Congruence Relations Connecting Tate-Shafarevich Groups with Hurwitz Numbers

Abstract

Let p > 3 be a rational prime congruent to 3 modulo 4, and h(−p) be the class number of the imaginary quadratic field Q(\\sqrt{-p}). Then h(−p)≡−2B\\frac{p+1}{2}modp, where Bn is the n-th Bernoulli number. This is a quite classical congruence. Under the full BSD conjecture, we provide an easy method to obtain the natural explicit generalization of this, which is a congruence between the conjectural order of the Tate-Shafarevich group for certain elliptic curve with Mordell-Weil rank 0 and a coefficient of power series expansion of an elliptic function associating the elliptic curve.