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Suppose that $M$ is an even lattice with dual $M^{*}$ and level $N$. Then thegroup $Mp_{2}(\mathbb{Z})$, which is the unique non-trivial double cover of$SL_{2}(\mathbb{Z})$, admits a representation $\rho_{M}$, called the Weilrepresentation, on the space $\mathbb{C}[M^{*}/M]$. The main aim of this paperis to show how the formulae for the $\rho_{M}$-action of a general element of$Mp_{2}(\mathbb{Z})$ can be obtained by a direct evaluation which does notdepend on ``external objects'' such as theta functions. We decompose the Weilrepresentation $\rho_{M}$ into $p$-parts, in which each $p$-part can be seen assubspace of the Schwartz functions on the $p$-adic vector space$M_{\mathbb{Q}_{p}}$. Then we consider the Weil representation of$Mp_{2}(\mathbb{Q}_{p})$ on the space of Schwartz functions on$M_{\mathbb{Q}_{p}}$, and see that restricting to $Mp_{2}(\mathbb{Z})$ justgives the $p$-part of $\rho_{M}$ again. The operators attained by the Weilrepresentation are not always those appearing in the formulae from 1964, butare rather their multiples by certain roots of unity. For this, one has to findwhich pair of elements, lying over a matrix in $SL_{2}(\mathbb{Q}_{p})$, belongto the metaplectic double cover. Some other properties are also investigated.

A $p$-adic Approach to the Weil Representation of Discriminant Forms Arising from Even Lattices