General SectionKolyvagin derivatives of modular points on elliptic curves
Introduction
Let be an elliptic curve of conductor . Then due to the modularity theorem, there exists a surjective morphism defined over known as the modular parametrisation of E, where ∞ on the modular curve is mapped to O. There exists a subvariety of which is a moduli space of points where A is an elliptic curve and C is a cyclic subgroup of A of order . Fixing , the image of under is known as a modular point, which we will denote where is the field of definition of C. We denote the compositum of all such as . This is the smallest field K such that its absolute Galois group acts by scalars on . This has Galois group which can be identified as a subgroup of via the mapping
We can also define higher modular points above . These are points of the form for an elliptic curve B isogenous to A over and D a cyclic subgroup of B of order .
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 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 be an elliptic curve of conductor . Let be an elliptic curve such that the j invariant of A is not in and the degree of any isogeny of A defined over is coprime to . Then the modular points are of infinite order for all cyclic subgroups C of order in A.
From this, we can show that if p is a prime with specific conditions related to A and E, then is not divisible by p in as seen in Proposition 3.4. We also see that there exist relationships between the modular points. If d is a divisor of and B is a cyclic subgroup of A of order d, then 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 . Let D be a cyclic subgroup of A of order for a prime p and . We look at the higher modular point coming from where is the isogeny defined by D. We define . We see that the higher modular points form a trace-compatible system with where the sum is taken over the subgroups of A of order containing D and is the p-th Fourier coefficient for the modular form associated to the isogeny class of E. Using this, we can show that if 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 as seen in Proposition 4.3 and so if defines the compositum of all such , then rank( 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 for , we will see that we can relate the group generated by the higher modular points to certain -lattices in for p prime and a Borel subgroup of . We can view where . This contains the standard lattice which we can easily understand the cohomology of with respect to subgroups of . Hence, we take a look at the integral representation theory of 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 as shown in Proposition 3.4.
Suppose p is one of the following:
- •
A prime of split multiplicative reduction for E and ;
- •
A prime of non-split multiplicative reduction for E;
- •
A non-anomalous prime of good ordinary reduction for E,
Theorem Let and be elliptic curves of conductor and respectively. Let be a prime of multiplicative reduction for A such that and let be as defined above. Assume that: A is semistable; E has either split multiplicative reduction at p with , non-split multiplicative reduction at p, or good ordinary non-anomalous reduction at p; The degree of any isogeny of A defined over is coprime to ; is surjective; is surjective; Any prime ℓ of bad reduction for E and good reduction for A has square modulo p.
Then there exists an element of order in .
We are able to prove that if the first and fourth conditions in the theorem are true, then is surjective and so for all . The fifth point is essential in order to ensure for all . 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 are of infinite order. This condition is weak as the number of such isogenies is finite.
The final point ensures that the primes dividing but not split completely in . This is fundamental when showing that the image of the derivation map 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 is trivial or not in . This will require a more in-depth look at the derivation map to see whether this mapping is injective or not. Also, we still do not fully understand all potential -lattices that could be isomorphic to. Further research into the integral representation theory of 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 an integer, we let be the m-torsion subgroup of . We have acting on where is the absolute Galois group of K. This leads to a Galois representation Let be the p-adic Tate module of E for a prime p. Then acts on which leads to the Galois representation We define the mapping where is the quotient
Modular points on elliptic curves
In this section, we will see when the points are of infinite order. Further, when the prime p satisfies certain conditions with respect to A and E, we will show that the points are not divisible by p in . We prove the following.
Theorem 3.1 Let and be elliptic curves of conductor and respectively. Suppose the j invariant of A is not in . Then there exists a cyclic subgroup of order in A, denoted C, such that 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 .
Let and be elliptic
Representations of
We are interested in the representations that appear in the study of the higher modular points. We have seen that when is surjective, the group generated by the higher modular points are isomorphic to the -module or depending on the reduction type of p with respect to E. In this section, we will prove the following.
Theorem 5.1 Let be the standard -lattice in and let U be a -lattice of such that . Then .
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 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 -lattice the
Examples
We now look at applying Theorem 6.13 to a few examples. Due to the size of the field extensions , 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 Let be the elliptic curve with Cremona label and be the elliptic curve with Cremona label . Then if we let , this is a prime of good ordinary non-anomalous reduction for E. We also see that as , 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)
Geometric Algebra
(1957)- et al.
Similarity classes of matrices over a local principal ideal ring
Commun. Algebra
(2009) Representations and cohomology. I
- et al.
Self-points on elliptic curves of prime conductor
Int. J. Number Theory
(2009) - et al.
The splitting of primes in division fields of elliptic curves
Exp. Math.
(2003) Iwasawa theory, projective modules, and modular representations
Mem. Am. Math. Soc.
(2011)