Congruences for Level 1 cusp forms of half-integral weight

2020/12/19 by Dicks, Robert · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2012.10587

Abstract

Suppose that ℓ ≥ 5 is prime. For a positive integer N with 4 | N, previous works studied properties of half-integral weight modular forms on Γ₀(N) which are supported on finitely many square classes modulo ℓ, in some cases proving that these forms are congruent to the image of a single variable theta series under some number of iterations of the Ramanujan Θ-operator. Here, we study the analogous problem for modular forms of half-integral weight on SL₂(ℤ). Let η be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on SL₂(ℤ) which is supported on finitely many square classes modulo ℓ can be written modulo ℓ in terms of η^ℓ and an iterated derivative of η.

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