Bounds on the number of non-rational subfields of a function field
type: Item
instance of: scholarly article
title: Bounds on the number of non-rational subfields of a function field
zbMATH Open document ID: 0615.12017
DOI: 10.1007/BF01388797
author: Ernst Kani
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143364
review text: Let C be a smooth, geometrically connected, projective algebraic curve defined over a field \(K\). Denote by \(J_ C\) the Jacobian and by \(F=K(C)\) the function field of \(C\) over \(K\), and let \(g=g_ F\) be the genus of F. To every separable, nonconstant morphism \(f: C\to C'\) of \(C\) to an algebraic curve \(C'\) over \(K\) with function field \(F'=K(C')\) of genus g'\(\geq 2\), one can associate an invariant \(\gamma^*_ E\) in the K- endomorphism ring \(End_ K(J_ C)\) of \(J_ C\) by taking \[ \gamma^*_ E:=\gamma^*_ f:=f^*\circ f_*: J_ C\to^{f_*}J_{C'}\to^{f^*}J_ C\quad, \] where \(f_*\) (resp. \(f^*)\) is the direct (resp. inverse) image map acting on the respective Jacobian varieties and \(E\) is the image of \(F'\), \(E:=f^* F'\leq F\), a subfield of \(F\). The classical theorem of de Franchis states that a) the number of invariants \(\gamma^*_ E\in End_ K(J_ C)\) is finite, and b) for any fixed invariant \(\gamma^*_ E\in End_ K(J_ C)\), there are at most finitely many subfields \(E/K\) of \(F/K\). By modifying the classical proof of de Franchis in the setting of \textit{P. Samuel} [Lectures on old and new results on algebraic curves, Tata Inst. Fundam. Res. (Bombay 1966; Zbl 0165.241)], \textit{A. Howard} and \textit{A. J. Sommese} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 429-436 (1983; Zbl 0534.14016)], in the case of \(K={\mathbb{C}}\), gave an explicit upper bound for the number \(N_ F\) of non-rational subfields of \(F/K\) in terms of the genus g of \(F/K\). (An error in their result was corrected by the author.) In the present very interesting and important paper, an ''Angle theorem'' is proved and combined with a ''packing argument'' due to Mumford to obtain a new and effective proof of de Franchis' theorem. In fact the author establishes the following refined explicit bound. Let r be the rank of \(End_ K(J_ C)\). It is known that \(r\leq r(g)\) for \(r(g)=2g^ 2\) or \(=4g^ 2\), according as the characteristic of K is \(=0\) or \(\neq 0\),…
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14E99
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 3995831
zbMATH Keywords: number of non-rational subfields
zbMATH Keywords: number of separable subfields
zbMATH Keywords: number of morphisms of algebraic curves
zbMATH Keywords: Chow coordinates
zbMATH Keywords: theorem of the base
zbMATH Keywords: Jacobian
zbMATH Keywords: genus
zbMATH Keywords: function field
zbMATH Keywords: Angle theorem
zbMATH Keywords: de Franchis' theorem
reviewed by: Horst G. Zimmer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3995831