[Paper Review] Shifted convolutions and a conjecture by Mazur, Rubin and Stein

Nikolaos Diamantis, Jeffrey Hoffstein+2 more|arXiv (Cornell University)|Jul 6, 2018

Analytic Number Theory Research11 references3 citations

TL;DR

This paper proves a conjecture by Mazur, Rubin, and Stein on the asymptotic behavior of modular symbols averaged over rational points on the modular curve, extending previous results to arbitrary levels—not just square-free ones. Using novel bounds on additive twists of L-functions and explicit estimates for Fourier coefficients of contragredient forms, the authors establish convergence rates with precise error terms involving the level and common prime factors of M and q.

ABSTRACT

In this paper, a conjecture of Mazur, Rubin and Stein concerning certain averages of modular symbols is proved.

Motivation & Objective

• To resolve a long-standing conjecture by Mazur, Rubin, and Stein on the limiting distribution of modular symbols for weight 2 newforms at arbitrary levels.
• To extend previous results—previously limited to square-free levels or prime M—by developing tools valid for non-square-free levels.
• To establish sharp error terms in the asymptotic expansion of averaged modular symbols, incorporating the arithmetic structure of the level and the averaging parameter M.
• To provide explicit, uniform bounds on Fourier coefficients of twisted modular forms, even when the twist is not a newform, with dependence on level made fully explicit.

Proposed method

• Derives an approximate functional equation for additive twists of L-functions valid for all levels and additive conductors.
• Introduces a new method to bypass reliance on shifted convolution series, simplifying the proof compared to prior approaches.
• Uses Eisenstein series and modular symbols to express the average modular symbol as a sum involving Fourier coefficients of twisted forms.
• Applies a novel bound on antiderivatives of weight 2 newforms at rational points, valid for all levels and rational arguments.
• Establishes a precise estimate for the Fourier coefficients of the 'contragredient' form in the functional equation of additive twists, with explicit dependence on the level.
• Combines error estimates from multiple sources—Fourier decay, additive twist bounds, and level-dependent factors—using a parameter choice to balance error terms.

Experimental results

Research questions

• RQ1 What is the precise asymptotic limit of the averaged modular symbol $ G_M^\pm(x) $ as $ M \to \infty $ for arbitrary level $ q $, not just square-free ones?
• RQ2 How do the error terms in the convergence of $ G_M^\pm(x) $ depend on the arithmetic of $ q $ and $ M $, particularly when $ q $ is not square-free?
• RQ3 Can the dependence of Fourier coefficient bounds on the level be made explicit even when the twist is not a newform?
• RQ4 What is the optimal error term in the approximation of modular symbols via additive twists of L-functions, and how does it scale with $ M $ and $ q $?
• RQ5 Can the reliance on shifted convolution series be avoided in the proof of the conjecture, leading to a cleaner and more general argument?

Key findings

• The conjecture of Mazur, Rubin, and Stein is fully proved for all levels $ q $, all $ x \in [0,1] $, and all sequences $ M \to \infty $, not just square-free levels or prime $ M $.
• The main term in the asymptotic expansion is $ \frac{1}{2\pi} \sum_{n \geq 1} \frac{a(n) \sin(2\pi n x)}{n^2} $ for $ G_M^+(x) $, and $ \frac{1}{2\pi i} \sum_{n \geq 1} \frac{a(n)(\cos(2\pi n x) - 1)}{n^2} $ for $ G_M^-(x) $, matching the conjectured form.
• The error term is bounded by $ \mathcal{O}_\epsilon\left( (Mq)^\epsilon M^{-1/4} q^{1/4} \prod_{\substack{p \mid (q,M) \\ p^2 \mid q}} p^{1/2} \right) $, which captures the influence of common prime powers between $ q $ and $ M $.
• A new bound is established for the antiderivative $ \int_{\infty}^{a/d} f(z) \, dz $ at rational $ a/d $, valid for all levels $ q $, generalizing previous results.
• The paper provides an explicit, level-dependent bound for Fourier coefficients of Dirichlet twists of cusp forms, even when the twist is not a newform, resolving a key technical obstacle.
• The authors achieve a cleaner proof by avoiding shifted convolution series, relying instead on additive twist analysis and explicit Fourier coefficient estimates.