Faltings-Serre method on three dimensional selfdual representations

Faltings Serre method

The starting point of this method is Falting’s article in which he proves the Mordell-Weil theorem. He remarked and Serre turned it into a working method, the fact that the equivalence of two λ-adic representations is something that can be basically determined on some finite extension of the base field (even though the representations might not factor through a finite quotient). Let Oλ be the r...

A Refinement of the Faltings-serre Method

In recent years the classification of elliptic curves over Q of various conductors has been attempted. Many results have shown that elliptic curves of a certain conductor do not exist. Later methods have concentrated on small conductors, striving to find them all and hence to verify the Shimura-Taniyama-Weil conjecture for those conductors. A typical case is the conductor 11. In [1], Agrawal, C...

Selfdual and Non-selfdual 3-dimensional Galois Representations

On three-dimensional Galois representations

We give an irreducibility criterion for the Galois images of compatible l-adic systems of Galois representations. We apply this to the l-adic realizations of the van Geemen-Top motive M and prove an analogue of Serre’s theorem for M.

Serre Weights for Locally Reducible Two-dimensional Galois Representations

Let F be a totally real field, and v a place of F dividing an odd prime p. We study the weight part of Serre’s conjecture for continuous totally odd representations ρ : GF → GL2(Fp) that are reducible locally at v. Let W be the set of predicted Serre weights for the semisimplification of ρ|GFv . We prove that when ρ|GFv is generic, the Serre weights inW for which ρ is modular are exactly the on...