Conjectures in Arithmetic Algebraic Geometry : A Survey

Hulsbergen, Wilfred W. J.

ISBN 10: 3528064331 / ISBN 13: 9783528064334

Published by Vieweg+Teubner Verlag 1992-01, 1992

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Bibliographic Details

Title: Conjectures in Arithmetic Algebraic Geometry...

Publisher: Vieweg+Teubner Verlag 1992-01

Publication Date: 1992

Binding: PF

Condition: New

About this title

Synopsis:

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro­ duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich­ let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper­ ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza­ tion of Dirichlet's L-functions with a generalization of class field the­ ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.

Synopsis:

An introduction to and a survey of Beilison's conjectures on special values of L -functions of algebraic varieties defined over number fields and an introduction to Jannsen's work with respect to Hodge and Tate conjectures. It addresses mathematicians with some knowledge of algebraic number theory,