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On the inequality of Castelnuovo-Severi

On the inequality of Castelnuovo-Severi Oil the inequality of Castehluovo-Severi TO EMIL ART1N on his 60 th birthday by ARTHUR MATTUCK and JOHN TATE It gives us great pleasure to dedicate this paper to ARTIN. In his thesis, which was published in 1924 in the Mathematische Zeitschrift, ART1N investigated the arithmetic of hypereUiptic function fields over finite fields and called attention to the fact that the analog of RIEMAN~'S hypothesis seemed to be true for them. In three papers in Crelle, 1936, HASSE developed (from a purely arithmetic point of view) the theory of endomorphisms of elliptic curves over arbitrary ground fields and was able to prove the RIEMAN~ hypothesis in the elliptic case. Soon after, WEIL realized that the general case would follow from an inequality of CASTEL~UOVO and Sv, VERI concerning correspondences between algebraic curves. In the form (1) below this inequality appears in SEvwPs book, Trattato di Geometrica Algebrica (Zanichelli, Bologna, 1926) on p. 265, where it is deduced from a formula of CASTEL~UOVO concerning algebraic series of divisors on curves. By establishing (1) for arbitrary ground fields, W~IL proved the RI~.MANN hypothesis in general (C. R., Paris, 1940). The ideas involved are fully elaborated in his two mono- graphs on http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

On the inequality of Castelnuovo-Severi

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02941959
Publisher site
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Abstract

Oil the inequality of Castehluovo-Severi TO EMIL ART1N on his 60 th birthday by ARTHUR MATTUCK and JOHN TATE It gives us great pleasure to dedicate this paper to ARTIN. In his thesis, which was published in 1924 in the Mathematische Zeitschrift, ART1N investigated the arithmetic of hypereUiptic function fields over finite fields and called attention to the fact that the analog of RIEMAN~'S hypothesis seemed to be true for them. In three papers in Crelle, 1936, HASSE developed (from a purely arithmetic point of view) the theory of endomorphisms of elliptic curves over arbitrary ground fields and was able to prove the RIEMAN~ hypothesis in the elliptic case. Soon after, WEIL realized that the general case would follow from an inequality of CASTEL~UOVO and Sv, VERI concerning correspondences between algebraic curves. In the form (1) below this inequality appears in SEvwPs book, Trattato di Geometrica Algebrica (Zanichelli, Bologna, 1926) on p. 265, where it is deduced from a formula of CASTEL~UOVO concerning algebraic series of divisors on curves. By establishing (1) for arbitrary ground fields, W~IL proved the RI~.MANN hypothesis in general (C. R., Paris, 1940). The ideas involved are fully elaborated in his two mono- graphs on

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Aug 29, 2008

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