Castella, Francesc On the \(p\)-part of the Birch-Swinnerton-Dyer formula for multiplicative primes. (English) Zbl 1400.11139 Camb. J. Math. 6, No. 1, 1-23 (2018). For an elliptic curve \(E\) over \(\mathbb{Q}\), an important and difficult problem is to determine the leading term of the \(L\)-function of \(E\). Recently, there have been remarkable progresses in the \(p\)-part of the leading term of \(L'(E,1)\) (called the \(p\)-part of the Birch and Swinnerton-Dyer formula) when \(E\) is of analytic rank one.The paper under review gives a proof of the \(p\)-part of the Birch and Swinnerton-Dyer formula for a semistable elliptic curve \(E/\mathbb{Q}\) of conductor \(N\) when \(E[p]\) is irreducible (\(p\mid N\) and \(p>3\)). Though it is a difficult task to treat the prime \(p\) which divides the conductor \(N\), the paper is well written and gives a very neat proof. To overcome the difficulty in the case of \(p\mid N\), the key method of the paper is to prove a two-variable analog of the Iwasawa-Greenberg main conjecture over the Hida family and, by specialization, deduce the Iwasawa-Greenberg main conjecture for the anticyclotomic Selmer group of \(E\) (under mild hypotheses) which is itself really worthy. Reviewer: Kazuma Morita (Sapporo) Cited in 1 ReviewCited in 5 Documents MSC: 11R23 Iwasawa theory 11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G05 Elliptic curves over global fields Keywords:Birch and Swinnerton-Dyer formula; elliptic curves; multiplicative reduction Citations:Zbl 1401.11103 PDFBibTeX XMLCite \textit{F. Castella}, Camb. J. Math. 6, No. 1, 1--23 (2018; Zbl 1400.11139) Full Text: DOI arXiv