The Birch and Swinnerton-Dyer conjecture for the Mazur-Kitagawa p-adic L-function in the presence of an exceptional zero.

摘要

Starting with the work of Mazur, Tate and Teitelbaum [17], various p-adic analogues of the Birch and Swinnerton-Dyer conjecture have been formulated. The case of an elliptic curve with split multiplicative reduction at the prime p is of special interest. In this so called "exceptional zero" case, the order of vanishing of the Mazur-Swinnerton-Dyer p-adic L-function at the central point seems to be one higher than it is in the classical case. Greenberg and Stevens [10] proved results about this conjecture, using properties of the two variable Mazur-Kitagawa p-adic L-function Lp(E, k, s), which was defined in [14]. Their proof relies on the fact that the Mazur-Kitagawa p-adic L-function Lp( E, k, s) vanishes along the central critical line s = k2 , and the fact that the restriction to k = 2 is equal to the Mazur-Swinnerton-Dyer p-adic L-function attached to E. In the case where Lp( E, k, k2 ) is not identically zero, a formula of Bertolini and Darmon [3] gives a formula for its second derivative at k = 2. Their formula is also valid for twists Lp(E, chi, k, k2 ) of the L-function by quadratic characters chi, and their method of proof relies essentially on the fact that chi is quadratic. This thesis looks into possible generalizations of the result of Bertolini and Darmon in the case of twists by Dirichlet characters of higher order.