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Suppose that M is an even lattice with dual M * and level N . Then the group Mp 2 (Z), which is the unique non-trivial double cover of SL 2 (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 Mp 2 (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 M Q p . Then we consider the Weil representation of Mp 2 (Q p ) on the space of Schwartz functions on M Q p , and see that restricting to Mp 2 (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 SL 2 (Q p ), belong to the metaplectic double cover. Some other properties are also investigated.Résumé Soit M un treillis pair de dual M * et de niveau N . Alors le groupe Mp 2 (Z), qui est l'unique revêtement non-trivial double de SL 2 (Z), admet une représentation ρ M , dite la représentation de Weil, sur l'espace C[M * /M]. Le but premier de cet article est de montrer que les formules pour l'action ρ M d'un élément quelconque de Mp 2 (Z) peuvent être obtenues via une évaluation directe qui ne dépend pas " d'objets externes" tels les fonctions thêta. Nous décomposons la représentation ρ M de Weil en p-parties, chacune de ces p-parties pouvant être vue comme un sous-espace des fonctions de Schwartz sur l'espace vectoriel p-adique M Q p . Nous considérons alors la représentation de Weil de Mp 2 (Q p ) sur l'espace des fonctions de Schwartz sur M Q p , et constatons que nous restreindre à Mp 2 (Z) redonne précisément la ppartie de ρ M . Les opérateurs touchés par la représentation de Weil ne sont pas toujours ceux qui apparaissent dans les formules de 1964, mais sont plutôt leurs multiples par certaines B Shaul Zemel

X * := Hom F (X, F) for the dual space, and we have V * ∼ = V via the non-degenerate bilinear form. If U is an isotropic subspace of V (i.e., (u, w) = 0 for u and w ∈ U , or equivalently U ⊆ U ⊥ ), then we set P U to be the stabilizer of U in O(V ) := Aut F V, (·, ·) , as well as W = U ⊥ /U.They are the parabolic subgroup of O(V ) that is associated with U and a nondegenerate non-degenerate quadratic space of dimension dim V − 2 dim U respectively. The fact that elements of P U must also preserve U ⊥ immediately yield the following first result.

We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.