Iwasawa \(L\)-functions of elliptic curves with additive reduction
Let \(A\) be an elliptic curve defined over a number field \(K\) and let \(p\) be an odd prime. Several authors have studied Iwasawa \(p\)-adic \(L\)-functions for \(A\) in the case where \(A\) has semistable reduction at the primes above \(p\), but little has been done in the case of additive reduction. In the present paper, the author considers the cases of potentially good ordinary reduction and potentially multiplicative reduction. Let \(L/K\) be a \(\mathbb{Z}_p\)-extension. The author defines the \(p\)-adic \(L\)-function in terms of the characteristic power series of the Pontryagin dual of a suitable Selmer group for \(A\) over \(L\). He determines the order of vanishing at \(s=1\) and shows that the analogue of the conjecture of Birch and Swinnerton-Dyer holds. In the previous work of \textit{B. Mazur}, \textit{J. Tate}, and \textit{J. Teitelbaum} [Invent. Math. 84, 1-48 (1986; Zbl 0699.14028)]\ in the case where \(A\) has split multiplicative reduction at \(p\), an extra zero (i.e., a higher order of vanishing) appeared. It is interesting to note that this phenomenon does not happen in the present case, even when the reduction is potentially split multiplicative.
characteristic power series of the Pontryagin dual
scientific article; zbMATH DE number 749193