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Exceptional Congruences for Eta-quotient newforms

Eddie O'Sullivan, Henry Stone, Swati, Xiaolan Jin

TL;DR

The paper completes the classification of congruences for eta-quotient newforms by extending Swinnerton-Dyer’s Type I/II framework to eta-quotients and proving these congruences modulo primes and their powers. It develops a systematic approach using modular forms modulo $\ell$, theta-operators, and filtration arguments to identify exceptional primes and derive explicit coefficient congruences in $S_k(\Gamma_0(N), \chi)$. The work also provides prime-power extensions with detailed congruence families and residue-class tables, and discusses CM and dihedral cases (Type III) in the eta-quotient setting. This advances the understanding of arithmetic properties of eta-quotient modular forms and their Galois-representation connections, with implications for congruences across levels and characters.

Abstract

In 1973, Swinnerton-Dyer completely classified all congruences for coefficients of normalized eigenforms in weights $k \in \{12, 16, 18, 20, 22, 26\}$ on $Γ_{0}(1) = \operatorname{SL}_{2}(\mathbb{Z})$ 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 $k \geq 2$, we prove them using the theory of modular forms modulo primes. We also prove extensions of these congruences modulo prime powers.

Exceptional Congruences for Eta-quotient newforms

TL;DR

The paper completes the classification of congruences for eta-quotient newforms by extending Swinnerton-Dyer’s Type I/II framework to eta-quotients and proving these congruences modulo primes and their powers. It develops a systematic approach using modular forms modulo , theta-operators, and filtration arguments to identify exceptional primes and derive explicit coefficient congruences in . The work also provides prime-power extensions with detailed congruence families and residue-class tables, and discusses CM and dihedral cases (Type III) in the eta-quotient setting. This advances the understanding of arithmetic properties of eta-quotient modular forms and their Galois-representation connections, with implications for congruences across levels and characters.

Abstract

In 1973, Swinnerton-Dyer completely classified all congruences for coefficients of normalized eigenforms in weights on 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 . When , we prove them using the theory of modular forms modulo primes. We also prove extensions of these congruences modulo prime powers.

Paper Structure

This paper contains 10 sections, 14 theorems, 94 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction and Statement of Results
  2. Introduction
  3. Statement of Results
  4. Extension to prime power modulus
  5. Plan for the paper
  6. Acknowledgment
  7. Background
  8. Proofs of Theorem \ref{['type1_cong']} and Theorem \ref{['type2_cong']}
  9. Proof of Type I Congruences
  10. Proof of Type II Congruences

Key Result

Theorem 1.1

Let $k \geq 2$ be an integer. Suppose that $f = \sum_{n \geq 1} a_{f}(n) q^{n} \in S_{k}(\Gamma_{0}(N), \chi)$ is a normalized eigenform with $\ell$-adic integer coefficients. Then there exists a continuous homomorphism for each prime $\ell$, depending on $f$, such that $\rho_{\ell, f}(Frob_{p})$ satisfies the characteristic polynomial for each $p \neq N \ell$.

Theorems & Definitions (19)

  • Theorem 1.1: Serre-Deligne
  • Lemma 1.2
  • Conjecture 1.3: Serre-Swinnerton-Dyer
  • Theorem 1.4: Boylan
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 3.1: Sturm
  • Proposition 3.2
  • ...and 9 more