On the dimensions of spaces of siegel modular forms of weight one

Li, Jian-Shu

doi:10.1007/bf02249262

Cited by 5 publications

(3 citation statements)

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“…for all t < n/2, and we may deduce similar results in the case of Sp 2n (O F ) for arbitrary F . In the case of singular forms of rank 2 on Sp 2n (Z), m(π, N) was exactly determined by Li in [9]. In particular, he proves that m(π, N) = 0 unless 4|N or p|N with p ≡ −1 (4), which demonstrates that the upper bound in our Theorem is not always attained.…”

Section: Introductionmentioning

confidence: 83%

“…The proposition will follow easily from the fact that π q is discrete series, and therefore contributes to the cohomology of certain automorphic vector bundles over Y ′ (c ′ ). More precisely if (τ, F 2 ) is an irreducible, finite dimensional representation of G ′ (R) with the same infinitessmal character as π q and q = (dim G ′ (R) − dim K ′ ∞ )/2, it is well known (see chapter 2, section 5 of [2]) that (9) H q (g, K; π q ⊗ F 2 ) ≃ C.…”

Section: Application Of the Local Theta Correspondencementioning

confidence: 99%

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Theta Lifting and Cohomology Growth in p-adic Towers

Cossutta¹,

Marshall

2012

International Mathematics Research Notices

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We use the theta lift to study the multiplicity with which certain automorphic representations of cohomological type occur in a family of congruence covers of an arithmetic manifold. When the family of covers is a so-called 'p-adic congruence tower' we obtain sharp asymptotics for the number of representations which occur as lifts. When combined with theorems on the surjectivity of the theta lift due to Howe and Li, and Bergeron, Millson and Moeglin, this allows us to verify certain cases of a conjecture of Sarnak and Xue.The second author is supported by NSF grant this ensures that the metaplectic cover of Sp(W ) splits on restriction to G × G ′ by a result of Kudla [6]. Let L ⊂ V be an O E lattice, whose completion in V v we shall assume to be self dual whenever V is unramified, and Γ ⊂ G(R) the arithmetic group stabilising L. Let r be the dimension of the maximal isotropic subspace of V , and define

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Section: Introductionmentioning

confidence: 83%

Section: Application Of the Local Theta Correspondencementioning

confidence: 99%

Theta Lifting and Cohomology Growth in p-adic Towers

Cossutta¹,

Marshall

2012

International Mathematics Research Notices

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“…We prove Proposition 3.5 by using this fact and properties of elliptic modular forms of weight 1, which we consider in Sect. 2. We remark that if g ≥ 3, Li already determined the dimension of M 1 (Γ g ( p r )) [7]. Moreover, by the theory of singular forms, studied by Resnikoff [8] and Freitag [2,3], we know that M 1 (Γ g 0 ( p), ψ) is generated by theta functions of quadratic forms, if g ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

The dimension of the space of Siegel–Eisenstein series of weight one

Gunji

2007

Math. Z.

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The dimensions of spaces of Siegel cusp forms of general degree

Wakatsuki

2018

Advances in Mathematics

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In this paper, we give a dimension formula for spaces of Siegel cusp forms of general degree with respect to neat arithmetic subgroups. The formula was conjectured before by several researchers (cf. [Ib1, IS2, IS3, Tsu4]). The dimensions are expressed by special values of Shintani zeta functions for spaces of symmetric matrices at non-positive integers. This formula was given by Shintani for only a small part of the geometric side of the trace formula (see [Shin]). To be precise, it is the contribution of unipotent elements corresponding to the partitions (2 j , 1 2n−2j ), where n denotes the degree and 0 ≤ j ≤ n. Hence, our work is to show that all the other contributions vanish. In addition, one finds that Shintani's formula means the dimension itself. Combining our formula and an explicit formula of the Shintani zeta functions, which was discovered by Ibukiyama and Saito (cf. [IS1, IK, Sai]), we can derive an explicit dimension formula for the principal congruence subgroups of level greater than two (cf. [IS2, IS3]). In this explicit dimension formula, the dimensions are described by degree n, weight k, level N , and the Bernoulli numbers B m .

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On the dimensions of spaces of siegel modular forms of weight one

Cited by 5 publications

References 19 publications

Theta Lifting and Cohomology Growth in p-adic Towers

Theta Lifting and Cohomology Growth in p-adic Towers

The dimension of the space of Siegel–Eisenstein series of weight one

The dimensions of spaces of Siegel cusp forms of general degree

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