Class numbers of quadratic fields and Shimura's correspondence

Abstract

It is well-known (see Fueter [8] and Aigner [1]) that Fermat’s equation X+Y 3 = 1 has no solutions in a quadratic field K = Q( √ D) ( for |D| ≡ 1 (mod 3)), provided the class number of K is not divisible by 3. This subject was further investigated by Frey [6], who gave estimates for the 3-rank of the class group of K in terms of 3-descent on a curve Y 2 = X +D. In this paper we consider curves ED : DY 2 = 4X − 27 ,which are quadratic twists of the Fermat’s curve X +Y 3 = 1. We give a precise formula for the rank of the Selmer group corresponding to the complex multiplication √ −3 : ED −→ E−3D in terms of the 3-rank of the class group of Q( √ D) resp. Q( √ −3D). This result may be considered as a quantitative version of [1] and [8]. We discuss also Birch and Swinnerton-Dyer’s conjecture for curves ED. According to a theorem of Waldspurger [22], [23], natural rational factor of L(ED/Q, 1) may be expressed in terms of coefficients of certain modular forms of weight 3/2. We identify these forms explicitely and verify (mod 3)-part of Birch and SwinnertonDyer’s conjecture for ED in most cases (for D of density 5/6 ).