Divisor moments of polynomials in Fourier coefficients of modular forms

2026/07/21 by Wonwoong Lee
Mathematics · #math.NT

paper · pdf

Abstract

We study higher moments of the divisor function evaluated at polynomial expressions in the Fourier coefficients of a non-CM newform. The logarithmic exponent appearing in our estimates depends only on the number of irreducible factors of the polynomial and remains unchanged under a Sato--Tate restriction. The proof combines an effective Chebotarev theorem, or an effective Chebotarev--Sato--Tate theorem, with the arithmetic of joint cycle types and a mean value estimation for multivariable multiplicative functions with Frobenian support.

Related

• Logarithmic vector-valued modular forms and polynomial-growth estimates of their Fourier coefficients
• Recovering Fourier coefficients of modular forms and factoring of integers
• Congruence classes for modular forms over small sets
• Lower Bounds on High Moments of Twisted Fourier coefficients of Modular Forms
• Determining newforms via arithmetic relations among Fourier coefficients