PUBLIC VIVA-VOCE NOTIFICATION
Subject : Mathematics
Date and Time : Wednesday, 20 July 2022, 10.00 am
Venue : Google meet (online mode)
Finiteness Theorems for potentially equivalent Galois representations: extensions of Faltings's finiteness criteria
Plawan Das
Chennai Mathematical Institute.
20-07-22

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Abstract

We will discuss the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $\rho$ is determined by an equality of an $m$-power character $g \mapsto \operatorname{Tr}\left(\rho\left(g^{m}\right)\right)$ for some natural number $m$. Using this, we extend Faltings' finiteness criteria to determine the equivalence of two $\ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We will also discuss finiteness results for twist unramified representations. We will prove that up to potential isogeny, there are only finitely many abelian varieties defined over a number field $K$ of dimension $g$, such that for any finite place $v$ outside a fixed set of places $S$ of $K$, it has either good reduction or totally bad reduction and good reduction over a quadratic extension of the completion of $K$ at $v$.