Thanasis Bouganis

Algebraicity of L-values for elliptic curves in a false Tate curve tower.

Let E be an elliptic curve over , and τ an Artin representation over that factors through the non-abelian extension , where p is an odd prime and n, m are positive integers. We show that L(E,τ,1), the special value at s=1 of the L-function of the twist of E by τ, divided by the classical transcendental period Ω+ d+ |Ω− d− |e(τ) is algebraic and Galois-equivariant, as predicted by Delignes conjecture.

A geometric view of decoding AG codes

We investigate the use of vector bundles over finite fields to obtain a geometric view of decoding algebraic-geometric codes. Building on ideas of Trygve Johnsen, who revealed a connection between the errors in a received word and certain vector bundles on the underlying curve, we give explicit constructions of the relevant geometric objects and efficient algorithms for some general computations needed in the constructions. The use of vector bundles to understand decoding as a geometric process is the first application of these objects to coding theory.

On Special L-Values Attached to Siegel Modular Forms

In his admirable book “Arithmeticity in the Theory of Automorphic Forms” Shimura establishes various algebraicity results concerning special values of Siegel modular forms. These results are all stated over an algebraic closure of \(\mathbb{Q}\). In this article we work out the field of definition of these special values. In this way we extend some previous results obtained by Sturm, Harris, Panchishkin, and Bocherer-Schmidt.

Error correcting codes over algebraic surfaces

We study error correcting codes over algebraic surfaces. We give a construction of linear error correcting codes over an arbitrary algebraic surface and then we focus on linear codes over ruled surfaces. At the end we discuss another approach to getting codes over algebraic surfaces using sections of rank two bundles. The new codes are not linear but do have a group structure.

p-adic Measures for Hermitian Modular Forms and the Rankin–Selberg Method

In this work we construct p-adic measures associated to an ordinary Hermitian modular form using the Rankin–Selberg method.

The Möbius–Wall congruences for p-adic L-functions of CM elliptic curves.

In this paper we prove, under a technical assumption, the so-called “Mobius–Wall” congruences between abelian p-adic L-functions of CM elliptic curves. These congruences are the analogue of those shown by Ritter and Weiss for the Tate motive, and offer strong evidences in favor of the existence of non-abelian p-adic L-functions for CM elliptic curves.