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The Second Moment Theory of Families of $L$-Functions–The Case of Twisted Hecke $L$-Functions
About this Title
Valentin Blomer, Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, Djordje Milićević and Will Sawin
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 282, Number 1394
ISBNs: 978-1-4704-5678-8 (print); 978-1-4704-7350-1 (online)
DOI: https://doi.org/10.1090/memo/1394
Published electronically: January 3, 2023
Keywords: $L$-functions,
modular forms,
special values of $L$-functions,
moments,
mollification,
analytic rank,
shifted convolution sums,
root number,
Kloosterman sums,
resonator method
Table of Contents
Chapters
- 1. The second moment theory of families of $L$-functions
- 2. Preliminaries
- 3. Algebraic exponential sums
- 4. Computation of the first twisted moment
- 5. Computation of the second twisted moment
- 6. Non-vanishing at the central point
- 7. Extreme values of twisted $L$-functions
- 8. Upper bounds for the analytic rank
- 9. A conjecture of Mazur-Rubin concerning modular symbols
- Notation index
Abstract
For a fairly general family of $L$-functions, we survey the known consequences of the existence of asymptotic formulas with power-saving error term for the (twisted) first and second moments of the central values in the family.
We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime $q$, and prove that it satisfies such formulas. We derive arithmetic consequences:
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We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.