Idempotent relations and the conjecture of Birch and Swinnerton-Dyer
type: Item
instance of: scholarly article
title: Idempotent relations and the conjecture of Birch and Swinnerton-Dyer
zbMATH Open document ID: 1089.11036
author: Hoseog Yu
published in: Mathematische Annalen
publication date: 2003-11-04T00:00:00Z
full work available at URL: http://rave.ohiolink.edu/etdc/view
review text: Let \(A\) be an abelian variety defined over a number field \(K\), and let \(L(A,s)\) denote its global \(L\)-function. If \(L(A,s)\) has an analytic continuation around \(s=1\) and if the Shafarevich-Tate group is finite, then the leading coefficient of the Taylor expansion of the \(L\)-function is conjecturally given by the so-called constant \(C(A/K)\) of Birch and Swinnerton-Dyer. The author proves various identities concerning the constants \(C(A/K')\) as \(K'\) varies over the subfields \(K'\) of a finite Galois extension \(L/K\) of \(K\) with Galois group \(G\). These identities arise from idempotent relations in the group ring \(\mathbb Q[G]\) in the following way: Let \(\varepsilon_H = \frac{1}{| H| } \sum_{\sigma\in H}\,\sigma\) denote the idempotent attached to the subgroup \(H\). The author shows that under the assumption that the Shafarevich-Tate group is finite a relation \(\sum_H n_H \varepsilon_H = 0\) between the idempotents implies the relation \(\prod_H \,C(A/L^H)^{n_H} = 1.\) In the special case of elliptic curves this was conjectured by \textit{H. Park} [``Idempotent relations and the conjecture of Birch and Swinnerton-Dyer'', in: Algebra and Topology (Taejon, 1990), 97--125 (1990; Zbl 0736.11029)]. The author then considers the more general situation of a twisted group ring \(\text{End}_L(A)[G]\), introduces twisted idempotent relations and proves similar results involving twists of abelian varieties.
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 2001048
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
zbMATH Keywords: abelian varieties
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: idempotent relations
reviewed by: Manfred Kolster
MaRDI profile type: Publication
OpenAlex ID: W2050697968
Wikidata QID: Q123002688
cites work: Néron Models
cites work: Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer.
cites work: Q3996909
cites work: On the conjecture of Birch and Swinnerton-Dyer
cites work: Relations Between the Genera and Between the Hasse-Witt Invariants of Galois Coverings of Curves
cites work: Idempotent relations and factors of Jacobians
cites work: Idempotent relations among arithmetic invariants attached to number fields and algebraic varieties
cites work: Galois descent and twists of an abelian variety
cites work: On the arithmetic of abelian varieties
cites work: Q5628266
cites work: Q3973943
cites work: Q3977592
cites work: Q3710637
cites work: Q5598116
cites work: Q5728890
cites work: Q5553838
DOI: 10.1007/S00208-003-0427-8
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description: scientific article; zbMATH DE number 2001048