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Central values of additive twists of Maaß forms $L$-functions

Sary Drappeau, Asbjørn Christian Nordentoft

Abstract

In the present paper we study the central values of additive twists of Maaß forms $L$-series. In the case of the modular group, we show that the additive twists (when averaged over denominators) are asymptotically normally distributed. This supplements the recent work of Petridis--Risager which settled an averaged version of a conjecture of Mazur--Rubin concerning modular symbols. The methods of the present paper combine dynamical input due to Bettin and the first named author with the new fact that the additive twists define quantum modular forms in the sense of Zagier. This latter property is shown for a general discrete, co-finite group with cusps. Our results also has a number of arithmetic applications; in the case of Hecke congruence groups the quantum modularity implies certain reciprocity relations for twisted moments of twisted ${\rm GL}_2$-automorphic $L$-functions, extending results of Conrey and the second named author. In the case of cuspidal Maaß forms for the modular group, we also obtain a calculation of certain wide moments of twists of the $L$-function of the Maaß form.

Central values of additive twists of Maaß forms $L$-functions

Abstract

In the present paper we study the central values of additive twists of Maaß forms -series. In the case of the modular group, we show that the additive twists (when averaged over denominators) are asymptotically normally distributed. This supplements the recent work of Petridis--Risager which settled an averaged version of a conjecture of Mazur--Rubin concerning modular symbols. The methods of the present paper combine dynamical input due to Bettin and the first named author with the new fact that the additive twists define quantum modular forms in the sense of Zagier. This latter property is shown for a general discrete, co-finite group with cusps. Our results also has a number of arithmetic applications; in the case of Hecke congruence groups the quantum modularity implies certain reciprocity relations for twisted moments of twisted -automorphic -functions, extending results of Conrey and the second named author. In the case of cuspidal Maaß forms for the modular group, we also obtain a calculation of certain wide moments of twists of the -function of the Maaß form.
Paper Structure (40 sections, 47 theorems, 402 equations, 2 figures)

This paper contains 40 sections, 47 theorems, 402 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Central values
  3. Statement of results: regularity and growth of h-gamma in general
  4. Normal distribution for SL(2,Z)
  5. Arithmetic applications
  6. Reciprocity formulæ
  7. Wide moments of Dirichlet twists
  8. Context and overview
  9. Notation
  10. Background
  11. Maaß forms
  12. The Eichler integral
  13. Quantum modularity for the Eichler integral
  14. A geometric proof of quantum modularity
  15. Quantum modularity in the general case
  16. ...and 25 more sections

Key Result

Theorem 1.2

Let $f$ be a holomorphic Hecke cusp form of integer weight $k$ for $\Gamma$ discrete co-finite with cusps, and $\phi(z) = (\operatorname{Im} z)^{k/2} f(z)$. Then the multi-sets become asymptotically normally distributed as $q\rightarrow \infty$.

Figures (2)

  • Figure 1: Hyperbolic triangle with vertices $x,x_0,\infty$
  • Figure 3: The hyperbolic triangle $\mathcal{F}_{x, 0}$ for large $x$

Theorems & Definitions (94)

  • Conjecture 1.1: Additive twists conjecture
  • Theorem 1.2: Additive twists on average for holomorphic forms, Nordentoft2021
  • Theorem 1.3: Regularity of $h_\gamma$
  • Theorem 1.4: Growth of $h_\gamma$
  • Theorem 1.5: Additive twists on average for $\mathop{\mathrm{SL}}\nolimits(2,\mathbb{Z})$
  • Conjecture 1.6
  • Remark 1.7
  • Remark 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 84 more