Moments and distribution of central L-values of quadratic twists of elliptic curves

Abstract

We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real) moments and a one-sided central limit theorem holds. We illustrate our method for the family of quadratic twists of an elliptic curve, obtaining sharp upper bounds for all moments below the first. We also establish a one sided central limit theorem supporting a conjecture of Keating and Snaith. Our work leads to a conjecture on the distribution of the order of the Tate-Shafarevich group for rank zero quadratic twists of an elliptic curve, and establishes the upper bound part of this conjecture (assuming the Birch-Swinnerton-Dyer conjecture).

28 pages

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• The Quantum Variance of the Modular Surface
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Cited by in corpus (24)

• Lower bounds for moments of zeta and $L$-functions revisited
• Quantum Unique Ergodicity for half-integral weight forms
• The second moment theory of families of L-functions
• Moments of quadratic twists of elliptic curve L-functions over function fields
• An upper bound for discrete moments of the derivative of the Riemann zeta-function
• Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics
• Large Fourier coefficients of half-integer weight modular forms
• Non-vanishing for cubic $L$--functions
• A weighted one-level density of families of $L$-functions
• Moments of quadratic Dirichlet $L$-functions over Function Fields
• On the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture
• Extreme central $L$-values of almost prime quadratic twists of elliptic curves
• Bounds for moments of cubic and quartic Dirichlet $L$-functions
• On fundamental Fourier coefficients of Siegel cusp forms of degree 2
• Lower bounds for moments of quadratic twisted self-dual $\text{GL}(3)$ central $L$-values
• Lower bounds for moments of quadratic twists of modular $L$-functions
• The Liouville function in short intervals [after Matomaki and Radziwill]
• The first moment of quadratic twists of modular $L$-functions
• Twisted first moment of quadratic and quadratic twist $L$-functions
• Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions
• Mass equidistribution for Saito-Kurokawa lifts
• Explicit mean value theorems for toric periods and automorphic $L$-functions
• Extreme central values of quadratic Dirichlet $L$-functions with prime conductors
• Lower bounds for moments of the derivative of the Riemann zeta function