Explicit Heegner Points: Kolyvagin's Conjecture and Non-trivial Elements in the Shafarevich-Tate Group
aa r X i v : . [ m a t h . N T ] J un Explicit Heegner Points: Kolyvagin’s Conjecture andNon-trivial Elements in the Shafarevich-Tate Group
Dimitar Jetchev Kristin Lauter William Stein
Abstract
Kolyvagin used Heegner points to associate a system of cohomology classes toan elliptic curve over Q and conjectured that the system contains a non-trivialclass. His conjecture has profound implications on the structure of Selmer groups.We provide new computational and theoretical evidence for Kolyvagin’s conjecture.More precisely, we explicitly compute Heegner points over ring class fields and usethese points to verify the conjecture for specific elliptic curves of rank two. Weexplain how Kolyvagin’s conjecture implies that if the analytic rank of an ellipticcurve is at least two then the Z p -corank of the corresponding Selmer group is atleast two as well. We also use explicitly computed Heegner points to producenon-trivial classes in the Shafarevich-Tate group. Contents y c . . . . . . . . . . . . . . . . . . . . 32.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 E = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 y c L -functions L ( f, χ, s ) and L ( π, s ) . . . . . . . . . . . . 165.2 Zhang’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3 Computing special values of derivatives of automorphic L -functions . . . 175.4 Asymptotic estimates on the canonical heights b h ( y c ) . . . . . . . . . . . 195.5 Height difference bounds and the main estimates . . . . . . . . . . . . . 201 Introduction
Let E /F be an elliptic curve over a number field F . The analytic rank r an ( E/F ) of E is the order of vanishing of the L -function L ( E /F , s ) at s = 1. The Mordell-Weil rank r MW ( E/F ) is the rank of the Mordell-Weil group E ( F ). The conjecture of Birch andSwinnerton-Dyer is the assertion that r an ( E/F ) = r MW ( E/F ).Kolyvagin constructed explicit cohomology classes from Heegner points over cer-tain abelian extensions of quadratic imaginary fields and used these classes to boundthe size of the Selmer groups for elliptic curves over Q of analytic rank at most one(see [Kol90], [Kol91b] and [Gro91]). His results, together with the Gross-Zagier formula(see [GZ86]), imply the following theorem: Theorem 1.1 (Gross-Zagier, Kolyvagin) . Let E / Q be an elliptic curve which satisfies r an ( E/ Q ) ≤ . Then r an ( E/ Q ) = r MW ( E/ Q ) . Unfortunately, very little is known about the Birch and Swinnerton-Dyer conjecturefor elliptic curves E / Q with r an ( E/ Q ) ≥
2. Still, it implies the following conjecture:
Conjecture 1.2. If r an ( E/ Q ) ≥ then r MW ( E/ Q ) ≥ . As far as we know, nothing has been proved towards the above assertion. A weakerconjecture can be formulated in the language of Selmer coranks. The Selmer corank r p ( E/F ) of E /F is the Z p -corank of the Selmer group Sel p ∞ ( E/F ). Using Kummertheory, one shows that r p ( E/ Q ) ≥ r MW ( E/ Q ) with an equality occuring if and only ifthe p -primary part of the Shafarevich-Tate group X ( E/ Q ) is finite. Thus, one obtainsthe following weaker conjecture: Conjecture 1.3. If r an ( E/ Q ) ≥ then r p ( E/ Q ) ≥ . For elliptic curves E of arbitrary analytic rank, Kolyvagin was able to explain theexact structure of the Selmer group Sel p ∞ ( E/ Q ) in terms of Heegner points and theassociated cohomology classes under a conjecture about the non-triviality of these classes(see [Kol91a, Conj.A]). Unfortunately, Kolyvagin’s conjecture appears to be extremelydifficult to prove. Until the present paper, there has been no example of an ellipticcurve over Q of rank at least 2 for which the conjecture has been verified.In this paper, we present a complete algorithm to compute Kolyvagin’s cohomol-ogy classes by explicitly computing the corresponding Heegner points over ring classfields. We use this algorithm to verify Kolyvagin’s conjecture for the first time for el-liptic curves of analytic rank two. We also explain (see Corollary 3.5) how Kolyvagin’sconjecture implies Conjecture 1.3. In addition, we use methods of Christophe Cornut(see [Cor02]) to provide theoretical evidence for Kolyvagin’s conjecture. Finally, as aseparate application of the explicit computation of Heegner points, we construct non-trivial cohomology classes in the Shafarevich-Tate group X ( E/K ) of elliptic curves E over certain quadratic imaginary fields.The paper is organized as follows. Section 2 introduces Heegner points over ringclass fields and Kolyvagin cohomology classes. We explain the method of computationand illustrate them with two examples. In Section 3 we state Kolyvagin’s conjecture,discuss Kolyvagin’s work on Selmer groups and establish Conjecture 1.3 as a corollary.Moreover, we present a proof of the theoretical evidence following closely Cornut’s ar-guments. Section 3.6 contains the essential examples for which we manage to explicitelyverify the conjecture. Finally, in Section 4 we apply our computational techniques toproduce explicit non-trivial elements in the Shafarevich-Tate groups for specific ellipticcurves. 2 cknowledgment. We are indebted to Christophe Cornut for sharing his thoughtson the Kolyvagin’s conjecture. We would like to thank Jan Nekov´aˇr for his helpfuldiscussions on height bounds, Stephen D. Miller and Mark Watkins for pointing outanalytic methods for estimating special values of derivatives of automorphic L -functions,Henri Darmon and David Jao for useful conversations on Heegner points computations,and Ken Ribet for reading the preliminary draft and for useful conversations. We discuss Heegner points over ring class fields in Section 2.1 and describe a methodfor computing them in Section 2.2. Height estimates for these points are given in theappendix. We illustrate the method with some examples in Section 2.3. The standardreferences are [Gro91], [Kol90] and [McC91].
Let E be an elliptic curve over Q of conductor N and let K = Q ( √− D ) for somefundamental discriminant − D < D = 3 ,
4, such that all prime factors of N are splitin K . We refer to such a discriminant as a Heegner discriminant for E/ Q . Let O K be the ring of integers of K . It follows that N O K = N ¯ N for an ideal N of O K with O K / N ≃ Z /N Z .By the modularity theorem (see [BCDT01]), there exists a modular parameterization ϕ : X ( N ) → E . Let N − be the fractional ideal of O K for which N N − = O K . Weview O K and N as Z -lattices of rank 2 in C and observe that C / O K → C / N − is acyclic isogeny of degree N between the elliptic curves C / O K and C / N − . This isogenycorresponds to a complex point x ∈ X ( N )( C ). According to the theory of complexmultiplication [Sil94, Ch.II], the point x is defined over the Hilbert class field H K of K . More generally, for an integer c , let O c = Z + c O K be the order of conductor c in O K and let N c = N ∩ O c , which is an invertible ideal of O c . Then O c / N c ≃ Z /N Z and the map C / O c → C / N − c is a cyclic isogeny of degree N . Thus, it defines a point x c ∈ X ( N )( C ). By the theory of complex multiplication, this point is defined over thering class field K [ c ] of conductor c over K (that is, the unique abelian extension of K corresponding to the norm subgroup c O c × K × ⊂ b K × ; e.g., if c = 1 then K [1] = H K ).We use the parameterization ϕ : X ( N ) → E to obtain points y c = ϕ ( x c ) ∈ E ( K [ c ]) . Let y K = Tr H K /K ( y ). We refer to y K as the Heegner point for the discriminant D ,even though it is only well defined up to sign and torsion (if N ′ is another ideal with O / N ′ ≃ Z /N Z then the new Heegner point differs from y K by at most a sign changeand a rational torsion point). y c Significant work has been done on explicit calculations of Heegner points on ellipticcurves (see [Coh07], [Del02], [Elk94], [Wat04]). Yet, all of these compute only thepoints y and y K . In [EJL] explicit computations of the points y c were considered inseveral examples and some difficulties were outlined.3o compute the point y c = [ C / O c → C / N − c ] ∈ E ( K [ c ]) we let f ∈ S (Γ ( N ))be the newform corresponding to the elliptic curve E and Λ be the complex lattice(defined up to homothety), such that E ∼ = C / Λ. Let h × = h ∪ P ( Q ) ∪ { i ∞} , where h = { z ∈ C : Im( z ) > } , equipped with the action of Γ ( N ) by linear fractionaltransformations. The modular parametrization ϕ : X ( N ) → E is then given by thefunction ϕ : h × → C / Λ ϕ ( τ ) = Z i ∞ τ f ( z ) dz = X n ≥ a n n e πinτ , (2.1)where f = ∞ X n =1 a n q n is the Fourier expansion of the modular form f .We first compute ideal class representatives a , a , . . . , a h c for the Picard groupPic( O c ) ∼ = Gal( K [ c ] /K ), where h c = O c ). Let σ i ∈ Gal( K [ c ] /K ) be the image ofthe the ideal class of a i under the Artin map. Thus, we can use the ideal a i to computea complex number τ i ∈ h representing the CM point σ i ( x c ) for each i = 1 , . . . , h c (since X ( N ) = Γ ( N ) \ h × ). Explicitly, the Galois conjugates of x c are σ i ( x c ) = [ C / a − i → C / a − i N − c ] , ∀ i = 1 , . . . , h c . Next, we can use (2.1) to approximate ϕ ( σ i ( x c )) as an element of C / Λ by truncatingthe infinite series. Finally, the image of ϕ ( τ i ) + Λ under the Weierstrass ℘ -functiongives us an approximation of the x -coordinate of the point y c on the Weierstrass modelof the elliptic curve E . On the other hand, this coordinate is K [ c ]-rational. Thus, ifwe compute the map (2.1) with sufficiently many terms and up to high enough floatingpoint accuracy, we must be able to recognize the correct x -coordinate of y c on theWeierstrass model as an element of K [ c ].To implement the last step, we use the upper bound established on the logarithmicheight of the Heegner point y c (given in the appendix). The bound on the logarithmicheight comes from a bound on the canonical height combined with bounds on the heightdifference (see the appendix for complete details). Once we have a height bound, weestimate the floating point accuracy required for the computation. Finally, we estimatethe number of terms of (2.1) necessary to compute the point y c up to the correspondingaccuracy (see [Coh07, p.591]). Remark 2.1.
In practice, there are two ways to implement the above algorithm. Thefirst approach is to compute an approximation x i of the x -coordinates of y σ i c for every i = 1 , . . . , c and form the polynomial F ( z ) = Q h c i =1 ( z − x i ). The coefficients of thispolynomial are very close to the rational coefficients of the minimal polynomial of theactual x -coordinate of y c . Thus, one can try to recognize the coefficients of F ( z ) byusing the continued fractions method. The second approach is to search for the τ i withthe largest imaginary part (which will make the convergence of the corresponding series(2.1) defining the modular parametrization fast) and then try to search for an algebraicdependence of degree [ K [ c ] : K ] using standard algorithms implemented in PARI/GP.Indeed, computing a conjugate with a smaller imaginary part might be significantlyharder since the infinite series in (2.1) will converge slower and one will need moreterms to compute the image up to the required accuracy. Remark 2.2.
We did not actually implement an algorithm for computing bounds onheights of Heegner points as described in the appendix of this paper. Thus the com-putations below are not provably correct, though we did many consistency checks, and4ur computational observations are almost certainly correct. The primary goal of theexamples and practical implementation of our algorithm is to provide tools and datafor improving our theoretical understanding of Kolyvagin’s conjecture, and not makingthe computations below provably correct does not detract from either of these goals.
We compute the Heegner points y c for specific elliptic curves and choices of quadraticimaginary fields. : Let E / Q be the elliptic curve with label in Cremona’s database. Explicitly, E is the curve y + xy + y = x − x . Let D = 43 and c = 5. The conductor of E is 53which is split in K = Q ( √ D ), so D is a Heegner discriminant for E . The modular formassociated to E is f E ( q ) = q − q − q − q + 3 q − q + 3 q + 6 q + · · · . One applies themethods from Section 2.2 to compute the minimal polynomial of the x -coordinate of y for the above model F ( x ) = x − x + 1980 x − x + 6930 x − x + 864 . Since F ( x ) is an irreducible polynomial over K , it generates the ring class field K [5] /K ,i.e., K [5] = K [ α ] ∼ = K [ x ] / h F ( x ) i , where α is one of the roots. To find the y -coordinate of y we substitute α into the equation of E and factor the resulting quadratic polynomialover K [5] to obtain that the point y is equal to ` α, − / α + 43 / α − / α + 2167 / α − / α + 544 / ´ ∈ E ( K [5]) . The elliptic curve with label is y + y = x + x − x and the associatedmodular form f E ( q ) = q − q − q + 2 q − q + 4 q − q + q + 6 q + · · · . Let D = 7(which is a Heegner discriminant for E ) and c = 5. As above, we compute the minimalpolynomial of the x -coordinate of y F ( x ) = x + 107 x − x − x + 314835 x − x + 487711225 . If α is a root of F ( x ) then y = ( α, β ) where β = 2807761 √− α + 10307761 √− α − √− α − √− α + 7056554327 √− α + − − √− . The curve with equation y + y = x − x − x has associated modularform f E ( q ) = q − q − q + 2 q − q + 2 q − q − q + · · · . Let D = 7 (a Heegner dis-criminant for E ) and c = 5. The minimal polynomial of the x -coordinate of y is F ( x ) = · · ` x − x − x − x + 2627410 x + 18136030 x + 339921 ´ , and if α is a root of x then y = ( α, β ) β = 34114562822 √− α − √− α − √− α + 71098971319262 √− α ++ 397565891319262 √− α + − √− . The curve has equation y + xy + y = x − x with associated modularform f E ( q ) = q − q − q + q − q +2 q − q − q + q +3 q + . . . . Again, for D = 7 and c = 55e find F ( x ) = · ` x + 12400 x + 32200 x + 78960 x + 289120 x + 622560 x + 472896 ´ and y = ( α, β ) with β = 1633512271 √− α + 20652536813 √− α + 549955259 √− α + 39053212271 √− α ++ − √− α + − √− . We briefly recall Kolyvagin’s construction of the cohomology classes in Section 3.2 andstate Kolyvagin’s conjecture in Section 3.3. Section 3.4 is devoted to the proof ofthe promised consequence regarding the Z p -corank of the Selmer group of an ellipticcurve with large analytic rank. In Section 3.5 we provide Cornut’s arguments for thetheoretical evidence for Kolyvagin’s conjecture and finally, in Section 3.6 we verifyKolyvagin’s conjecture for particular elliptic curves. Throughout the entire section weassume that E / Q is an elliptic curve of conductor N , D is a Heegner discriminant for E and p ∤ N D is a prime such that the mod p Galois representation ρ E,p : Gal( Q / Q ) → Aut( E [ p ]) is surjective. Most of this section follows the exposition in [Gro91], [McC91] and [Kol91c].
1. Kolyvagin primes.
We refer to a prime number ℓ as a Kolyvagin prime if ℓ is inertin K and p divides both a ℓ and ℓ + 1). For a Kolyvagin prime ℓ let M ( ℓ ) = ord p (gcd( a ℓ , ℓ + 1)) . We denote by Λ r the set of all square-free products of exactly r Kolyvagin primes andlet Λ = [ r Λ r . For any c ∈ Λ, let M ( c ) = min ℓ | c M ( ℓ ). Finally, letΛ rm = { c ∈ Λ r : M ( c ) ≥ m } and let Λ m = [ r Λ rm .
2. Kolyvagin derivative operators.
Let G c = Gal( K [ c ] /K ) and G c = Gal( K [ c ] /K [1]).For each ℓ ∈ Λ , the group G ℓ is cyclic of order ℓ + 1. Indeed, G ℓ ≃ ( O K /ℓ O K ) × / ( Z /ℓ Z ) × ≃ F × λ / F × ℓ . Moreover, G c ∼ = Y ℓ | c G ℓ (since Gal( K [ c ] /K [ c/ℓ ]) ∼ = G ℓ ). Next, fix a generator σ ℓ of G ℓ for each ℓ ∈ Λ . Define D ℓ = P ℓi =1 iσ iℓ ∈ Z [ G ℓ ] and let D c = Y ℓ | c D ℓ ∈ Z [ G c ] . Note that ( σ ℓ − D ℓ = 1 + ℓ − Tr K [ ℓ ] /K [1] .6e refer to D c as the Kolyvagin derivative operators . Finally, let S be a set of cosetrepresentatives for the subgroup G c ⊆ G c . Define P c = X s ∈ S sD c y c ∈ E ( K [ c ]) . The points P c are derived from the points y c , so we will refer to them as derived Heegnerpoints .
3. The function m : Λ → Z and the sequence { m r } r ≥ . For any c ∈ Λ let m ′ ( c ) be thelargest positive integer such that P c ∈ p m ′ ( c ) E ( K [ c ]) (if P c is torsion then m ′ ( c ) = ∞ ).Define a function m : Λ → Z by m ( c ) = (cid:26) m ′ ( c ) if m ′ ( c ) ≤ M ( c ) , ∞ otherwise.Finally, let m r = min c ∈ Λ r m ( c ). Proposition 3.1.
The sequence { m r } r ≥ is non-increasing, i.e., m r ≥ m r +1 .Proof. This is proved in [Kol91c, Thm.C].
Kolyvagin uses the points P c to construct classes κ c,m ∈ H ( K, E [ p m ]) for any c ∈ Λ m .For the details of the construction, we refer to [Gro91, pp.241-242]) and [McC91, § κ c,m is explicit, in the sense that it is represented by the 1-cocycle σ σ (cid:18) P c p m (cid:19) − P c p m − ( σ − P c p m , (3.1)where ( σ − P c p m is the unique p m -division point of ( σ − P c in E ( K [ c ]) (see [McC91,Lem. 4.1]). The class κ c,m is non-trivial if and only if P c / ∈ p m E ( K [ c ]) (which isequivalent to m > m ( c )).Finally, let − ε be the sign of the functional equation corresponding to E . For each c ∈ Λ m , let ε ( c ) = ε · ( − f c where f c = { ℓ : ℓ | c } (e.g., f = 0). It follows from [Gro91,Prop.5.4(ii)] that κ c,m lies in the ε ( c )-eigenspace for the action of complex conjugationon H ( K, E [ p m ]). We are interested in m ∞ = min c ∈ Λ m ( c ) = lim r →∞ m r . In the case when the Heegner point P = y K has infinite order in E ( K ), the Gross-Zagier formula (see [GZ86]) implies that E ( K ) has rank 1, i.e., m < ∞ as it is ord p ([ E ( K ) : Z y K ]). In that case, m ∞ < ∞ , sothe system of cohomology classes T = { κ c,m : m ≤ M ( c ) } is nonzero. A much more interesting and subtle is the case of an elliptic curves E over K of rank at least 2. Kolyvagin conjectured (see [Kol91a, Conj.C]) that in all cases T is non-trivial. 7 onjecture 3.2 (Kolyvagin’s conjecture) . We have m ∞ < ∞ , i.e., T is non-trivial. Remark 3.3.
Kolyvagin’s conjecture is obvious in the case of elliptic curves of analyticrank one over K since m < ∞ (which follows from Gross-Zagier’s formula). Still,it turns out that the p -part of the Birch and Swinnerton-Dyer conjectural formula isequivalent to m ∞ = ord p Y q | N c q , where c q is the Tamagawa number of E / Q at q .See [Jet07] for some new results related to this question which imply (in many cases)the exact upper bounds on the p -primary part of the Shafarevich-Tate group as predictedby the BSD formula. Theorem 3.4 (Kolyvagin) . Assume Conjecture 3.2 and let f be the smallest nonnega-tive integer for which m f < ∞ . Then Sel p ∞ ( E/K ) ε ( − f +1 ∼ = ( Q p / Z p ) f +1 ⊕ (a finite group)and Sel p ∞ ( E/K ) ε ( − f ∼ = ( Q p / Z p ) r ⊕ (a finite group)where r ≤ f and f − r is even. The above structure theorem of Kolyvagin has the following consequence whichstrongly supports Conjecture 1.3.
Corollary 3.5.
Assume Conjecture 3.2. Then (i) If r an ( E/ Q ) is even and nonzerothen r p ( E/ Q ) ≥ . (ii) If r an ( E/ Q ) is odd and strictly larger than then r p ( E/ Q ) ≥ . Proof. (i) By using [BFH90] or [MM97] one can choose a quadratic imaginary field K = Q ( √− D ), such that the derivative L ′ ( E D/ Q , s ) of the L -function L ( E D/ Q , s ) of thetwist E D of E by the quadratic character associated to K does not vanish at s = 1. Thismeans (by Gross-Zagier’s formula [GZ86]) that the basic Heegner point y K has infiniteorder and thus, by Kolyvagin’s work, the Selmer group Sel p ∞ ( E D / Q ) has corank one,i.e., r − p ( E/K ) = 1. We want to show that r p ( E/K ) ≥
3, i.e., r + p ( E/K ) = r p ( E/ Q ) ≥ r + p ( E/K ) ≤
1. Then, according to Theorem 3.4, r = 0. Since f has the same parity as r , we conclude that f = 0 as well, i.e., the Heegner point y K has infinite order in E ( K ) and hence (by the Gross-Zagier formula) the L -function van-ishes to order 1 which is a contradiction, since by hypothesis r an ( E/ Q ) >
0. Therefore r p ( E/ Q ) = r + p ( E/K ) ≥ K = Q ( √− D ), such that the L -function of the twist E D satisfies L ( E D , = 0. This means that r p ( E D / Q ) = 0, i.e., r + p ( E/K ) = 0. Thus,by Theorem 3.4 we obtain r = 0 and f is even ( r and f are as in Theorem 3.4). If f > r p ( E/K ) ≥
3. If f = 0, we use the same argument asin (i) to arrive at a contradiction. Therefore, r p ( E/ Q ) = r + p ( E/K ) ≥ . .5 Cornut’s theoretical evidence for Kolyvagin’s conjecture The following evidence for Conjecture 3.2 was proven by Christophe Cornut.
Proposition 3.6.
For all but finitely many c ∈ Λ there exists a choice R of liftingsfor the elements of Gal( K [1] /K ) into Gal( K ab /K ) , such that if P c = D D c y c is theHeegner point defined in terms of this choice of liftings (i.e, if D = X σ ∈ R σ ), then P c isnon-torsion. Remark 3.7.
For a nontorsion point P c , let e c denotes the minimal exponent e , suchthat P c / ∈ p e c E ( K [ c ]). Proposition 3.6 gives very little evidence towards the Kolyvaginconjecture. The reason is that even if one gets non-torsion points P c , it might stillhappen that for each such c we have e c > M ( c ) in which case all classes κ c,m with m ≤ M ( c ) will be trivial.Let K [ ∞ ] = [ c ∈ Λ K [ c ]. Lemma 3.8.
The group E ( K [ ∞ ]) tors is finite.Proof. Let q be any prime which is a prime of good reduction for E , which is inert in K and which is different from the primes in Λ . Let q be the unique prime of K over q . It follows from class field theory that the prime q splits completely in K [ ∞ ] since itsplits in each of the finite extensions K [ c ]. Thus, the completion of K [ ∞ ] at any primewhich lies over ℓ is isomorphic to K q and therefore, E ( K [ ∞ ]) tors ֒ → E ( K λ ) tors . The lastgroup is finite since it is isomorphic to an extension of Z q by a finite group (see [Mil86,Lem.I.3.3] or [Tat67, p.168-169]). Therefore, E ( K [ ∞ ] tors ) is finite.Let | E ( K [ ∞ ]) tors | = M < ∞ and let d ( c ) = Y ℓ | c ( ℓ + 1) for any c ∈ Λ. Let m E be the modular degree of E , i.e., the degree of an optimal modular parametrization π : X ( N ) → E . Lemma 3.9.
Suppose that c ∈ Λ satisfies d ( c ) > m E M . There exists a lifting R of Gal( K [1] /K ) in Gal( K [ c ] /K ) , such that D y c / ∈ E ( K [ c ]) tors , where D = X σ ∈ R σ .Proof. The Gal( K [ c ] /K [1])-orbit of the point x c ∈ X ( N )( K [ c ]) consists of d ( c ) distinctpoints, so there are at least d ( c ) /m E elements in the orbit Gal( K [ c ] /K [1]) y c . Choosea set of representatives R of Gal( K [ c ] /K ) / Gal( K [ c ] /K [1]) which contains the identityelement 1 ∈ Gal( K [ c ] /K ). For τ ∈ Gal( K [ c ] /K [1]) define R τ = ( R − { σ } ) ∪ { τ } . Let S = X σ ∈ R σy c and S τ = X σ ∈ R τ σy c . Then S τ − S = σy c − y c , which takes at least d ( c ) /m E > M distinct values. Therefore, there exists an automor-phism τ ∈ Gal( K [ c ] /K [1]), for which S τ / ∈ E ( K [ c ]) tors , which proves the lemma.9 roof of Proposition 3.6. Suppose that c ∈ Λ satisfies the statement of Lemma 3.9,i.e., D y c / ∈ E ( K [ c ]) tors . For any ring class character χ : Gal( K [ c ] /K ) → C × let e χ ∈ C [Gal( K [ c ] /K )] be the eidempotent projector corresponding to χ . Explicitly, e χ = 1 K [ c ] /K ) X σ ∈ Gal( K [ c ] /K ) χ − ( σ ) σ ∈ C [Gal( K [ c ] /K )] . Consider V = E ( K [ c ]) ⊗ C as a complex representation of Gal( K [ c ] /K ). Then thevector D y c ⊗ ∈ V is nontrivial and since V = M χ :Gal( K [ c ] /K ) → C × V χ , then there exists a ring class character χ , such that e χ D ( y c ⊗ = 0 (here, V χ is theeigenspace corresponding to the character χ ). Next, we consider the point D D c y c ∈ E ( K [ c ]).Finally, we claim that D D c y c ⊗ ∈ E ( K [ c ]) ⊗ C is nonzero, which is sufficient toconclude that P c = D D c y c / ∈ E ( K [ c ]) tors . We prove that e χ ( D D c y c ⊗ = 0. Indeed, e χ D D c ( y c ⊗
1) = e χ D c D ( y c ⊗
1) = Y ℓ | c ℓ X i =1 iσ iℓ ! e χ D ( y c ⊗
1) == Y ℓ | c ℓ X i =1 iχ ( σ ℓ ) i ! e χ D ( y c ⊗ , the last equality holding since τ e χ = χ ( τ ) e χ in C [Gal( K [ c ] /K )] for all τ ∈ Gal( K [ c ] /K ).Thus, it remains to compute ℓ X i =1 iχ ( σ ℓ ) i for every ℓ | c . It is not hard to show that ℓ X i =1 iχ ( σ ℓ ) i = ( ℓ +1 χ ( σ ℓ ) − if χ ( σ ℓ ) = 1 ℓ ( ℓ +1)2 if χ ( σ ℓ ) = 1 . Thus, e χ D D c ( y c ⊗ = 0 which means that P c = D D c y c / ∈ E ( K [ c ]) tors for any c satisfying D y c / ∈ E ( K [ c ]) tors . To complete the proof, notice that for all, but finitelymany c ∈ Λ, the hypothesis of Lemma 3.9 will be satisfied.
Consider the example E = with equation y + y = x + x − x . As in Section 2.3,let D = 7, ℓ = 5, and p = 3. Using the algorithm of [GJP + § p Galois representation ρ E,p is surjective. Next, we observe that ℓ = 5 is aKolyvagin prime for E, p and D . Let c = 5 and consider the class κ , ∈ H ( K, E [3]).We claim that κ , = 0 which will verify Kolyvagin’s conjecture. Proposition 3.10.
The class κ , = 0 . In other words, Kolyvagin’s conjecture holdsfor E = , D = 7 and p = 3 . Before proving the proposition, we recall some standard facts about division polyno-mials (see, e.g., [Sil92, Ex.3.7]). For an elliptic curve given in Weierstrass form over any10eld of characteristic different from 2 and 3, y = x + Ax + B, one defines a sequenceof polynomials ψ m ∈ Z [ A, B, x, y ] inductively as follows : ψ = 1 , ψ = 2 y,ψ = 3 x + 6 Ax + 12 Bx − A ,ψ = 4 y ( x + 5 Ax + 20 Bx − A x − ABx − B − A ) ,ψ m +1 = ψ m +2 ψ m − ψ m − ψ m +1 for m ≥ , yψ m = ψ m ( ψ m +2 ψ m − − ψ m − ψ m +1 ) for m ≥ . Define also polynomials φ m and ω m by φ m = xψ m − ψ m +1 ψ m − , yω m = ψ m +2 ψ m − − ψ m − ψ m +1 . After replacing y by x + Ax + B , the polynomials φ m and ψ m can be viewed as poly-nomials in x with leading terms x m and m x m − , respectively. Finally, multiplication-by- m is given by mP = „ φ m ( P ) ψ m ( P ) , ω m ( P ) ψ m ( P ) « . Proof of Proposition 3.10.
We already computed the Heegner point y on the model y + y = x + x − x in Section 2.3. The Weierstrass model for E is y = x − / x +107 / , so A = − / B = 107 / P = P i =1 iσ i ( y ) ∈ E ( K [5]) on the Weierstrass model, where σ is a generator of Gal( K [5] /K ). To showthat κ , = 0 we need to check that there is no point Q = ( x, y ), such that 3 Q = P .For the verification of this fact, we use the division polynomial ψ and the polynomial φ . Indeed, it follows from the recursive definitions that φ ( x ) = x − Ax − Bx + (30 A + 72 B ) x − ABx ++ (36 A + 144 AB − B ) x + 72 A Bx ++ (9 A − A B + 96 AB + 144 B ) x + 8 A B + 64 B . Consider the polynomial g ( x ) = φ ( x ) − X ( P ) ψ ( x ) , where X ( P ) is the x -coordinateof the point P on the Weierstrass model. We factor g ( x ) (which has degree 9) over thenumber field K [5] and check that it is irreducible. In particular, there is no root of g ( x )in K [5], i.e., there is no Q ∈ E ( K [5]), such that 3 Q = P . Thus, κ , = 0. Remark 3.11.
Using exactly the same method as above, we verify Kolyvagin’s conjec-ture for the other two elliptic curves of rank two from Section 2.3. For both E = and E = we use D = 7, p = 3 and ℓ = 5 (which are valid parameters), andverify that κ , = 0 in the two cases. For completeness, we provide all the data of eachcomputation in the three examples in the files , and . Throughout the entire section, let E / Q be a non-CM elliptic curve, K = Q ( √− D ),where D is a Heegner discriminant for E such that the Heegner point y K has infiniteorder in E ( K ) (which, by the Gross-Zagier formula and Kolyvagin’s result, means that E ( K ) has Mordell-Weil rank one) and let p be a prime, such that p ∤ DN and the mod p Galois representation ρ E,p is surjective. It is easy to check that these are polynomials. .1 Non-triviality of Kolyvagin classes. Under the above assumptions, the next proposition provides a criterion which guaranteesthat an explicit class in the Shafarevich-Tate group X ( E/K ) is non-zero.
Proposition 4.1.
Let c ∈ Λ m . Assume that the following hypotheses are satisfied:1. [Selmer hypothesis]: The class κ c,m ∈ H ( K, E [ p m ]) is an element of the Selmergroup Sel p m ( E/K ) .2. [Non-divisibility]: The derived Heegner point P c is not divisible by p m in E ( K [ c ]) ,i.e., P c / ∈ p m E ( K [ c ]) .3. [Parity]: The number f c = { ℓ : ℓ | c } is odd.Then the image κ ′ c,m ∈ H ( K, E )[ p m ] of κ c,m is a non-zero element of X ( E/K )[ p m ] .Proof. The first hypothesis implies that the image κ ′ c,m of κ c,m in H ( K, E )[ p m ] is anelement of the Shafarevich-Tate group X ( E/K ). The second one implies that κ c,m = 0.To show that κ ′ c,m = 0 we use the exact sequence0 → E ( K ) /p m E ( K ) → Sel p m ( E/K ) → X ( E/K )[ p m ] → → ( E ( K ) /p m E ( K )) ± → Sel p m ( E/K ) ± → X ( E/K ) ± [ p m ] → . According to [Gro91, Prop.5.4(2)], the class κ c,m lies in the ε c -eigenspace of the Selmergroup Sel p m ( E/K ) for the action of complex conjugation, where ε c = ε ( − f c = − f c is odd by the third hypothesis and ε = 1 since − ε is the sign of the functional equationfor E /K which is − y K = P lies in the ε -eigenspace of complex conjugation (again, by [Gro91, Prop.5.4(2)]) where ε = ε ( − f = 1. Since E ( K ) has rank one, the group E ( K ) − is torsion and since E ( K )[ p ] = 0, we obtain that ( E ( K ) /p m E ( K )) − = 0. Therefore,Sel p m ( E/K ) − ∼ = X ( E/K ) − [ p m ] , which implies κ ′ c,m = 0. E = The Weierstrass equation for the curve E = is y = x + 405 x + 16038 and E has rank one over Q . The Fourier coefficient a ( f ) ≡ ≡ ℓ = 5 isa Kolyvagin prime for E , the discriminant D = 43 and the prime p = 3. Kolyvagin’sconstruction exhibits a class κ , ∈ H ( K, E [3]). We will prove the following proposition:
Proposition 4.2.
The cohomology class κ , ∈ H ( K, E [3]) lies in the Selmer group
Sel ( E/K ) and its image κ ′ , in the Shafarevich-Tate group X ( E/K ) is a nonzero -torsion element. Remark 4.3.
Since
E/K has analytic rank one, Kolyvagin’s conjecture is automatic(since m < ∞ by Gross-Zagier’s formula) and one knows (see [McC91, Thm. 5.8]) thatthere exist Kolyvagin classes κ ′ c,m which generate X ( E/K )[ p ∞ ]. Yet, this result is notexplicit in the sense that one does not know any particular Kolyvagin class which isnon-trivial. The above proposition exhibits an explicit non-zero cohomology class inthe p -primary part of the Shafarevich-Tate group X ( E/K ).12 roof.
Using the data computed in Section 2.3 for this curve, we apply the Kolyvaginderivative to compute the point P . In order to do this, one needs a generator of theGalois group Gal( K [5] /K ). Such a generator is determined by the image of α , whichwill be another root of f ( x ) in K [5]. We check that the automorphism σ defined by α √− α + 12401980 ( − − √− α ++ 1600495 (34507457 + 40541607 √− α + 14803960 (102487877 − √− α ++ 1400330 ( − √− α + 1200165 (18971815 − √− is a generator (we found this automorphism by factoring the defining polynomial of thenumber field over the number field K [5]). Thus, we can compute P = X i =1 iσ i ( y ).Note that we are computing the point on the Weierstrass model of E rather thanon the original model. The cohomology class κ , is represented by the cocycle σ
7→ − ( σ − P σ P − P which is trivial if and only if P ∈ E ( K [5]). To show that P / ∈ E ( K [5]) we repeatthe argument of Proposition 3.10 and verify (using any factorization algorithm for poly-nomials over number fields) that the polynomial g ( x ) = φ ( x ) − X ( P ) ψ ( x ) has nolinear factors over K [5] (here, X ( P ) is the x -coordinate of P ). This means that thereis no point Q = ( x, y ) ∈ E ( K [5]), such that 3 Q = P , i.e., κ , = 0. Finally, usingProposition 4.1 we conclude that the class κ ′ , ∈ X ( E/K )[3] is non-trivial.
Remark 4.4.
For completeness, all the computational data is provided (with the ap-propriate explanations) in the file . We verified the irreducibility of g ( x ) usingMAGMA and PARI/GP independently. References [BCDT01] C. Breuil, B. Conrad, F. Diamond, and R. Taylor,
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Gross-Zagier formula for GL , Asian J. Math. (2001), no. 2,183–290. y c We explain how to compute an upper bound on the logarithmic height h ( y c ). Themethod first relates the canonical height b h ( y c ) to special values of the first derivativesof certain automorphic L -functions via Zhang’s generalization of the Gross-Zagier for-mula. Then we either compute the special values up to arbitrary precision using awell-known algorithm (recently implemented by Dokchitser) or use effective asymptoticupper bounds (convexity bounds) on the special values and Cauchy’s integral formula.Finally, using some known bounds on the difference between the canonical and the log-arithmic heights, we obtain explicit upper bounds on the logarithmic height h ( y c ). Weprovide a summary of the asymptotic bounds in Section 5.4 and refer the reader to [Jet]for complete details. 15 .1 The automorphic L -functions L ( f, χ, s ) and L ( π, s ) Let d c = c D and let f = X n ≥ a n q n be the new eigenform of level N and weight twocorresponding to E . Let χ : Gal( K [ c ] /K ) → C × be a ring class character.
1. The theta series θ χ . Recall that ideal classes for Pic( O c ) correspond to primitive,reduced binary quadratic forms of discriminants d c . To each ideal class A we considerthe corresponding binary quadratic form Q A and the theta series θ Q A associated to itvia θ Q A = X M e πizQ A ( M ) which is a modular form for Γ ( d c ) of weight one with character ε (the quadratic char-acter of K ) according to Weil’s converse theorem (see [Shi71] for details). This allowsus to define a cusp form θ χ = X A∈ Pic( O c ) χ − ( A ) θ Q A ∈ S (Γ ( d c ) , ε ) . Here, we view χ − as a character of Pic( O c ) via the isomorphism Pic( O c ) ∼ = Gal( K [ c ] /K ).Let θ χ = P m ≥ b m q m be the Fourier expansion. By L ( f, χ, s ) we will always mean theRankin L -function L ( f ⊗ θ χ , s ) (equivalently, the L -function associated to the auto- morphic representation π = f ⊗ θ χ of GL ).
2. The functional equation of L ( f, χ, s ) . We recall some basic facts about the Rankin L -series L ( f ⊗ θ χ , s ) following [Gro84, § III]. Since (
N, D ) = 1, the conductor of L ( f ⊗ θ χ , s )is Q = N d c . The Euler factor at infinity (the gamma factor) is L ∞ ( f ⊗ θ χ , s ) = Γ C ( s ) .If we set Λ( f ⊗ θ χ , s ) = Q s/ L ∞ ( f ⊗ θ χ , s ) L ( f ⊗ θ χ , s )then the function Λ has a holomorphic continuation to the entire complex plane andsatisfies the functional equationΛ( f ⊗ θ χ , s ) = − Λ( f ⊗ θ χ , − s ) . In particular, the order of vanishing of L ( f ⊗ θ χ , s ) at s = 1 is non-negative and odd,i.e., L ( f ⊗ θ χ ,
1) = 0.
3. The shifted L -function L ( π, s ) . In order to center the critical line at Re( s ) = 12instead of Re( s ) = 1 (which is consistent with Langlands convention), we will be lookingat the shifted automorphic L -function L ( π, s ) = L (cid:18) f ⊗ θ χ , s + 12 (cid:19) Moreover, L ( π, s ) satisfies a functional equation relating the values at s and 1 − s . Let L ( π, s ) = X n ≥ λ π ( n ) n s = Y p (1 − α π, ( p ) p − s ) − . . . (1 − α π,d ( p ) p − s ) − be the Dirichlet series and the Euler product of L ( π, s ) (which are absolutely convergentfor Re( s ) > Put a reference for Rankin L -functions! .2 Zhang’s formula For a character χ of Gal( K [ c ] /K ), let e χ = 1 K [ c ] /K ) X σ ∈ Gal( K [ c ] /K ) χ − ( σ ) σ ∈ C [Gal( K [ c ] /K )]be the associated eidempotent. The canonical height b h ( e χ y c ) is related via the general-ized Gross-Zagier formula of Zhang to a special value of the derivative of the L -function L ( f, χ, s ) at s = 1 (see [Zha01, Thm.1.2.1]). More precisely, Theorem 5.1 (Zhang) . If ( , ) denotes the Petersson inner product on S (Γ ( N )) then L ′ ( f, χ,
1) = 4 √ D ( f, f ) b h ( e χ y c ) . Since h e χ ′ y c , e χ ′′ y c i = 0 whenever χ ′ = χ ′′ (here, h , i denotes the N´eron-Tate heightpairing for E ) and since b h ( x ) = h x, x i then b h ( y c ) = b h X χ e χ y c ! = X χ b h ( e χ y c ) . (5.1)Thus, we will have an upper bound on the canonical height b h ( y c ) if we have upperbounds on the special values L ′ ( f, χ,
1) for every character χ of Gal( K [ c ] /K ). L -functions For simplicity, let γ ( s ) = L ∞ ( f ⊗ θ χ , s + 1 /
2) be the gamma factor of the L -function L ( π, s ). This means that if λ ( π, s ) = Q s/ γ ( s ) L ( π, s ) then Λ( π, s ) satisfies the functionalequation Λ( π, s ) = Λ( π, − s ). We will describe a classical algorithm to computethe value of L ( k ) ( π, s ) at s = s up to arbitrary precision. The algorithm and itsimplementation is discussed in a greater generality in [Dok04]. The main idea is toexpress Λ( π, s ) as an infinite series with rapid convergence which is usually done in thefollowing sequence of steps:1. Consider the inverse Mellin transform of the gamma factor γ ( s ), i.e., the function φ ( t ) which satisfies γ ( s ) = Z ∞ φ ( t ) t s dtt . One can show (see [Dok04, § φ ( t ) decays exponentially for large t . Hence,the sum Θ( t ) = ∞ X n =1 λ π ( n ) φ (cid:18) nt √ Q (cid:19) converges exponentially fast. The function φ ( t ) can be computed numerically asexplained in [Dok04, § t ) is exactly the function Λ( π, s ). Indeed, Z ∞ Θ( t ) t s dtt = Z ∞ ∞ X n =1 λ π ( n ) φ (cid:18) nt √ Q (cid:19) t s dtt = ∞ X n =1 λ π ( n ) Z ∞ φ (cid:18) nt √ Q (cid:19) t s dtt == ∞ X n =1 λ π ( n ) (cid:18) √ Qn (cid:19) s Z ∞ φ ( t ′ ) t ′ s dt ′ t ′ = Q s/ γ ( s ) L ( π, s ) = Λ( π, s ) .
17. Next, we obtain a functional equation for Θ( t ) which relates Θ( t ) to Θ(1 /t ). In-deed, since Λ( π, s ) is holomorphic, Mellin’s inversion formula implies thatΘ( t ) = Z c + i ∞ c − i ∞ Λ( π, s ) t − s ds, ∀ c. Therefore,Θ(1 /t ) = Z c + i ∞ c − i ∞ Λ( π, s )(1 /t ) − s ds = − t Z c + i ∞ c − i ∞ Λ( π, − s ) t − (1 − s ) ds == − t Z c + i ∞ c − i ∞ Λ( π, s ′ ) t − s ′ ds ′ = − t Θ( t ) . Thus, Θ( t ) satisfies the functional equation Θ(1 /t ) = − t Θ( t ).4. Next, we consider the incomplete Mellin transform G s ( t ) = t − s Z ∞ t φ ( x ) x s dxx , t > φ ( t ). The function G s ( t ) satisfies lim t → t s G s ( t ) = γ ( s ) and it decays exponentially.Moreover, it can be computed numerically (see [Dok04, § t ) to obtainΛ( π, s ) = Z ∞ Θ( t ) t s dtt = Z Θ( t ) t s dtt + Z ∞ Θ( t ) t s dtt == Z ∞ Θ(1 /t ′ ) t ′− s dt ′ t ′ + Z ∞ Θ( t ) t s dtt == − Z ∞ Θ( t ′ ) t ′ − s dt ′ t ′ + Z ∞ Θ( t ) t s dtt .
6. Finally, we compute Z ∞ Θ( t ) t s dtt = Z ∞ ∞ X n =1 λ π ( n ) φ (cid:18) nt √ Q (cid:19) t s dtt = ∞ X n =1 λ π ( n ) Z ∞ φ (cid:18) nt √ Q (cid:19) t s dtt == ∞ X n =1 λ π ( n ) Z ∞ n √ Q φ ( t ′ ) (cid:18) √ Qt ′ n (cid:19) s = ∞ X n =1 λ π ( n ) G s (cid:18) n √ Q (cid:19) . Thus, Λ( π, s ) = ∞ X n =1 λ π ( n ) G s (cid:18) n √ Q (cid:19) − ∞ X n =1 λ π ( n ) G − s (cid:18) n √ Q (cid:19) is the desired expansion. From here, we obtain a formula for the k -th derivative ∂ k ∂s k Λ( π, s ) = ∞ X n =1 λ π ( n ) ∂ k ∂s k G s (cid:18) n √ Q (cid:19) − ∞ X n =1 λ π ( n ) ∂ k ∂s k G − s (cid:18) n √ Q (cid:19) . The computation of the derivatives of G s ( x ) is explained in [Dok04, § .4 Asymptotic estimates on the canonical heights b h ( y c ) In this section we provide an asymptotic bound on the canonical height b h ( y c ) by usingconvexity bounds on the special values of the automorphic L -functions L ( π, s ) defined inSection 5.1. We only outline the basic techniques used to prove the asymptotic boundsand refer the reader to [Jet] for the complete details. Asymptotic bounds on heights ofHeegner points are obtained in [RV], but these bounds are of significantly different typethan ours. In our case, we fix the elliptic curve E and let the fundamental discriminant D and the conductor c of the ring class field both vary. The result that we obtain is thefollowing Proposition 5.2.
Fix the elliptic curve E and let the fundamental discriminant D andthe conductor c vary. For any ε > the following asymptotic bound holds b h ( y c ) ≪ ε,f h D D ε c ε , where h D is the class number of the quadratic imaginary field K = Q ( √− D ) . Moreover,the implied constant depends only on ε and the cusp form f . One proves the proposition by combining the formula of Zhang with convexitybounds on special values of automorphic L -functions. The latter are conveniently ex-pressed in terms of a quantity known as the analytic conductor associated to the auto-morphic representation π (see [Mic02, p.12]). It is a function Q π ( t ) over the real line,which is defined as Q π ( t ) = Q · d Y i =1 (1 + | it − µ π,i | ) , ∀ t ∈ R , where µ π,i are obtained from the gamma factor L ∞ ( π, s ) = d Y i =1 Γ R ( s − µ π,i ) , Γ R ( s ) = π − s/ Γ( s/ . In our situation, d = 4 and µ π, = µ π, = 0, µ π, = µ π, = 1 (see [Mic02, § §
3] for discussions of local factors at archimedian places). Moreover, we let Q π = Q π (0).The main idea is to prove that for a fixed f , | L ′ ( π f ⊗ θ χ , / | ≪ ε,f Q / επ f ⊗ θχ , where theimplied constant only depends on f and ε (and is independent of χ and the discriminant D ). To establish the bound, we first prove an asymptotic bound for the L -function L ( π f ⊗ θ χ , s ) on the vertical line Re( s ) = 1 + ε by either using the Ramanujan-Peterssonconjecture or a method of Iwaniec (see [Mic02, p.26]). This gives us the estimate | L ( π f ⊗ θ χ , ε + it ) | ≪ ε,f Q π f ⊗ θχ ( t ) ε . Then, by the functional equation for L ( π f ⊗ θ χ , s )and Stirling’s approximation formula, we deduce an upper bound for the L -functionon the vertical line Re( s ) = − ε , i.e., | L ( π f ⊗ θ χ , − ε + t ) | ≪ ε,f Q π f ⊗ θχ ( t ) / ε . Next,we apply Phragmen-Lindel¨of’s convexity principle (see [IK04, Thm.5.53]) to obtain thebound | L ( π f ⊗ θ χ , / it ) | ≪ ε,f Q π ( t ) / ε (also known as convexity bound ). Finally,by applying Cauchy’s integral formula for a small circle centered at s = 1 /
2, we obtainthe asymptotic estimate | L ′ ( π f ⊗ θ χ , / | ≪ ε,f Q / επ f ⊗ θχ . Since Q = N d c = N D c inour situation and since [ K [ c ] : K ] = h D Q ℓ | c ( ℓ + 1), Zhang’s formula (Theorem 5.1) andequation (5.1) imply that for any ε > b h ( y c ) ≪ ε,f h D D ε c ε . emark 5.3. In the above situation (the Rankin-Selberg L -function of two cusp formsof levels N and d c = c D ), one can even prove a subconvexity bound | L ′ ( π f ⊗ θ χ , / | ≪ f D / − / c − / , where the implied constant depends only on f and is independentof χ (see [Mic04, Thm.2]). Yet, the proof relies on much more involved analytic numbertheory techniques than the convexity principle, so we do not discuss it here. To estimate h ( y c ) we need a bound on the difference between the canonical and thelogarithmic heights. Such a bound has been established in [Sil90] and [CPS06] and iseffective.Let F be a number field. For any non-archimedian place v of K , let E ( F v ) denotethe points of E ( F v ) which specialize to the identity component of the N´eron model of E over the ring of integers O v of F v . Moreover, let n v = [ F v : Q v ] and let M ∞ F denote theset of all archimedian places of F . A slightly weakened (but easier to compute) boundson the height difference are provided by the following result of [CPS06, Thm.2] Theorem 5.4 (Cremona-Prickett-Siksek) . Let P ∈ E ( F ) and suppose that P ∈ E ( F v ) for every non-archimedian place v of F . Then F : Q ] X v ∈ M ∞ F n v log δ v ≤ h ( P ) − b h ( P ) ≤ F : Q ] X v ∈ M ∞ F n v log ε v , where ε v and δ v are defined in [CPS06, § Remark 5.5.
All of the points y c in our particular examples satisfies the condition y c ∈ E ( K [ c ] v ) for all non-archimedian places v of K [ c ]. Indeed, according to [GZ86, § III.3] (see also [Jet07, Cor.3.2]) the point y c lies in E ( K [ c ] v ) up to a rational torsionpoint . Since E ( Q ) tor is trivial for all the curves that we are considering, the aboveproposition is applicable. In general, one does not need this assumption in order tocompute height bounds (see [CPS06, Thm.1] for the general case). Remark 5.6.
A method for computing ε v and δ v up to arbitrary precision for real andcomplex archimedian places is provided in [CPS06, § See also [Jet07] for another application of this local property of the points y c ..