### A correspondence for the generalized Hecke algebra of the metaplectic cover $\overline{SL(2,F)}$, $F$$p$-adic.

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In this paper, we extend the results of Ribet and Taylor on level-raising for algebraic modular forms on the multiplicative group of a definite quaternion algebra over a totally real field $F$. We do this for automorphic representations of an arbitrary reductive group $G$ over $F$, which is compact at infinity. In the special case where $G$ is an inner form of $\mathrm{GSp}\left(4\right)$ over $\mathbb{Q}$, we use this to produce congruences between Saito-Kurokawa forms and forms with a generic local component.

Lafforgue has proposed a new approach to the principle of functoriality in a test case, namely, the case of automorphic induction from an idele class character of a quadratic extension. For technical reasons, he considers only the case of function fields and assumes the data is unramified. In this paper, we show that his method applies without these restrictions. The ground field is a number field or a function field and the data may be ramified.

Given a representation $\pi $ of a local unitary group $G$ and another local unitary group $H$, either the Theta correspondence provides a representation ${\theta}_{H}\left(\pi \right)$ of $H$ or we set ${\theta}_{H}\left(\pi \right)=0$. If $G$ is fixed and $H$ varies in a Witt tower, a natural question is: for which $H$ is ${\theta}_{H}\left(\pi \right)\ne 0$ ? For given dimension $m$ there are exactly two isometry classes of unitary spaces that we denote ${H}_{m}^{\pm}$. For $\epsilon \in \{0,1\}$ let us denote ${m}_{\epsilon}^{\pm}\left(\pi \right)$ the minimal $m$ of the same parity of $\epsilon $ such that ${\theta}_{{H}_{m}^{\pm}}\left(\pi \right)\ne 0$, then we prove that ${m}_{\epsilon}^{+}\left(\pi \right)+{m}_{\epsilon}^{-}\left(\pi \right)\ge 2n+2$ where $n$ is the dimension of $\pi $.