ) , p ] a p [( p, − γ m ) − a p , p ] − . One can then check locally that I = JI ′ . (cid:3) .4. Consequences for the heuristics.
In the case when m = ± p is prime, we have a consequence that Proposition 2.11. If m is prime (possibly negative) then the map ι ∗ : Cl + ( Z [ γ m ]) → Cl + ( R m ) is anisomorphism. More generally, oriented ideal classes of Z [ γ m ] are in bijection with oriented ideal classes of R m via [ I, κ ] [ IR m , κ ] . Translating back into the language of quadratic forms, we also have the important corollary:
Corollary 2.12.
Two quadratic forms over Z of determinant − p are SL ( Z [ p ]) -equivalent if and only ifthey are SL ( Z ) -equivalent. This corollary is also implied by Trotter’s work in [16].We can now easily prove Theorem 1.2:
Proof of Theorem 1.2.
We show the equivalent statement, that the total is ≫ X / − ǫ / log X for any ǫ > ǫ >
0. By the Brauer-Siegel theorem plus the formula for class number of non-maximal orders,there exists some constant c ǫ such that the size | Cl + ( Z [ γ m ]) | ≥ c ǫ X / − ǫ for every m ∈ [0 , X ]. We have just seen that | Cl + ( Z [ γ m ]) | = | Cl + ( R m ) | when m is prime. Hence the totalcontribution of the ∼ X/ log X primes in [0 , X ] is ≫ X / − ǫ / log X . (cid:3) When m is not prime, for any p dividing m the ideals ( p, γ m ) and ( p, − γ m ) represent two nontrivialdistinct ideal classes, as can be checked with reduction theory. Hence the class [( p, γ m ) , p ][( p, − γ m ) , p ] − =[( p, γ m ) , p ] is always a nontrivial element of Cl + ( Z [ γ m ]) . Note that these ideals satisfy one relation, comingfrom the identity Y p | m ( p, γ m ) v p ( m ) = ( γ m ) . This motivates the following heuristics
Heuristic 4.
When m runs through all positive integers of the form p n · · · p n k k , the distribution of the finiteabelian groups Cl + ( R m ) agrees with the distribution of G/ h g , . . . , g k i , where G is a finite abelian groupselected from the Cohen-Lenstra distribution for narrow class groups of imaginary quadratic fields [3] and g , . . . g k are randomly chosen elements of G subject to the constraint that Q g kn k = 1 .When m runs through all positive integers of the form p n · · · p n k k , the distribution of the finite abeliangroups Cl + ( R m ) agrees with the distribution of G/ h g , . . . , g k i , where G is a finite abelian group selected fromthe Cohen-Lenstra distribution for narrow class groups of real quadratic fields [3] and g , . . . g k are randomlychosen elements of G subject to the constraint that Q g kn k = 1 . These heuristics are fairly na ive and it is worth investigating them further for accuracy, but I conjecturethat they at least give the correct order of magnitude for the average sizes of these groups. An importantspecial case: if m = p p , this is essentially the Cohen-Lenstra distribution for narrow class groups of realquadratic fields, and we should expect similar behavior, namely that on average the class group should havesize about (log m ) , and the total contribution of all m = p p ≤ X will be O ( X log X ).We now consider the case of general m . By Erd¨os-Kac, most integers ≤ X have on the order of log log X prime factors. On the other hand, Cohen-Lenstra distribution is heavily biased towards groups G where G is generated by a small number of elements: for every n , the probability of G being generated by ≤ n elements is positive, and goes to 1 rapidly as n → ∞ .Hence we expect that, for a density 1 subset of m , we haveCl + ( R m ) ∼ = Cl + ( Z [ γ ]) / Cl + ( Z [ γ ]) . (By genus theory for binary quadratic forms, this implies that two quadratic forms of discriminant 1 − m are Z [ m ]-equivalent if and only if they are Z -equivalent.) .5. Proof of Theorem 1.1 by Sieving.
By Corollaries 2.7 and 2.12, it’s enough to show
Proposition 2.13.
The total number of SL ( Z ) -equivalence classes of binary quadratic forms ax + bxy + cy with prime discriminant of the form p = 1 − m for m ∈ [0 , X ] is bounded above by O ( X / log X ) . We’ll actually show that the total for m ∈ [ X, X ] is also O ( X / log X ), and the proposition will follow bysumming. As well, we will only count the positive definite quadratic forms, as the count of negative definiteforms is the same.We follow the approach of Rosser’s sieve [7], modifying the terminology to suit our approach. We introducean auxiliary parameter Z ≤ X whose value will be chosen later, and let P ( Z ) denote the product of all primesup to Z . Let F = { ( α, β, γ ∈ R | | β | ≤ γ ≤ α } be the standard fundamental domain for SL ( Z ) acting on positive definite binary quadratic forms.Then the total we wish to bound is at most: S ( X, Z ) := X X ≤ m ≤ X ( m,P ( Z ))=1 { ( a, b, c ) ∈ Z ∩ F : b − ac = 1 − m } Note here that b − ac = 1 − m implies b odd: we write b = 2 b ′ + 1 and let F ′ be the preimage of F underthe affine transformation ( α, β, γ ) ( α, β + 1 , γ ). Using this change of variables S ( X, Z ) = X X ≤ m ≤ X ( m,P ( Z ))=1 { ( a, b ′ , c ) ∈ Z ∩ F ′ : ac − b ′ ( b ′ + 1) = m } To apply the sieve, we need estimates on the following quantities for all squarefree d ≤ X :(1) S d ( X ) := X m ∈ [ X, X ] d | m { ( a, b ′ , c ) ∈ Z ∩ F ′ : ac − b ′ ( b ′ + 1) = m } . Lemma 2.14.
For a positive integer s , let ρ ( s ) = Q p | s p +1 p . If d is a squarefree positive integer ≤ X , thereexist explicit real constants c , c such that (2) S d ( X ) = c ρ ( d ) X / + R d ( X ) , and the error term R d ( X ) is bounded by (3) | R d ( X ) | ≤ c Xρ ( d )(max( d, log X )) . Proof.
We observe that for all squarefree d , S d ( X ) counts the number of points in the intersection of theregion R X = { ( α, β ′ , γ ) ∈ F ′ | | αγ − β ′ ( β ′ + 1) | ≤ X with the union of the cosets of ( d Z ) on which the function ac − b ( b + 1) vanishes modulo d .We wish to apply: Lemma 2.15 (Davenport) . [4] Let R be a bounded semi-algebraic region in R n , defined by k polynomialinequalities of degree at most ℓ . Then the number of points ( a, b, c ) ∈ Z n ∩ R can be asymptotically expressedas vol( R ) + ǫ ( R ) with the error term ǫ ( R ) bounded in size by ǫ ( R ) < κ max(vol( R ) , where R runs over all projections of R onto subspaces of R n spanned by coordinate axes, and κ = κ ( n, m, k, ℓ ) is some explicit constant dependingonly on n, m, k , and ℓ . We cannot apply Davenport’s lemma directly because R X goes off to infinity. Instead, we truncate thecusp: for a positive real parameter R , define R X,T = { ( x, y, z ) ∈ R X | z < T } . We observe that any lattice point ( a, b, c ) ∈ R X has c ≤ X , so also belongs to R X, X . ne can calculate that the largest 1-dimensional projection of R X, X has length 2 X , while the largest2-dimensional projection of R X, X has area c X log X for an explicit constant c .Now let L , . . . , L n be the cosets of ( d Z ) for which ac − b ′ ( b ′ + 1) ≡ d ) for all ( a, b ′ , c ) ∈ L i .The number n is equal to the number of solutions to ac − b ′ ( b ′ + 1) = 0 in ( Z /d Z ) : a calculation with theChinese remainder theorem gives n = ρ ( d ) d .Applying Davenport’s lemma to R X,t rescaled by d − and translated appropriately, we obtain that thereexists a real number κ such that for each i R X, X ∩ L i ) − c d − X / ≤ κ (max( d − X log X, d − X, κd − X max(log X, d )(4)where the last step uses d ≤ X .Summing over all ρ ( d ) d values of i and applying the triangle inequality, we obtain S d ( X ) − ρ ( d ) c d − X / = X i ( R X, X ∩ L i ) − c d − X / ) ≤ ρ ( d ) κd − X max(log X, d )as desired. (cid:3)
We are now ready to prove Proposition 2.13.
Proof.
We apply Rosser’s sieve. First we calculate the “sieving density,” also known as the “dimension”.The following inequality is analogous to (1.3) in [7]: for all
Z > W ≥ Y W ≤ p 1, the main term in (6) is O ( X / / log X ) and theerror term is o ( X / / log X ), giving the desired asymptotic. (cid:3) Proof of Theorem 1.3. We now prove Theorem 1.3. We will prove it in the equivalent form Theorem 2.16. If T ( X ) is the number of SL ( Z [ m ]) -equivalence classes of binary quadratic forms of dis-criminant − m as m runs through all integers in the range [ − X, X ] , then lim X →∞ T ( X ) X / = 0Gauss’s bound for the total number of SL ( Z )-equivalence classes of binary quadratic forms with dis-criminant in this range is O ( X / ), so this theorem says that strengthening the equivalence relation toSL ( Z [ m ])-equivalence decreases the order of growth.We can instead think of our total T ( X ) as counting SL ( Z )-equivalence classes of binary quadratic formsweighted as follows: Definition. The weight w ( Q ) of a binary quadratic form Q of discriminant 1 − m is equal to n where n is the number of distinct SL ( Z )-equivalence classes of binary quadratic forms comprising the SL ( Z [ m ])-equivalence class of Q .Then T ( X ) = X SL ( Z )-equivalence classes [ Q ]disc[ Q ]=1 − m with m ∈ [ − X,X ] w ( Q ) ecall that the content of a binary quadratic form ax + bxy + cy is defined to be content( Q ) = gcd( a, b, c ).In order to bound our weighted count we will first need to divide up by content. Definition. Let T d ( X ) be the number of SL ( Z [ m ])-equivalence classes of binary quadratic forms ax + bxy + cy of content d with discriminant 1 − m as m runs through all integers in the range [ − X, X ].As before, we have(11) T d ( X ) = X SL ( Z )-equivalence classes [ Q ]disc[ Q ]=1 − m with m ∈ [ − X,X ] content( Q )= d w ( Q ) . The key result we need here is that Lemma 2.17. If Q is a binary quadratic form with content d and discriminant − m , then we have anupper bound w ( Q ) ≤ ǫ d ( ω ( m )) Where ǫ d ( n ) = d − − n and ω ( m ) denotes the number of distinct prime factors of m .Proof. We translate this question over to the language of oriented ideal classes. Our hypothesis becomes: if[ I, α ] is an oriented ideal class of Z [ γ m ], and the endomorphism ring of the ideal I Ø I = { λ ∈ Q [ γ m ] | λI ⊂ I } , equals Z [ √ m/d ], or equivalently, Z [ γ m ] = Z + d Ø I , We must show that there are at least ǫ d ( n ) distinctideal classes [ I ′ , α ′ ] of Ø I such that the [ I ′ R f , α ′ ] and [ IR f , α ] are equivalent oriented ideal classes of R f .We first do the case d = 1.Let [ I Q , κ Q ] be the oriented ideal class of Z [ γ m ] corresponding to the quadratic form Q . Because d = 1,the ideal class I Q is invertible. Hence if [ J, λ ] runs through the elements of ker ι ∗ , the ideal classes [ JI Q , λκ Q ]are all distinct but becomes equivalent classes of λ .So it suffices for f = 1 to show that | ker ι ∗ | ≥ ω ( m ) − . For this, note that if we let s run through thedivisors of m with s > √ m , the CM points γs = √ − m s all lie in the fundamental domain of the upper-halfplane. Hence the ideal classes ( m, γ ) are all distinct, and they also all lie in ker ι ∗ . We conclude that | ker ι ∗ | ≥ (cid:22) d ( m )2 (cid:23) ≥ ω ( m ) − . We can use the same argument for d > 1. However, here I Q is no longer invertible, so the map [ J, λ ] [ JI Q , λκ Q ] is no longer injective. Indeed, by the theory of ideals in quadratic orders, the fibers of this mapare the same as the fibers of the homomorphism φ : Cl + ( Z [ γ m ]) → Cl + (Ø I ) . If we show | ker( φ ) | ≤ d we will then get the desired bound.Note that Z [ γ m ] has index d in O I , so Z [ γ m ] = Ø I + d Ø I . Then there is a surjective homomorphism(Ø I /d Ø I ) × → ker φ .When Z [ γ m ] is a maximal order, this is (exercise in Cox). Explicitly this is given by: for a class [ α ] ∈ (Ø I /d Ø I ) × , choose a representative α ∈ Ø I with αα > α relatively prime to 1 − m . Then we map[ α ] to the narrow ideal class [ α Ø I ∩ Z [ γ m ] , αα ] . Hence | ker φ | ≤ | (Ø I /d Ø I ) × | < d as desired. (In fact, one can do better using the precise formulas for class numbers of orders.) (cid:3) Proof of Theorem 2.16. We divide up our count of quadratic forms according to the content. Note that X − / T ( X ) = P d ≥ X − / T d ( X ).We claim that the series P d ≥ X − / T d ( X ) satisfies the conditions of the dominated convergence theorem.Indeed, for every d , T d ( X ) is at most the number of SL -equivalence classes of primitive binary quadraticforms with odd discriminant in the range [ − d − X, d − X ]. By Gauss’s count of binary quadratic forms, the atter is bounded above by cd − X / for some uniform constant c . Hence X − / T d ( X ) ≤ cd − for all X ,and so the series P d ≥ X − / T d ( X ) is dominated by the convergent series P d ≥ cd − .So it suffices to prove lim X →∞ X − / T d ( X ) = 0 for any fixed d . We’ll show that lim X →∞ X − / T d ( X ) < ǫ for any ǫ > T d ( X ) using equation (11).Let S = { ( α, β ′ , γ ∈ R | | β ′ + 1 | ≤ | γ | ≤ | α |} . By reduction theory of binary quadratic forms, every SL ( Z )-orbit of binary quadratic forms has at leastone representative of the form ax + (2 b ′ + 1) xy + cy with ( a, b ′ , c ) ∈ S (possibly more than one if the formis indefinite or if one of the representatives lies on the boundary of S ).Using this and Lemma 2.17, we bound the right hand side of (11) as T d ( X ) ≤ X Q = ax +(2 b ′ +1) xy + cy ( a,b ′ ,c ) ∈S∩ Z m = b ′ ( b ′ +1) − ac ∈ [ − X,X ]gcd( a, b ′ +1 ,c )= d w ( Q ) ≤ X ( a,b ′ ,c ) ∈S∩ Z b ′ ( b ′ +1) − ac ∈ [ − X,X ]gcd( a, b ′ +1 ,c )= d max(1 , d − ω ( b ′ ( b ′ +1) − ac ) )Now, we choose a parameter N , and split the sum into the contribution from triples ( a, b ′ , c ) with ω ( b ′ ( b ′ + 1) − ac ) ≥ N and those triples with ω ( b ′ ( b ′ + 1) − ac ) < N. The total contribution of the triples with ω ( b ′ ( b ′ + 1) − ac ) ≥ N is at most(12) d − N { a, b ′ , c ) ∈ S ∩ Z | gcd( a, b ′ + 1 , c ) = d, b ′ ( b ′ + 1) − ac ∈ [ − X, X ] } ≤ − N κ ( d ) X / for some real constant κ ( d ) depending on d , by the same Davenport’s Lemma argument used in the proofof (4).On the other hand, since all weights are ≤ 1, the contribution of the triples ( a, b ′ , c ) such that ω ( b ′ ( b ′ + 1) − ac ) < N is at most { a, b ′ , c ) ∈ S ∩ Z | gcd( a, b ′ + 1 , c ) = d, b ′ ( b ′ + 1) − ac ∈ [ − X, X ] has at most r prime factors } However, a na¨ıve sieving argument shows that the triples ( a, b ′ , c ) with ω ( b ′ ( b ′ + 1) − ac ) ≥ N have density 0 in S ∩ Z . To be more precise, if we pick a real parameter r , then the set A ( N, r ) = { a, b ′ , c ) ∈ S ∩ Z | , b ′ ( b ′ + 1) − ac ∈ [ − X, X ] has < N prime factors ≤ r } is a union of sublattices of Z . By a standard sieving argument, the density ρ ( N, r ) of A ( N, r ) in Z goes to0 as r → ∞ for fixed N .Applying Davenport’s lemma to this union of sublattices, we obtain { a, b ′ , c ) ∈ S ∩ Z | gcd( a, b ′ + 1 , c ) = d, b ′ ( b ′ + 1) − ac ∈ [ − X, X ] has at most r prime factors }≤ { a, b ′ , c ) ∈ S ∩ A ( N, r ) | b ′ ( b ′ + 1) − ac ∈ [ − X, X ] } = ρ ( N, r ) κ X / + ǫ N,R ( X )(13)where the error term ǫ N,r ( X ) is O ( X log X ) for fixed N and r . ombining equations (12) and (13) we get T d ( X ) ≤ − N κ ( d ) X / + ρ ( N, r ) κ X / + ǫ N,r ( X )so lim X →∞ X − / T d ( X ) ≤ − N κ ( d ) + ρ ( N, r ) κ . But we can make the right hand side arbitrarily small by first choosing N arbitrarily large and then choosing r arbitrarily large depending on N .Hence lim X →∞ X − / T d ( X ) = 0 as desired. (cid:3) References [1] E. Bayer and F. Michel. Finitude du nombre des classes d’isomorphisme des structures isometriques enti`eres. Comment.Math. Helv. , 54(3):378–396, 1979.[2] M. 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