Theoretical Statistics and Mathematics Unit, ISI Delhi

October 14, 2015 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Devika Sharma,
Indian Statistical Institute, Delhi

Title:
Modular Galois representations

Abstract of Talk

Let $p$ be a prime and let $f$ be a modular form. Let $\rho_f$ be the two dimensional $p$-adic Galois representation attached to $f$. We are interested in the (local) behaviour of $\rho_f$ when $f$ is a $p$-ordinary form of weight at least $2$. A result of Wiles and Mazur-Wiles says that when $f$ is $p$-ordinary, $\rho_f$ restricted to the decomposition group $G_p$ at $p$ is reducible.
Greenberg asked the natural question; when does $\rho_f \lvert_{G_p}$ split?
It is not too hard to see that $\rho_f \lvert_{G_p}$ splits if $f$ has complex multiplication (CM). In this talk, we will discuss the converse, i.e.,
$$ \rho_f \lvert_{G_p} \hbox{ splits} \stackrel{?}{\Longrightarrow} f \hbox{ has CM}. $$
We use deformation theory of Galois representations and the theory of $p$-adic families of modular forms (Hida families) to generate various non-trivial examples in support of the converse. I will describe these ideas in sufficient detail. This is joint work with Eknath Ghate.