國立清華大學 數學系

Abstract: 
As a refinement of Goldfeld’s conjecture, there is a conjecture of Keating-Snaith asserting log L(1/2,E_d) of certain quadratic twists E_d of an elliptic curve E behave like a normal random variable. In light of this, Radziwill and Soundararajan conjectured that the distribution of log(|Sha(E_d)|/|d|^0.5) is also approximately Gaussian for these E_d, and proved that the conjectures of Keating-Snaith and theirs are both valid “from above”. More recently, under GRH, they further established a lower bound for the involving distribution towards Keating-Snaith’s conjecture.
In this talk, after briefly reviewing some background knowledge, we shall discuss the joint distribution of central values and orders of Sha groups of E_d and how to adapt Radziwill-Soundararajan’s methods to study upper bound and lower bounds for such a joint distribution if time allows.