Non-ordinary primes and congruences for weakly holomorphic modular forms of level 2

Vladimir Nechaev

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arXiv · 2607.16336

Non-ordinary primes and congruences for weakly holomorphic modular forms of level 2

Vladimir Nechaev

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Abstract

In this paper, we investigate the arithmetic properties of the Fourier coefficients of weakly holomorphic modular forms of weight $k$ for the congruence subgroup $\Gamma_0(2)$, which have poles exclusively at the cusp $\infty$. Using the structure of canonical bases and the explicit duality between the coefficients of spaces of weakly holomorphic modular forms, which have a pole only at $\infty$, we prove a general family of congruences modulo primes $p \ge 5$ and derive several corollaries as examples of the theorem's application.

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Vladimir Nechaev. 2026-07-16. Non-ordinary primes and congruences for weakly holomorphic modular forms of level 2. https://arxiv.org/abs/2607.16336

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