Iwasawa \(L\)-functions of elliptic curves with additive reduction
type: Item
instance of: scholarly article
title: Iwasawa \(L\)-functions of elliptic curves with additive reduction
zbMATH Open document ID: 0844.11042
author: John W. Jones
published in: Journal of Number Theory
publication date: 1996-09-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 749193
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
zbMATH Keywords: elliptic curve
zbMATH Keywords: Iwasawa \(p\)-adic \(L\)-functions
zbMATH Keywords: additive reduction
zbMATH Keywords: potentially good ordinary reduction
zbMATH Keywords: potentially multiplicative reduction
zbMATH Keywords: characteristic power series of the Pontryagin dual
zbMATH Keywords: order of vanishing
reviewed by: Lawrence C. Washington
MaRDI profile type: Publication
full work available at URL: jnth.1995.1037
OpenAlex ID: W2092434516
DOI: 10.1006/JNTH.1995.1037
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description: scientific article; zbMATH DE number 749193