| Résume | I will begin my talk with Shimura's theorem on the critical values of the standard $L$-function attached to a holomorphic Hilbert modular form. Then, I will recast Shimura's theorem into a more representation-theoretic context by talking about the critical values of $L$-functions attached to cohomological cuspidal automorphic representations for $GL(2)$ over a totally real number field $F$. With this as the back-drop I will present results of some joint work with Harald Grobner concerning the critical values of $L$-functions attached to cohomological cuspidal automorphic representations of $GL(2n)$ over $F$ which admit Shalika models. Putting $n=1$ retrieves Shimura's theorem. I will also mention applications to symmetric cube $L$-function of a Hilbert modular form, and the degree four $L$-function of a Siegel modular form. The latter part of my talk will be more technical and I will (i) show how to define certain periods by comparing rational structures on Shalika models and cohomological models; (ii) analyze the behaviour of periods upon twisting the representation by characters; (iii) sketch a proof of why these periods appear in the critical values of standard $L$-functions for $GL(2n)$. |
