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Book Synopsis The Second Moment Theory of Families of $L$-Functions–The Case of Twisted Hecke $L$-Functions by : Valentin Blomer
Download or read book The Second Moment Theory of Families of $L$-Functions–The Case of Twisted Hecke $L$-Functions written by Valentin Blomer and published by American Mathematical Society. This book was released on 2023-02-13 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
"For a fairly general family of L-functions, we survey the known consequences of the existence of asymptomatic 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 teh 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 times] 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 molecular symbols."--Page v.
Book Synopsis The Second Moment Theory of Families of L-functions by : Valentin Blomer
Download or read book The Second Moment Theory of Families of L-functions written by Valentin Blomer and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "For a fairly general family of L-functions, we survey the known consequences of the existence of asymptomatic 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 teh 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 times] 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 molecular symbols."--Page v.
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