A construction of antisymmetric modular forms for Weil representations
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Mathematische Zeitschrift
A construction of antisymmetric modular forms for Weil representations Brandon Williams1 Received: 13 November 2018 / Accepted: 14 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We give coefficient formulas for antisymmetric vector-valued cusp forms with rational Fourier coefficients for the Weil representation associated to a finite quadratic module. The forms we construct always span all cusp forms in weight at least three. These formulas are useful for computing explicitly with theta lifts. Mathematics Subject Classification 11F27 · 11F55
1 Introduction This note is an extended version of chapter 7 of the author’s dissertation and is in some sense a continuation of [15]. Its purpose is to give formulas for a spanning set of vector-valued cusp forms with rational Fourier coefficients for the (dual) Weil representation ρ ∗ attached to a finite quadratic module (A, Q) which are antisymmetric under the action of −I ∈ SL2 (Z). Equivalently the weight k of these cusp forms is such that k + sig(A, Q)/2 is odd, where sig(A, Q) is the signature of (A, Q). Bases of modular forms with rational coefficients are known to exist due to the work of McGraw [8]. On the other hand, all algorithms to compute such bases in the literature that the author is aware of (e.g. [11,15]) assume that k +sig(A, Q)/2 is even. Computing antisymmetric modular forms has received less attention; the first effective formula to compute the space of Eisenstein series in antisymmetric weights for arbitrary (A, Q) was given in [13]. The computation of cusp forms here complements this. The most important application of antisymmetric vector-valued modular forms is that they are mapped to orthogonal modular forms under the additive theta lift (of Gritsenko, Kudla, Oda, Rallis–Schiffmann and many others). Our main results are the two theorems below. (The terms and notation are explained in section two.) Theorem 1.1 Let (A, Q) be a finite quadratic module, and let k ≥ 3 be a weight for which k + sig(A, Q)/2 is an odd integer. For any β ∈ A and m ∈ Z − Q(β), m > 0, let Rk,m,β be the cusp form defined through the Petersson scalar product by
B 1
Brandon Williams [email protected] Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
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B. Williams
( f , Rk,m,β ) = 2 ·
(k − 1) L m,β ( f , 2k − 1) for all cusp forms f , (4πm)k−1
where L m,β is essentially a rescaled symmetric square L-function: L m,β ( f , s) =
∞ c(λ2 m, λβ) λ=1
λs
if f (τ ) =
c(n, γ )q n eγ , q = e2πiτ .
γ ∈A n∈Z−Q(γ )
Then all Rk,m,β have rational Fourier coefficients, and there is a finite collection of indices (m, β) for which the forms Rk,m,β span the entire cusp space Sk (ρ ∗ ). Theorem 1.2 Let ( , Q) be an even lattice which realizes the discriminant group A = / , and let k ≥ 4 be a weight for which k + sig( )/2 is odd. For any β ∈ and m ∈ Z − Q(β), m > 0, let m,β denote the even lattice with underlying group ⊕ Z and quadratic form Q m,β (v, λ) = Q(v
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