Person:

Li, Chao

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 $p$.