Abstract: Admissible representations of locally $p$-adic Lie groups and automorphic representations of adelic Lie groups are important ingredients in modern number theory. This is because of their deep relationship to modular forms, L-series and Galois representations. This talk will explain how to construct these representations as objects (monomial resolutions) in a (non-abelian) derived category. This construction applies to representations defined over an algebraically closed field of any characteristic. The intended advantage of this point of view lies in the fact that in this derived category L-series and base base change have elementary descriptions.

Abstract: Let $Y$ denote the space of holomorphic functions in a planar domain $\Omega$, such that the derivatives of all orders extend continuously to the closure of $\Omega$ in the plane $\mathbb{C}$. We endow $Y$ with its natural topology and let $X$ denote the closure in $Y$ of all rational functions with poles off the closure of $\Omega$. Some universality results concerning Taylor series or Pade approximants are generic in $X$. In order to strengthen the above results we give a sufficient condition of geometric nature assuring that $X$=$Y$. In addition to this, if a Jordan domain $\Omega$ satisfies the above condition, then the primitive $F$ of a holomorphic function $f$ in $\Omega$ is at least as smooth on the boundary as $f$, even if the boundary of $\Omega$ has infinite length. This led us to construct a Jordan domain $\Omega$ supporting a holomorphic function $f$ which extends continuously on the closure of $\Omega$, such that its primitive $F$ is even not bounded in $\Omega$. Finally we extend the last result in generic form to more general Volterra operators.

This is based on a joint work with Ilias Zadik.