The equivariant Tamagawa number conjecture and modular symbols
type: Item
instance of: scholarly article
title: The equivariant Tamagawa number conjecture and modular symbols
zbMATH Open document ID: 1288.11066
published in: Mathematische Annalen
publication date: 2013-05-30T00:00:00Z
review text: Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\), \(K/\mathbb{Q}\) a finite Galois extension with \(G:=\mathrm{Gal}(K/\mathbb{Q})\) and \(E_K\) the base change of \(E\) to \(K\). The paper deals with the \(\ell\)-part of the Equivariant Tamagawa Number Conjecture (ETNC\(_\ell\) from now on), as formulated in [\textit{D. Burns} and \textit{M. Flach}, Doc. Math., J. DMV 6, 501--570 (2001; Zbl 1052.11077)], for the motive \(h^1(E_K)(1)\) with the action of \(\mathbb{Q}[G]\) and for \(\ell\)-\textit{arithmetically trivial pairs} \((E,K)\). A pair \((E,K)\) is called \(\ell\)-arithmetically trivial if it verifies: \noindent a) \([K:\mathbb{Q}]\) is a power of \(\ell\); \noindent b) \((d_K,\ell)=(N_E,\ell)=(d_K,N_E)=1\) (\(d_K\) is the discriminant of \(K\), \(N_E\) is the conductor of \(E\)); \noindent c) \(\ell \nmid \# E(\mathbb{Q})_{\mathrm{tors}} \prod_{p\mid d_K} \# \overline{E}(\mathbb{F}_p)\) (\(\overline{E}\) is the reduced curve); \noindent d) \(\ell \nmid \prod_{p\mid N_E} c_p(E,\mathbb{Q})\) (\(c_p(E,\mathbb{Q})\) is the Tamagawa factor at \(p\)); \noindent e) rank\(_{\mathbb{Z}}\) \((E(K))=0\); \noindent f) \(\ell\) does not divide the order of the Tate-Shafarevich group of \(E\) over \(K\). \noindent In a previous paper [Math. Comput. 81, No. 279, 1681--1705 (2012; Zbl 1277.11071)] the author proved the equivalence between the ETNC\(_\ell\) and a congruence, modulo \(\mathbb{Z}_\ell[G]^*\,\), between the product on primes dividing \(d_K\) of local Euler factors (twisted by the absolutely irreducible characters \(\chi\) of \(G\)) for \(E/\mathbb{Q}\) at \(s=1\) and the leading coefficients for the (twisted) \(L\)-functions \((L(E/\mathbb{Q},\overline{\chi},s))_\chi\) at \(s=1\). The rationality and an explicit description of the last object are the main features of the ETNC and the ETNC\(_\ell\) simply considers the same problem modulo \(\ell\)-adic units. \noindent In the paper under review the author proves first the congruence mentioned above (he…
reviewed by: Andrea Bandini
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11R65
zbMATH DE Number: 6168467
zbMATH Keywords: equivariant Tamagawa number conjecture
zbMATH Keywords: modular symbol
zbMATH Keywords: \(L\)-function
author: Werner Bley
MaRDI profile type: Publication
full work available at URL: s00208-012-0837-6
OpenAlex ID: W2006794958
Wikidata QID: Q123018547
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DOI: 10.1007/S00208-012-0837-6
description: scientific article; zbMATH DE number 6168467