>3. Vytautas Paskunas (Univ. Duisburg-Essen)

> 

>14:00–15:00

> 

>Title: An application of a patched module (after Shen-Ning Tung)

> 

>Abstract: Let $F$ be a finite extension of $\mathbb{Q}_p$. Together with Caraiani, Emerton, Gee, Geraghty and Shin we have constructed a patched module $M_{\infty}$ which has commuting actions by $GL_n(F)$ and a deformation ring of an $n$-dimensional Galois representation of the absolute Galois group of $F$. In the talk I will explain recent results of my PhD student Shen-Ning Tung, who uses $M_{\infty}$ in the special case $F=\mathbb{Q}_p$ and $n=2$ to give a new proof of Fontaine-Mazur conjecture for  odd $2$-dimensional representations of Galois groups of totally real fields.