STNB 2016 (30th edition)
An extension of the Faltings-Serre method.
Presenters
Abstract
Faltings showed that a finite amount of computation is
enough to decide whether two $\ell$-adic representations
of the absolute Galois group of a number field K into $GL_n(\mathbb{Z}_p)$ are isomorphic.
Serre turned this into a practical algorithm and when $n=p=2$
and applied it to show there is only one isogeny class of elliptic curves
of conductor 11. Later Dieulefait--Guerberoff--Pacetti fully automatized
and implemented Serre's ideas and applied it to prove modularity
of many elliptic curves over imaginary quadratic fields.
In this talk we will discuss possible extensions of the Faltings-Serre
method like working with $p > 3$ or $n=4$, motivated by applications to
modularity of abelian surfaces. This is ongoing work with Lassina Dembele
and Luis Dieulefait.
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