The Weil Representation of and Some Applications

Scheithauer, Nils R.

doi:10.1093/imrn/rnn166

Cited by 65 publications

(118 citation statements)

References 15 publications

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“…Multiplying the expressions mentioned in the previous paragraph by the roots of unity described here completes the evaluation of ρ M (A) for arbitrary matrix A ∈ Mp 2 (Z). This reproduces the formulae of [19,23] for the case of even M, as well as the formula of [22]. The results for odd lattices seem to be new.…”

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confidence: 79%

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

Zemel

2015

Ann. Math. Québec

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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

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supporting

confidence: 79%

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

Zemel

2015

Ann. Math. Québec

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“…In the present paper, by generalizing Bruinier and Bundschuh's idea in [3] and applying Scheithauer's formulas for the Weil representations in [12], we establish such a correspondence in the case when the discriminant form is given by the norm form of a real quadratic field. In particular, the level N is a positive fundamental discriminant.…”

Section: Introductionmentioning

confidence: 76%

“…Then since F s is T -invariant, that is F s (τ + 1) = F s , by Lemma 4.2 above and Theorem 4.7 in [12],…”

Section: Proposition 43 W = Wmentioning

confidence: 88%

“…Define another map ψ : Proof To consider either one, we may drop the operator W (N ). It is easy to verify that ψ is well defined and that for φ follows from Proposition 4.5 in [12] applied to γ = 0.…”

Section: Proposition 43 W = Wmentioning

confidence: 93%

“…where we applied Lemma 2.1 and the -condition of f to f | k U (m)η m , and used Theorem 4.7 in[12] for the concrete computation of ρ D (γ −1 m )e 0 , e 0 . Since there are in total 2 ω(N ) positive divisors of N and they correspond bijectively to nonequivalent cusps, we are done with the proof in the case of N 1 ≡ 1 mod 4.…”

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confidence: 99%

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An isomorphism between scalar-valued modular forms and modular forms for Weil representations

Zhang

2014

Ramanujan J

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In this paper, we consider discriminant forms that are given by the norm form of real quadratic fields and their induced Weil representations. We prove that there exists an isomorphism between the space of vector-valued modular forms for the Weil representations that are invariant under the action of the automorphism group and the space of scalar-valued modular forms that satisfy some -condition, with which we translate Borcherds's theorem of obstructions to scalar-valued modular forms. In the end, we consider an example in the case of level 12.

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Lie Algebras, Vertex Algebras, and Automorphic Forms

Scheithauer¹

2010

Progress in Mathematics

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The Weil Representation of and Some Applications

Cited by 65 publications

References 15 publications

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

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

An isomorphism between scalar-valued modular forms and modular forms for Weil representations

Lie Algebras, Vertex Algebras, and Automorphic Forms

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