Extra rank for odd parity quadratic twists
نویسنده
چکیده
We consider the question of giving an asymptotic estimate for the number of odd parity quadratic twists of a fixed elliptic curve that have (analytic) rank greater than 1. The use of Waldspurger’s formula and modular forms of weight 3/2 has allowed the accrual of a large amount of data for the even parity analogue of this, and we use the method of Heegner points to fortify the data in the odd parity case. We are able to provide strong evidence for the −3/2 exponent in the analogue of the ratios conjecture (here we fix a modulus p and consider the ratio of extra vanishings when comparing twists by squares to twists by nonsquares). Our data are not so compelling when considering just the counting function of rank 3 twists, but we are still able to make some comments about the likely validity of various theoretical models.
منابع مشابه
Twists of elliptic curves of rank at least four
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