Let $J_1$ be the Jacobian of the modular curve associated with ${\Gamma }_1\left(Np\right),(p,N)=1$ and $J_0$ the one associated with ${\Gamma }_1\left(N\right)\cap {\Gamma }_0\left(p\right)$. We study $J_1[p-1]$ as a Hecke and Galois-module. We relate a certain matrix of $p$-adic periods to the infinitesimal deformation of the $U_p$-operator.

It is known that with a non-unit Pisot substitution $\sigma$ one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization...

The Local Converse Problem is to determine how the family of the local gamma factors $\gamma (s,\pi \times \tau ,\psi )$ characterizes the isomorphism class of an irreducible admissible generic representation $\pi$ of ${\mathrm{GL}}_n\left(F\right)$, with $F$ a non-archimedean local field, where $\tau$ runs through all irreducible supercuspidal representations of ${\mathrm{GL}}_r\left(F\right)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,...,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture,...