No abstract

In this paper, under GRH for elliptic L-functions, we give an upper bound for the probability for an elliptic curve with analytic rank ≤ a for a ≥ 11, and also give an upper bound of n-th moments of analytic ranks of elliptic curves. These are applications of counting elliptic curves with local conditions, for example, having good reduction at p.

Under the parity conjecture, an infinite family of elliptic curves of rank 2 with a torsion subgroup of order 2 or 3 is constructed.

In this paper, we show nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the p-th Fermat curve where 2 is a quadratic residue modulo p, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.