2-Selmer groups, 2-class groups and rational points on elliptic curves

Abstract

Let E : y 2 = F ( x ) E: y^2=F(x) be an elliptic curve over Q \mathbb {Q} defined by a monic irreducible integral cubic polynomial F ( x ) F(x) with negative square-free discriminant − D -D . We determine its 2 2 -Selmer rank in terms of the 2-rank of the class group of the cubic field L = Q [ x ] / F ( x ) L=\mathbb {Q}[x]/F(x) . When the 2 2 -rank of the class group of L L is at most 1 1 and the root number of E E is − 1 -1 , the Birch and Swinnerton-Dyer conjecture predicts that E ( Q ) E(\mathbb {Q}) should have rank 1 1 . We construct a canonical point in E ( Q ) E(\mathbb {Q}) using a new Heegner point construction. We naturally conjecture it to be of infinite order. We verify this conjecture explicitly for the case D = 11 D=11 , and propose an approach towards the general case based on a mod 2 congruence between elliptic curves and Artin representations.