On Elliptic Curves and Special Values of L-functions
魏福村 博士清華大學數學系 下午 3:00–4:30 鴻經館 107 室摘要：
Let E be an elliptic curve over a global field k. The conjecture of Birch and Swinnerton-Dyer predicts that the vanishing order of the Hasse-Weil L-function L(E/k,s) at the central critical point s=1 is equal to the rank of the Mordell-Weil group E(k). when the base field k is the rational number field or a global function field, it is known that E(k) is finite if the special value L(E/k,1) is non-zero. On the other hand, the “modularity” of E assures that there exists a unique “weight 2” cusp form f_E such that its L-function coincides with L(E/k,s) In this talk, I will discuss two formulas of central critical L-values coming from these cusp forms. Consequently, these formulas give us criterions for the finiteness of E(k).