Rank determines semi-stable conductor.
type: Item
instance of: scholarly article
title: Rank determines semi-stable conductor.
zbMATH Open document ID: 1047.11053
DOI: 10.1016/S0022-314X(03)00157-4
author: D. Massart
published in: Journal of Number Theory
publication date: 2004-03-14T00:00:00Z
review text: The following question posed by Yu. Zarkhin motivates the study in this paper: Let \(X_1\) and \(X_2\) be Abelian varieties defined over \(k\). If \(\text{rank}_{\mathbb{Z}}(X_1(K))= \text{rank}_{\mathbb{Z}}(X_2(K))\) for every finite extension \(K/k\), then are \(X_1\) and \(X_2\) isogenous over \(k\)? The author proves the following: Let \(E_1\), \(E_2\) be elliptic curves defined over \(\mathbb{Q}\), with conductors \(N(E_1)\) and \(N(E_2)\). Suppose that for every number field \(K\) with \([K:\mathbb{Q}]\leq 2\), \(\text{ord}_{s=1}L(E_1/K,s)\equiv \text{ord}_{s=1} L(E_2/K,s)\pmod 2\) holds. Then \(N(E_1)\) and \(N(E_2)\) are equal up to square factors. Under the additional assumption that \(E_i\) are semi-stable and the 2-primary parts of the Tate-Shafarevich groups of \(E_i\) are finite for all such \(K\), he shows that if \(\text{rank}_{\mathbb{Z}}(E_1(K))\equiv \text{rank}_{\mathbb{Z}}(E_2(K))\pmod 2\), then \(N(E_1)= N(E_2)\). In particular, there are only a finite number of isogeny classes of such elliptic curves.
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 2056169
zbMATH Keywords: elliptic curves
zbMATH Keywords: Mordell-Weil group
describes a project that uses: ecdata
MaRDI profile type: Publication
cites work: Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120)
cites work: On the modularity of elliptic curves over \(\mathbb Q\): wild 3-adic exercises
cites work: Q4003857
cites work: Arithmetic of Elliptic Curves Upon Quadratic Extension
cites work: Q3960951
cites work: Generalizing the Birch-Stephens theorem. I: Modular curves
cites work: Q5625265
cites work: Ring-theoretic properties of certain Hecke algebras
cites work: Modular elliptic curves and Fermat's Last Theorem
Recommended article: The Regularity of the Conductor
Recommended article: Conductors in mixed characteristic
Recommended article: The Stability Theory of Charge Distributions on Conductors
Recommended article: Q3211736
Recommended article: Fluctuations of conductivity and current in conductors
description: scientific article; zbMATH DE number 2056169