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).Comment: 28 page
