### A Dirichlet series for Hermitian modular forms of degree 2

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We prove the recursive integral formula of class one $M$-Whittaker functions on SL$(n,\mathbb{R})$ conjectured and verified in case of $n=3,4$ by Stade.

Let $k$ be a positive integer divisible by 4, $p\>k$ a prime, $f$ an elliptic cuspidal eigenform (ordinary at $p$) of weight $k-1$, level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ${\text{ad}}^{0}M(-1)$ and ${\text{ad}}^{0}M\left(2\right)$, where $M$ is the motif attached to $f$. More precisely, we prove that under certain conditions the $p$-adic valuation of the algebraic part of the symmetric square $L$-function of $f$ evaluated at $k$ provides a lower bound for the $p$-adic valuation of the order of the Pontryagin...

Let $\ell \ne p$ be two different prime numbers, let $F$ be a local non archimedean field of residual characteristic $p$, and let ${\overline{\mathbf{Q}}}_{\ell},{\overline{\mathbf{Z}}}_{\ell},{\overline{\mathbf{F}}}_{\ell}$ be an algebraic closure of the field of $\ell $-adic numbers ${\mathbf{Q}}_{\ell}$, the ring of integers of ${\overline{\mathbf{Q}}}_{\ell}$, the residual field of ${\overline{\mathbf{Z}}}_{\ell}$. We proved the existence and the unicity of a Langlands local correspondence over ${\overline{\mathbf{F}}}_{\ell}$ for all $n\ge 2$, compatible with the reduction modulo $\ell $ in [V5], without using $L$ and $\u03f5$factors of pairs. We conjecture that the Langlands local correspondence over ${\overline{\mathbf{Q}}}_{\ell}$ respects congruences modulo $\ell $ between...

Mixed automorphic forms generalize elliptic modular forms, and they occur naturally as holomorphic forms of the highest degree on families of abelian varieties parametrized by a Riemann surface. We construct generalized Eisenstein series and Poincaré series, and prove that they are mixed automorphic forms.