Torsion points on curves and p-adic abelian integrals
type: Item
instance of: scholarly article
title: Torsion points on curves and p-adic abelian integrals
zbMATH Open document ID: 0578.14038
DOI: 10.2307/1971194
author: Robert F. Coleman
published in: Annals of Mathematics. Second Series
publication date: 1985-01-01T00:00:00Z
review text: For an elliptic curve C over a number field K, i.e. a curve of genus \(g=1\) having a rational point over K, the group \(T_ K\) of rational torsion points on C over K is finite by the Mordell-Weil theorem, and the (still unproved) boundedness conjecture implies that \(T_ K\) is bounded by a constant depending only on K. For (smooth) curves C of higher genus \(g\geq 2\) in an abelian variety J, the generalized Manin-Mumford conjecture states the finiteness of the set \(T_{\bar K}\) of torsion points of J arizing from C and being defined over the algebraic closure \(\bar K\) of K. This latter conjecture was proved by S. Lang for abelian varieties J admitting complex multiplication (CM) and by Raynaud for arbitrary abelian varieties. Assuming that the abelian variety J has potential complex multiplication, the author proves the following boundedness theorem: \(\#T_{\bar K}\leq pg,\) where p denotes the smallest prime of \({\mathbb{Q}}\) divisible by a prime in the set of primes \({\mathfrak p}\) of the number field K such that (i) \({\mathfrak p}\) does not divide 2 or 3, (ii) K is unramified at \({\mathfrak p}\), (iii) C has good ordinary reduction over K at \({\mathfrak p}.\) Generalizations of this theorem are also obtained. Moreover, the theorem can be applied to completely determining the torsion points on the Fermat curves \(F(m):X^ m+Y^ m+Z^ m=0\) provided that \(m+1\) is a prime and \(m\geq 10.\) As a tool for proving his theorem, the author develops a theory of p-adic abelian integrals based on Tate's rigid analysis and Monsky-Washnitzer's dagger analysis. In particular, the p-adic integrals of the first kind on an abelian variety turn out to satisfy an addition law as a result of which the torsion points on a curve C in its Jacobian J can be identified as the common zeros of these integrals. This establishes the connection between torsion points and p-adic abelian integrals. Some interesting examples of torsion points on genus \(g=2\) curves are given at the …
Mathematics Subject Classification ID: 14K20
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11S80
zbMATH DE Number: 3924990
zbMATH Keywords: boundedness conjecture for group of rational torsion points
zbMATH Keywords: elliptic curve
zbMATH Keywords: Mordell-Weil theorem
zbMATH Keywords: abelian variety
zbMATH Keywords: potential complex multiplication
zbMATH Keywords: torsion points on the Fermat curves
zbMATH Keywords: p-adic abelian integrals
Wikidata QID: Q56813943
reviewed by: Horst G. Zimmer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971194
OpenAlex ID: W2084600561
description: scientific article; zbMATH DE number 3924990