2-Selmer groups and Heegner points on elliptic curves

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/Q : y2 = 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 = 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 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 Neron model of their Jacobians.;For certain elliptic curves E/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.