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Let $F$ be a field which is, either local non archimedean, or finite, ofresidual charcateristic $p$ but of characteristic different from $2$. Let $W$be a symplectic space of finite dimension over $F$. Suppose $R$ is a field ofcharacteristic $\ell \neq p$ so that there exists a non trivial smooth additivecharacter $\psi : F \to R^\times$. Then the Stone-von Neumann theorem of theHeisenberg group $H(W)$ is still valid for representations with coefficients in$R$. It leads to a projective representation of the group $\text{Sp}(W)$ whichlifts to a genuine smooth representation of a central extension of$\text{Sp}(W)$ by $R^\times$: this is the modular Weil representation of themetaplectic group. For any dual pair $(H_1,H_2)$, their lifts to themetaplectic group may splitor not according to the different cases at stake.Eventually, computing the biggest isotypic quotient of the modular Weilrepresentation allows to define the $\Theta$-lift. Some new lines ofinvestigation are thus available with these new tools such as studying scalarextension and reduction modulo $\ell$.

Correspondance th\^eta locale $\ell$-modulaire I : groupe m\'etaplectique, repr\'esentation de Weil et $\Theta$-lift