TY - JOUR

T1 - Congruences for level four cusp forms

AU - Ahlgren, Scott

AU - Choi, Dohoon

AU - Rouse, Jeremy

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2009/7

Y1 - 2009/7

N2 - In this paper, we study congruences for modular forms of half-integral weight on Γ(4). Suppose that l ≤ 5 is prime, that K is a number field, and that v is a prime of K above l. Let Οo denote the ring of v-integral elements of K, and suppose that f(z) = Σ ∞n=1 a(n)qn ε Ο v[[q]] is a cusp form of weight λ + 1/2 on Γ 0(4) in Kohnen's plus space. We prove that if the coefficients of f are supported on finitely many square classes modulo v and λ + 1/2 < l(l + 1 + 1/2), then λ is even and f(z) = a(1)Σ∞ n=1nλqn2 (mod v). This result is a precise analogue of a characteristic zero theorem of Vignéras [22]. As an application, we study divisibility properties of the algebraic parts of the central critical values of modular L-functions.

AB - In this paper, we study congruences for modular forms of half-integral weight on Γ(4). Suppose that l ≤ 5 is prime, that K is a number field, and that v is a prime of K above l. Let Οo denote the ring of v-integral elements of K, and suppose that f(z) = Σ ∞n=1 a(n)qn ε Ο v[[q]] is a cusp form of weight λ + 1/2 on Γ 0(4) in Kohnen's plus space. We prove that if the coefficients of f are supported on finitely many square classes modulo v and λ + 1/2 < l(l + 1 + 1/2), then λ is even and f(z) = a(1)Σ∞ n=1nλqn2 (mod v). This result is a precise analogue of a characteristic zero theorem of Vignéras [22]. As an application, we study divisibility properties of the algebraic parts of the central critical values of modular L-functions.

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U2 - 10.4310/MRL.2009.v16.n4.a10

DO - 10.4310/MRL.2009.v16.n4.a10

M3 - Article

AN - SCOPUS:69949112759

VL - 16

SP - 683

EP - 701

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 4

ER -