A p-adic approach to the Weil representation of discriminant forms arising from even lattices

Suppose that M is an even lattice with dual M^∗ and level N. Then the group Mp₂(Z) , which is the unique non-trivial double cover of SL₂(Z) , admits a representation ρ_M, called the Weil representation, on the space C[ M^∗/ M]. The main aim of this paper is to show how the formulae for the ρ_M-action of a general element of Mp₂(Z) can be obtained by a direct evaluation which does not depend on “external objects” such as theta functions. We decompose the Weil representation ρ_M into p-parts, in which each p-part can be seen as subspace of the Schwartz functions on the p-adic vector space MQp. Then we consider the Weil representation of Mp₂(Q_p) on the space of Schwartz functions on MQp, and see that restricting to Mp₂(Z) just gives the p-part of ρ_M again. The operators attained by the Weil representation are not always those appearing in the formulae from 1964, but are rather their multiples by certain roots of unity. For this, one has to find which pair of elements, lying over a matrix in SL₂(Q_p) , belong to the metaplectic double cover. Some other properties are also investigated.