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On the \(p\)-part of the Birch-Swinnerton-Dyer formula for multiplicative primes. (English) Zbl 1400.11139

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.

MSC:

11R23 Iwasawa theory
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields

Citations:

Zbl 1401.11103
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