Geometry of \(p\)-jets
type: Item
instance of: scholarly article
title: Geometry of \(p\)-jets
zbMATH Open document ID: 0882.14007
DOI: 10.1215/S0012-7094-96-08216-2
published in: Duke Mathematical Journal
publication date: 1996-09-08T00:00:00Z
review text: In the author's previous work on diophantine problems over function fields, methods of differential algebra were used. In this present work, the author concentrates on number fields where, of course, no non-trivial derivations exist. The author replaces these with certain ``natural'' non-additive maps called ``\(p\)-derivations''. The main result shown is the following quantitative version of the Manin-Mumford conjecture: Theorem: Let \(X\) be a curve defined over a number field \(K\) and let \(X\to J\) be the associated Abel map into the Jacobian of \(X\). Let \({\mathfrak p}\) be a prime of \(K\) with \(p= \text{char} {\mathfrak p} >2g\) with \(g\) the genus of \(X\). Assume that \(K/ \mathbb{Q}\) is unramified at \({\mathfrak p}\) and that \(X/K\) has good reduction at \({\mathfrak p}\). Let \(K^{ac}\) be the algebraic closure of \(K\). Then \[ \# (X(K^{ac}) \cap J(K^{ac})_{\text{tors}}) \leq p^{4g} 3^g[p(2g-2) +6g]\cdot g!. \]
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 11G30
zbMATH DE Number: 908694
zbMATH Keywords: \(p\)-derivations
zbMATH Keywords: Manin-Mumford conjecture
zbMATH Keywords: Abel map
zbMATH Keywords: Jacobian
author: Alexandru Buium
reviewed by: David Goss
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 908694