A Siegel cusp form of degree 12 and weight 12

“…Classifying the linear relations among the theta series of the Niemeier lattices is a very interesting problem. The best results are due to Erokhin [9], see also [3], and we revisit these results in the light of our present estimates for weight 12 cusp forms.…”

Section: Examples and Discussionsupporting

confidence: 61%

Linear dependence among Siegel modular forms

Poor¹,

Yuen²

2000

Mathematische Annalen

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Theorems are given which describe when high enough vanishing at the cusps implies that a Siegel modular cusp form is zero. Formerly impractical computations become practical and examples are given in degree four. Vanishing order is described by kernels, a type of polyhedral convex hull. Subject Classification (1991): 11F46, 11F27, 11H55 Mathematics IntroductionThis paper extends to Siegel modular forms certain practical computational techniques available for modular forms on the upper half plane. Two modular forms are equal when enough of their Fourier coefficients agree; more generally, a linear dependence relation holds among modular forms when it holds among enough of their Fourier coefficients. For example, in [29] Schiemann shows that the theta series for two distinct classes of 4 × 4 integral positive definite quadratic forms are equal by showing that their first 375 Fourier coefficients agree. The type of theorem one requires is that a cusp form is zero if it vanishes to a sufficiently large order; in the above example the cusp form in question is given by the difference of the theta series. In the case of Siegel modular forms, Siegel provided a version of the following result for the full modular group.Theorem (Siegel). Let f ∈ S k n have the Fourier expansion f (Ω) = s>0 a s e(tr (sΩ)). The following conditions are equivalent.(1) f = 0.(2) For all s such that tr(s) ≤ κ n k 4π , we have a s = 0. (3) For all s such that tr(s) ≤ nµ n n 2 √ 3 k 4π , we have a s = 0. (4) For all s such that det(s) 1/n ≤ µ n n 2 √ 3 k 4π , we have a s = 0.

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“…The article [5] gives a quite general construction of a cusp form of degree k. Let Λ be a 2k-dimensional even lattice and choose some prime p such that the quadratic space (Λ/pΛ, Q p ) (where Q p (x) := 1 2 (x, x) + pZ) is isometric to the sum of k hyperbolic planes. Fix a totally isotropic subspace F of Λ/pΛ of dimension k.…”

Section: The Borcherds-freitag-weissauer Cusp Formmentioning

confidence: 99%

“…By [5] the form BFW(Λ, p) is a linear combination of Siegel theta-series of lattices in the genus of Λ: For any k-dimensional totally isotropic subspace E of Λ/pΛ let Γ(E) := E, pΛ be the full preimage of E. Dividing the scalar product by p, one obtains a lattice 1/p Γ(E) :…”

Section: The Borcherds-freitag-weissauer Cusp Formmentioning

confidence: 99%