Metaplectic stacks and vector-valued modular forms of half-integral weight (original) (raw)
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We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over Q. This formula has applications to proving the nonvanishing of this lift and to an explicit version of the Rallis inner product formula.
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We study the graded space H(ρ) of holomorphic vector-valued modular forms of integral weight associated to a 2-dimensional irreducible representation ρ of SL(2, Z). When ρ(T ) is unitary, a complete description of H(ρ) is given: the Poincaré series is calculated and it is shown that H(ρ) is a free module of rank 2 over the ring of (classical) holomorphic modular forms on SL(2, Z).
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In this note, we show that the algebraicity of the Fourier coefficients of half-integral weight modular forms can be determined by checking the algebraicity of the first few of them. We also give a necessary and sufficient condition for a half-integral weight modular form to be in Kohnen's +-subspace by considering only finitely many terms.
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Arxiv preprint arXiv:1102.0006, 2011
Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. In spite of their relevance they have been studied essentially for genus g = 2, where correspond to suitable commutators of Siegel modular forms. We show that for g ≥ 4, a new class of vector-valued modular forms [i 1 , . . . , i K n |τ ] naturally appears from the Mumford form, a question directly related to the Schottky problem. In particular, the weight of [i 1 , . . . , i K n |τ ] is c n − g+n−1 n−1 , with c n := 6n 2 − 6n + 1 the power of the Hodge bundle in the Mumford isomorphism. In this framework we show that the discriminant of the quadric associated to the algebraic curve of g = 4 is proportional to the square root of the product of the Thetanullwerte, which is a proof of the recently rediscovered Klein "amazing formula". Furthermore, it turns out that the coefficients of the quadric are derivatives with respect to the period matrix of the Schottky-Igusa form, implying a new theta-relation relating the latter to the product of Thetanullwerte. As a byproduct we express the product of Thetanullwerte for g = 4 in terms of theta series corresponding to the even unimodular lattices Λ = E 8 and Λ = D + 16 .
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