Elsevier

Journal of Number Theory

Volume 229, December 2021, Pages 405-431
Journal of Number Theory

General Section
Kolyvagin derivatives of modular points on elliptic curves

https://doi.org/10.1016/j.jnt.2020.10.014Get rights and content

Abstract

Let E/Q and A/Q be elliptic curves. We can construct modular points derived from A via the modular parametrisation of E. With certain assumptions we can show that these points are of infinite order and are not divisible by a prime p. In particular, using Kolyvagin's construction of derivative classes, we can find elements in certain Shafarevich-Tate groups of order pn.

Introduction

Let E/Q be an elliptic curve of conductor NE. Then due to the modularity theorem, there exists a surjective morphismϕE:X0(NE)E defined over Q known as the modular parametrisation of E, where ∞ on the modular curve X0(NE) is mapped to O. There exists a subvariety Y0(NE) of X0(NE) which is a moduli space of points xA,C=(A,C) where A is an elliptic curve and C is a cyclic subgroup of A of order NE. Fixing A/Q, the image of xA,C under ϕE is known as a modular point, which we will denote PA,CE(Q(C)) where Q(C) is the field of definition of C. We denote the compositum of all such Q(C) as KNE. This is the smallest field K such that its absolute Galois group GK acts by scalars on A[NE]. This has Galois group GNE:=Gal(KNE/Q) which can be identified as a subgroup of PGL2(Z/NEZ) via the mappingτQ,A,NE:GQAut(A[NE])GL2(Z/NEZ)PGL2(Z/NEZ).

We can also define higher modular points above PA,C. These are points of the form ϕE(B,D) for an elliptic curve B isogenous to A over Q and D a cyclic subgroup of B of order NE.

In this paper, we use these points to bound Selmer groups using methods similar to that of Kolyvagin in [10] and Wuthrich in [13]. Kolyvagin initially looked at a specific type of modular point known as Heegner points and used them to bound Selmer groups. This involved creating cohomology classes coming from these points and using the classes to bound the Selmer groups from above. Wuthrich then worked on an analogue system to Kolyvagin's work where he uses a type of modular point known as self points to create derivative classes and finds lower bounds of Selmer groups over certain fields.

We look to extend the idea of self points to a generalised construction of modular point. In particular, we use more advanced methods in modular representation theory to show that Selmer groups over certain fields must contain points of prime power order when the higher modular points satisfy certain conditions.

In the first section, we look at the divisibility of the modular points in E(Q(C)) and understand the relationships between them. This will provide us with an understanding of the rank of the group generated by these points. Initially, we want to see when the modular points are of infinite order. We obtain the following result.

Theorem

Let E/Q be an elliptic curve of conductor NE. Let A/Q be an elliptic curve such that the j invariant of A is not in 12Z and the degree of any isogeny of A defined over Q is coprime to NE. Then the modular points PA,C are of infinite order for all cyclic subgroups C of order NE in A.

From this, we can show that if p is a prime with specific conditions related to A and E, then PA,C is not divisible by p in E(Q(C)) as seen in Proposition 3.4. We also see that there exist relationships between the modular points. If d is a divisor of NE and B is a cyclic subgroup of A of order d, thenCBPA,CE(Kd) is torsion. This reduces the rank of the group generated by these points.

We then take a look at a specific case of creating higher modular points for a prime p of either good ordinary or multiplicative reduction with respect to E. Here, we will look at the case where p is coprime to NE. Let D be a cyclic subgroup of A of order pn+1 for a prime p and n0. We look at the higher modular point coming from (A/D,ψ(C)) where ψ:AA/D is the isogeny defined by D. We define QA,D=ϕE(A/D,ψ(C))E(Q(C,D)). We see that the higher modular points form a trace-compatible system withap(E)QA,D=DDQA,D where the sum is taken over the subgroups D of A of order pn+2 containing D and ap(E) is the p-th Fourier coefficient for the modular form associated to the isogeny class of E. Using this, we can show that if PA,C is of infinite order, then we have created a tower of points which are also of infinite order. In particular, we show that the higher modular points generate a group of rank pn+1+pn as seen in Proposition 4.3 and so if Fn defines the compositum of all such Q(C,D), then rank(E(Fn))pn+1+pn by [7, Corollary 3.7]. Hence we are able to use this information to establish a link between the group generated by the higher modular points and the representation theory associated to the projective general linear group.

If we let Gn:=PGL2(Z/pn+1Z) for n0, we will see that we can relate the group generated by the higher modular points to certain Zp[Gn]-lattices inVn=ker(Qp[/BnGn]Qp) for p prime and Bn a Borel subgroup of Gn. We can viewVn={f:Pn1Qp:Cf(C)=0} where Pn1:=P1(Z/pn+1Z). This contains the standard latticeTn=ker(Zp[/BnGn]Zp), which we can easily understand the cohomology of with respect to subgroups of Gn. Hence, we take a look at the integral representation theory of Gn and will later look at the application of this to the modular points.

We then follow a similar method to Wuthrich in [13] from the ideas of Kolyvagin in [10]. We create derivative classes coming from higher modular points of infinite order which are not divisible by p in E(Q(C)) as shown in Proposition 3.4.

Suppose p is one of the following:

  • A prime of split multiplicative reduction for E and pordp(ΔE);

  • A prime of non-split multiplicative reduction for E;

  • A non-anomalous prime of good ordinary reduction for E,

where ΔE is the minimal discriminant of E. This ensures that the higher modular points are not divisible by p in E(Q(C)). LetFn:={Kpn+1NEif pNE,KpnNEif p||NE, for n1 with F:=F1. We assumeτF,A,p:Gal(F/F)PGL2(Zp) is surjective giving Gal(Fn/F)Gn. We let An be a non-split Cartan subgroup of Gn. This is a cyclic subgroup of order pn+1+pn. Then we define Ln to be the subfield of Fn fixed by An. We are able to construct a mapping where Sn denotes the saturated group generated by the higher modular points in E(Fn). With the conditions on p, we can show that the source of δn is a cyclic group of order pn. This result derives from the link between the construction of the derivative classes and integral representation theory. The group Sn defined earlier is isomorphic to a Zp[Gn]-lattice containing Tn. Due to the structure of Sn, we are able to show that SnTn under the conditions we have stated and as we understand the cohomology of Tn associated to subgroups of Gn. This leads to the following.

Theorem

Let E/Q and A/Q be elliptic curves of conductor NE and NA respectively. Let p>3 be a prime of multiplicative reduction for A such that ordp(ΔA)=1 and let Fn be as defined above. Assume that:

  • 1.

    A is semistable;

  • 2.

    E has either split multiplicative reduction at p with pordp(ΔE), non-split multiplicative reduction at p, or good ordinary non-anomalous reduction at p;

  • 3.

    The degree of any isogeny of A defined over Q is coprime to NE;

  • 4.

    ρQ,A,p:GQAut(A[p])GL2(Fp) is surjective;

  • 5.

    ρQ,E,p:GQAut(E[p])GL2(Fp) is surjective;

  • 6.

    Any prime ℓ of bad reduction for E and good reduction for A has a(A)24 square modulo p.

Then there exists an element of order pn in Selpn(E/Ln).

We are able to prove that if the first and fourth conditions in the theorem are true, then τF,A,p is surjective and so Gal(Fn/F)Gn for all n0. The fifth point is essential in order to ensure E(Fn)[p]=0 for all n0. We know that this condition, together with the fourth condition, excludes only a finite number of primes if A and E have no complex multiplication. Therefore, these two conditions aren't very strong. The third condition is vital to ensure all the points PA,C are of infinite order. This condition is weak as the number of such isogenies is finite.

The final point ensures that the primes dividing NE but not NA split completely in Fn/Ln. This is fundamental when showing that the image of the derivation map δn lies in

. In particular, as this is only focusing on primes of bad reduction for E and good reduction for A, this condition only looks at a finite number of primes.

However, we are unsure as to whether the image of this element in Selpn(E/Ln) is trivial or not in

. This will require a more in-depth look at the derivation map δn to see whether this mapping is injective or not. Also, we still do not fully understand all potential Zp[Gn]-lattices that Sn could be isomorphic to. Further research into the integral representation theory of Zp[Gn] would improve our understanding of the structure of the saturated group of higher modular points and further still, understand the properties of the derivative classes constructed.

Section snippets

Preliminaries

Let K be a number field. For an elliptic curve E over K and m>1 an integer, we let E[m] be the m-torsion subgroup of E(K). We have GK acting on E[m] where GK:=Gal(K/K) is the absolute Galois group of K. This leads to a Galois representationρK,E,m:GKAut(E[m])GL2(Z/mZ). Let TpE=limnE[pn] be the p-adic Tate module of E for a prime p. Then GK acts on TpE which leads to the Galois representationρK,E,p:GKAut(TpE)GL2(Zp). We define the mapping τK,E,m:=smρK,E,m where sm is the quotient

Modular points on elliptic curves

In this section, we will see when the points PA,C are of infinite order. Further, when the prime p satisfies certain conditions with respect to A and E, we will show that the points PA,C are not divisible by p in E(Q(C)). We prove the following.

Theorem 3.1

Let A/Q and E/Q be elliptic curves of conductor NA and NE respectively. Suppose the j invariant of A is not in 12Z. Then there exists a cyclic subgroup of order NE in A, denoted C, such that PA,CE(Qp) is non-torsion.

Proof

The following has been adapted from

Higher modular points

We now extend the construction of modular points to higher modular points by following the construction as in [13] in the more generalised setting. We will create a tower of number fields in which the higher modular points will be defined and will show that they satisfy certain trace relations similar to that in Kolyvagin's construction of Heegner points. In particular, the relations between the points show that a link can be established with representations of Gn.

Let E/Q and A/Q be elliptic

Representations of PGL2(Z/pn+1Z)

We are interested in the representations that appear in the study of the higher modular points. We have seen that when τF,A,p is surjective, the group generated by the higher modular points are isomorphic to the Qp[Gn]-module V(n) or Vn depending on the reduction type of p with respect to E. In this section, we will prove the following.

Theorem 5.1

Let Tn:=ker(Zp[/BnGn]Zp) be the standard Zp[Gn]-lattice in Vn and let U be a Zp[Gn]-lattice of Vn such that UBn=TnBn. Then UTn.

This will be crucial in the

Derivatives

We now look at creating derivative classes associated to the higher modular points with the method proposed by Wuthrich in [13] in order to create points of prime power order in certain Selmer groups. To do this, we will use the link created between the higher modular points and representations of Gn in section 4 as well as Theorem 5.1. We will show that under the conditions of the prime p with respect to A and E as outlined in Proposition 3.4, we understand the type of Zp[Gn]-lattice the

Examples

We now look at applying Theorem 6.13 to a few examples. Due to the size of the field extensions Ln/Q, it would be very difficult to verify the calculations in the examples. For information on specific elliptic curves, we obtained our data from [12].

Example

LetE:y2+y=x3x2x2 be the elliptic curve with Cremona label 143a1 andA:y2+y=x3+x2x be the elliptic curve with Cremona label 35a3. Then if we let p=7, this is a prime of good ordinary non-anomalous reduction for E. We also see that as ΔA=35, then 7

Acknowledgments

I would like to thank Chris Wuthrich for all his help on this topic.

This work was supported by the Engineering and Physical Sciences Research Council (Grant No. EP/N50970X/1).

References (13)

  • E. Artin

    Geometric Algebra

    (1957)
  • N. Avni et al.

    Similarity classes of 3×3 matrices over a local principal ideal ring

    Commun. Algebra

    (2009)
  • D.J. Benson

    Representations and cohomology. I

  • C. Delaunay et al.

    Self-points on elliptic curves of prime conductor

    Int. J. Number Theory

    (2009)
  • W. Duke et al.

    The splitting of primes in division fields of elliptic curves

    Exp. Math.

    (2003)
  • R. Greenberg

    Iwasawa theory, projective modules, and modular representations

    Mem. Am. Math. Soc.

    (2011)
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