A Theorem to Reduce Certain Modular Form Relations Modulo Primes

Abstract

Let p be a prime. Let $\mathcal {A}_k^1(N)$ be the set of meromorphic modular forms of weight k for the group Γ₁(N) and $\mathcal {A}^1(N):=\oplus _{k=-\infty }^{\infty } \mathcal {A}_k^1(N)$. Let $f_j(\tau )=\sum _{n=m_j}^{\infty }a_j(n)q^n\in \mathcal {A}^1(N)$, j = 0, …, ν and q = e ^2πiτ such that a_j(n) are integers. The main result of this paper is that

$$\displaystyle f_0(\tau )+qf_1(\tau )+q^2f_2(\tau )+\cdots +q^{\nu }f_\nu (\tau )\equiv 0\pmod {p} $$

iff

$$\displaystyle f_0(\tau )\equiv f_1(\tau )\equiv f_2(\tau )\equiv \cdots \equiv f_{\nu }(\tau )\equiv 0\pmod {p}. $$

Dedicated to my advisor and friend Peter Paule on the occasion of his 60th birthday

The research was funded by the Austrian Science Fund (FWF), W1214-N15, project DK6 and by grant P2016-N18. The research was supported by the strategic program “Innovatives 2010 plus” by the Upper Austrian Government.