Generation of Hecke fields by squares of cyclotomic twists of modular $L$-values

TL;DR

The paper examines when Hecke fields attached to a non-CM holomorphic newform are generated by squares of central L-values twisted by Dirichlet characters of p-power conductor. By proving a non-vanishing of a certain second twisted moment over the Galois orbit and combining approximate functional equations, Voronoi summation, and a refined unipotent-mixing analysis, the authors show that ${f Q}_f(\chi)$ is generated by ${\bf Q}(\mu_p, |L_f(\chi)|^2)$ (over ${\bf Q}(\mu_p)$) for all but finitely many wild characters; under additional hypotheses on $p$ and the degree $[{f Q}_f:{\bf Q}]$, one even obtains ${\bf Q}_f\subseteq {\bf Q}(\mu_p, |L_f(\chi)|^2, \chi)$. The work extends Luo–Ramakrishnan and Sun by replacing $L_f(\chi)$ with $|L_f(\chi)|^2$ and leverages Shimura reciprocity, a determinant-type non-vanishing argument, and a detailed analysis of diagonal and off-diagonal contributions. The results illuminate how arithmetical invariants of modular forms, via central L-values, govern the fields of definition and showcase powerful analytic techniques for moments of twisted L-values.

Abstract

Let $f$ be a non-CM elliptic newform without a quadratic inner twist, $p$ an odd prime and $χ$ a Dirichlet character of $p$-power order and sufficiently large $p$-power conductor. We show that the compositum $\mathbb{Q}_{f}(χ)$ of the Hecke fields associated to $f$ and $χ$ is generated by the square of the absolute value of the corresponding central $L$-value $L^{\rm alg}(1/2, f \otimes χ)$ over $\mathbb{Q}(μ_p)$. The proof is based among other things on techniques used for the recent resolution of unipotent mixing conjecture by the first and third named authors.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.1
• Remark 1.2
• Corollary 1.4
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.5