The Second Moment Theory of Families of L-functions-the Case of Twisted Hecke L-functions (Memoirs of the American Mathematical Society, 282) - Softcover

Blomer, Valentin; Kowalski, Emmanuel; Michel, Philippe G.; Milicevic, Djordje; Sawin, Will

Softcover

ISBN 10: 1470456788 ISBN 13: 9781470456788

Publisher: Amer Mathematical Society, 2023

Synopsis

"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: a positive proportion of central values L(f [circled x] X, 1/2) are non-zero, and indeed bounded from below; there exist many characters X for which the central L-value is very large; the probability of a large analytic rank decays exponentially fast. 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"--

Edition details

• Publisher: Amer Mathematical Society
• Publication date: 2023
• Language: English
• ISBN 10: 1470456788
• ISBN 13: 9781470456788
• Binding: Paperback
• Number of pages: 154