2-Selmer groups and Heegner points on elliptic curves

Abstract

This thesis studies several aspects of the arithmetic of elliptic curves. In particular, we explore the prediction of the Birch and Swinnerton-Dyer conjecture when the 2-Selmer group has rank one. For certain elliptic curves $E/\mathbb{Q}: y^2=F(x)$ with additive reduction at 2, we determine their 2-Selmer ranks in terms of the 2-rank of the class group of the cubic field $L=\mathbb{Q}[x]/F(x)$. We then interpret this result as a mod 2 congruence between the Hasse-Weil $L$-function of $E$ and a degree two Artin $L$-function associated to the cubic field $L$. When the class number of $L$ is odd, the Birch and Swinnerton-Dyer conjecture predicts that $E$ should have rank one over $\mathbb{Q}(i)$. To construct such a point on $E$, we study Heegner points on Shimura curves with non-maximal level at a prime $p$ ramified in the quaternion algebra (in the special case when $p = 2$). These curves have a $p$-adic uniformization by a tame \etale covering of Drinfeld's $p$-adic half-plane. We use the covering to describe the geometry of their reduction mod $p$ and compute the N'eron model of their Jacobians. For certain elliptic curves $E/\mathbb{Q}$ with good or multiplicative reduction at 2, we study their 2-Selmer groups over imaginary quadratic fields using the method of level raising of modular forms mod $p=2$. We prove a parity result (predicted by the Birch and Swinnerton-Dyer conjecture) for 2-Selmer ranks. We also show that there is an obstruction for lowering the 2-Selmer ranks, revealing a different phenomenon compared to odd $p$.