The Eisenstein descent on \(J_ 0(N)\)
type: Item
instance of: scholarly article
title: The Eisenstein descent on \(J_ 0(N)\)
zbMATH Open document ID: 0594.14027
DOI: 10.1007/BF01388965
author: Jonathan Lubin
author: Benedict H. Gross
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143314
review text: The paper under review deals with the general question of how to compute the rank of an abelian variety over a number field K. The standard approach uses the injection \(A(K)/n\cdot A(K)\to H^ 1(0_ k,A[n])\) (flat cohomology): One tries to find an upper bound for the righthand side, and a lower bound for the left hand side which coincides with it. Here the authors deal with A \(= Eisenstein\)-quotient of \(J_ 0(N) = Jacobian\) of the modular curve \(X_ 0(N)\), \(N=p\) or \(N=p^ 2\) (p a prime. The case \(K={\mathbb{Q}}\), \(N=p\) has been treated extensively by B. Mazur). Under certain hypotheses (on Bernoulli-numbers) the authors can show that either the right hand side vanishes, or that the theory of Heegner points suffices to account for all of it. The details of their arguments are a little bit involved, so we prefer not to comment on them.
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14K30
zbMATH DE Number: 3955108
zbMATH Keywords: descent
zbMATH Keywords: Eisenstein-quotient of Jacobian of modular curve
zbMATH Keywords: rank of an abelian variety
zbMATH Keywords: Eisenstein-quotient of
zbMATH Keywords: Jacobian of the modular curve
zbMATH Keywords: Heegner points
MaRDI profile type: Publication
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reviewed by: Gerd Faltings
description: scientific article; zbMATH DE number 3955108