Integral differentials of elliptic function fields
type: Item
instance of: scholarly article
title: Integral differentials of elliptic function fields
zbMATH Open document ID: 1073.11039
DOI: 10.1007/BF02941539
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2005-02-28T00:00:00Z
review text: Let \(K\) be an arithmetic function field of transcendence degree \(r\) over a number field \(k\) and denote by \(A\) the ring of integers of \(k\). In his fundamental work ``Geometria aritmetica'' published in [Ann. Mat. Pura Appl., IV. Ser. 45 (1958; Zbl 0142.18101)], which laid the foundations for an algebraic differential calculus on arithmetic varieties, \textit{E. Kähler} has introduced the so-called module of integral differential forms of \(K\) with respect to the different. This graded \(A\)-module \(D({K\over\vartheta}|A)\) has recently been reconsidered by the authors of the paper under review who gave a geometric-cohomological interpretation of it in terms of a proper and normal model scheme \(X\) of the function field \(K\) [Abh. Math. Semin. Univ. Hamb. 73, 297--310 (2003; Zbl 1045.14009)]. On the other hand, as a variation of Kähler's original differential module, J.-B. Bost has introduced the graded ring \(D_s(K):= \bigcap_{R\in V_s} \Omega_{R/A}\), where \(V_s\) denotes the set of all discrete valuation rings with quotient field \(K\) which are smooth over \(A\) [cf., \textit{J.-B. Bost}, A neglected aspect of Kähler's work on arithmetic geometry birational invariants of algebraic varieties over number fields; in: Erich Kähler, Mathematical Works (R. Berndt and O. Riemenschneider (eds.), De Gruyter Verlag, Berlin, 854--869 (2003; Zbl 1039.01013)]. In the present note, the authors supplement these recent developments by a comparison theorem in the special case of elliptic function fields. More precisely, let \(K\) be the arithmetic function field of an elliptic curve \(E\) defined over the number field \(k\) and having a \(k\)-rational point. Using a Weierstrass model for \(E\), the authors construct a module \(W(E/k)\) describing, in terms of differentials, the reciprocal of the Weierstrass class of \(E\). Then their main theorem establishes the equalities \[ D\biggl({K\over\vartheta} \Bigl| A\biggr)= W(E/k)= D_s(K) \] for elliptic function fields \…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 2139437
zbMATH Keywords: elliptic curves over global fields
zbMATH Keywords: arithmetic function fields
zbMATH Keywords: sheaves of differentials
zbMATH Keywords: Kähler differentials
zbMATH Keywords: arithmetic schemes
zbMATH Keywords: valuation rings
author: E. Kunz
author: R. Waldi
MaRDI profile type: Publication
cites work: A Neglected Aspect of Kähler's Work on Arithmetic Geometry: Birational Invariants of Algebraic Varieties Over Number Fields
cites work: Courbes elliptiques: Formulaire
cites work: On Kähler's integral differential forms of arithmetic function fields
cites work: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux
cites work: Q3333177
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description: scientific article; zbMATH DE number 2139437