Proportion of Atkin-Lehner sign patterns and Hecke Eigenvalue Equidistribution

TL;DR

The paper analyzes the distribution of Atkin–Lehner sign patterns in spaces of cusp forms and establishes $\mu_p$-equidistribution of Hecke eigenvalues on the refined sign-pattern subspaces $S_k^{\sigma}(N)$ and $S_k^{\mathrm{new},\sigma}(N)$ as $N+k \to \infty$. By developing trace formulas for $\mathrm{Tr}_{S_k(N)} T'_m \circ W_Q$ and their sign-pattern analogues, the authors derive exact proportions for both global signs and general sign patterns, including explicit correction factors depending on the level structure. They show that, for admissible patterns, eigenvalues of $T'_p$ are distributed according to the $p$-adic Plancherel measure $\mu_p$, extending Serre’s equidistribution results to the finest sign-pattern decompositions and yielding consequences for Galois orbits, coefficient-field degrees, and $J_0(N)$-factor dimensions. The results provide a quantitative, structure-aware view of Hecke data across Atkin–Lehner decompositions, linking symmetry patterns to arithmetic and geometric properties of modular forms and curves. Together, these findings offer a robust framework for understanding refined symmetries in modular forms and their arithmetic implications, including a form of Maeda-type behavior in high level.

Abstract

Let $N \ge 1$, $k \ge 2$ even, and $σ$ denote a sign pattern for $N$. In this paper, we first determine the exact proportion of forms in $S_k(N)$ and $S_k^\mathrm{new}(N)$ with a given Atkin-Lehner sign pattern $σ$. Then we study the asymptotic behavior of the Hecke operators $T_p$ over the subspaces of $S_k(N)$ and $S_k^{\mathrm{new}}(N)$ with Atkin-Lehner sign pattern $σ$. In particular, for the $p$-adic Plancherel measure $μ_p$, we show that the Hecke eigenvalues for $T_p$ over these subspaces are $μ_p$-equidistributed as $N+k \to \infty$.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1: Skoruppa-Zagier Skoruppa1988, correction by Assaf Assaf
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Lemma 3.1
• proof