Christian Wuthrich

Algorithms for the arithmetic of elliptic curves using Iwasawa theory

We explain how to use results from Iwasawa theory to obtain information about p-parts of Tate-Shafarevich groups of specific elliptic curves over Q. Our method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when the rank of the Mordell-Weil group is greater than 1. We apply our results along with a computer calculation to show that X(E/Q)[p] = 0 for the 1,534,422 pairs (E, p) consisting of a non-CM elliptic curve E over Q with conductor ≤ 30,000, rank ≥ 2, and good ordinary primes p with 5 ≤ p < 1000 and surjective mod-p representation.

Iwasawa theory of the fine Selmer group

The fine Selmer group of an elliptic curve E over a number field K is obtained as a subgroup of the usual Selmer group by imposing stronger conditions at places above p. We prove a formula for the Euler-characteristic of the fine Selmer group over a Zp-extension and use it to compute explicit examples.

A moduli interpretation for the non-split Cartan modular curve

Modular curves like X_0(N) and X_1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL_2(Z), they allow for a more arithmetic description as a solution to a moduli problem. This description turns out to be very useful in many applications. We wish to give such a moduli description for two modular curves: those associated to non-split Cartan subgroups and their normaliser in GL_2(F_p). These modular curves appear for instance in Serres problem of classifying all possible Galois structures of p-torsion points on elliptic curves over number fields. Some classical results about the geometry of those curves can be proven using this moduli description. For instance, we can count the number of elliptic points, describe the cusps and the degeneracy maps. We also give a moduli-theoretic interpretation and a new proof of a result of Chen.

Vanishing of some galois cohomology groups for elliptic curves

Let \(E/\mathbb {Q}\) be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of \(\mathbb {Q}\) obtained by adjoining the coordinates of the p-torsion points on E. We determine all cases when the Galois cohomology group \(H^1\bigl ( G, E[p]\bigr )\) does not vanish, and investigate the analogous question for \(E[p^i]\) when \(i>1\). We include an application to the verification of certain cases of the Birch and Swinnerton-Dyer conjecture, and another application to the Grunwald–Wang problem for elliptic curves.

Parity conjectures for elliptic curves over global fields of positive characteristic

We prove the

Numerical modular symbols for elliptic curves

p

SELF-POINTS ON ELLIPTIC CURVES OF PRIME CONDUCTOR

-parity conjecture for elliptic curves over global fields of characteristic

The fine Tate¿Shafarevich group

p > 3

Overview of Some Iwasawa Theory

. We also present partial results on the

On the Integrality of Modular Symbols and Kato's Euler System for Elliptic Curves

\ell