Home » Conjectures In Arithmetic Algebraic Geometry: A Survey by Wilfred W. J. Hulsbergen
Conjectures In Arithmetic Algebraic Geometry: A Survey Wilfred W. J. Hulsbergen

Conjectures In Arithmetic Algebraic Geometry: A Survey

Wilfred W. J. Hulsbergen

Published January 1st 1992
ISBN : 9783528064334
Paperback
240 pages
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 About the Book 

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 Fermats Last Theorem one is naturally ledMoreIn 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 Fermats 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 Fermats 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 Riemanns 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 theoristss view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlets 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 Dirichlets 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.