Exceptional Congruences for Eta-quotient newforms

2025/11/20 by O'Sullivan, Eddie, Stone, Henry, Swati +1
#11F11 #11F33 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.16039

Abstract

In 1973, Swinnerton-Dyer completely classified all congruences for coefficients of normalized eigenforms in weights k ∈ \12, 16, 18, 20, 22, 26\ on Γ₀(1) = SL₂(ℤ) using the theory of modular Galois representations. In this paper, we classify congruences of Type I and Type II considered by Swinnerton-Dyer for the coefficients of eta-quotient newforms in S_k(N, χ). When k ≥ 2, we prove them using the theory of modular forms modulo primes. We also prove extensions of these congruences modulo prime powers.

Citations

• Computing Congruences of Modular Forms and Galois Representations Modulo\n Prime Powers
• CM newforms with rational coefficients

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