**Pierre Colmez**

Institut de Mathematiques de Jussieu, France

December 19, 2013

**Modular forms and $p$-adic representations**:
The p-adic local Langlands correspondence for ${\rm GL}_2({\bf Q}_p)$
attaches a $p$-adic representation $\Pi$ of ${\rm GL}_2({\bf Q}_p)$ to each $2$-dimensionnal representation $V$ of the absolute Galois group of ${\bf Q}_p$. Il $V$ comes from a modular form $f$ of weight $k$, one can read on $\Pi$ the (locally constant) representation of $GL_2({\bf Q}_p)$ obtained by letting ${\rm GL}_2({\bf Q})$ act on $f$ (compatibility between the classical and $p$-adic local Langlands coreespondences): the recipe is quite simple if $k\geq 2$, but more subtle if $k=1$."