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

Li, Chao

doi:10.1090/tran/7373

2-Selmer groups, 2-class groups and rational points on elliptic curves
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Trans. Amer. Math. Soc. 371 (2019), 4631-4653 Request permission

Abstract:

Let $E: y^2=F(x)$ be an elliptic curve over $\mathbb {Q}$ defined by a monic irreducible integral cubic polynomial $F(x)$ with negative square-free discriminant $-D$. We determine its $2$-Selmer rank in terms of the 2-rank of the class group of the cubic field $L=\mathbb {Q}[x]/F(x)$.

When the $2$-rank of the class group of $L$ is at most $1$ and the root number of $E$ is $-1$, the Birch and Swinnerton-Dyer conjecture predicts that $E(\mathbb {Q})$ should have rank $1$. We construct a canonical point in $E(\mathbb {Q})$ using a new Heegner point construction. We naturally conjecture it to be of infinite order. We verify this conjecture explicitly for the case $D=11$, and propose an approach towards the general case based on a mod 2 congruence between elliptic curves and Artin representations.

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