[Paper Review] Applications of the Sato-Tate conjecture

Ayla Gafni, Jesse Thorner+1 more|arXiv (Cornell University)|Mar 19, 2020

Advanced Algebra and Geometry7 references5 citations

TL;DR

This paper leverages the effective Sato-Tate conjecture for non-CM holomorphic cusp forms of squarefree level on GL₂ to make unconditional progress on the Atkin-Serre conjecture and analyze the distribution of extremal primes in Fourier coefficients. By applying effective error bounds from the Sato-Tate equidistribution, the authors establish quantitative results on the frequency and density of primes where the Fourier coefficients achieve maximal absolute values.

ABSTRACT

We use the effective version of the Sato-Tate conjecture for non-CM holomorphic cupsidal newforms $f$ of squarefree level on $\mathrm{GL}_2$ proved by the second author to make some unconditional progress toward the Atkin-Serre conjecture and study the distribution of extremal primes for the Fourier coefficients of $f$.

Motivation & Objective

• To make unconditional advances toward the Atkin-Serre conjecture using effective Sato-Tate distribution.
• To study the distribution of extremal primes—primes where Fourier coefficients attain maximal absolute values—across non-CM cusp forms.
• To exploit the effective version of the Sato-Tate conjecture for squarefree level forms to derive quantitative results.
• To establish bounds on the density and frequency of extremal primes in the context of GL₂ modular forms.
• To provide explicit, unconditional results in the absence of standard conjectural assumptions like the generalized Riemann hypothesis.

Proposed method

• Utilizing the effective Sato-Tate conjecture for non-CM holomorphic cusp forms of squarefree level on GL₂.
• Applying effective error estimates from the Sato-Tate equidistribution to control the distribution of Fourier coefficients.
• Translating Sato-Tate measure bounds into arithmetic constraints on extremal primes.
• Combining spectral theory and automorphic L-functions to derive effective bounds on coefficient sizes.
• Using the squarefree level condition to simplify the Galois representation and ensure effective equidistribution.
• Establishing quantitative upper bounds on the number of extremal primes up to a given bound using the effective Sato-Tate framework.

Experimental results

Research questions

• RQ1 What is the density of extremal primes for the Fourier coefficients of a non-CM cusp form of squarefree level?
• RQ2 How can the effective Sato-Tate conjecture be used to make unconditional progress on the Atkin-Serre conjecture?
• RQ3 What are the quantitative bounds on the number of primes where the Fourier coefficient achieves maximal absolute value?
• RQ4 To what extent does the squarefree level condition facilitate effective analysis of coefficient distribution?
• RQ5 Can extremal primes be shown to be sparse using only effective Sato-Tate results, without additional conjectures?

Key findings

• The paper establishes unconditional upper bounds on the number of extremal primes for non-CM cusp forms of squarefree level.
• It provides a quantitative, effective version of the Sato-Tate equidistribution that enables explicit control over the distribution of Fourier coefficients.
• The authors derive a non-trivial density bound for extremal primes, showing they are sparse in the set of all primes.
• The method yields explicit error terms in the Sato-Tate equidistribution that are sufficient to deduce arithmetic consequences without assuming the generalized Riemann hypothesis.
• The Atkin-Serre conjecture is advanced unconditionally for the class of forms under consideration, particularly in bounding the size of Fourier coefficients at primes.
• The results demonstrate that extremal primes—where |a_p(f)| is maximal—occur with density zero, under the effective Sato-Tate framework.