Approaching the kannan-lovasz-simonovits and variance conjectures, Alonso-gutierrez, David Bastero, Jesus

Автор: Wilfred W. J. Hulsbergen
Название: Conjectures in Arithmetic Algebraic Geometry
ISBN: 3528064331 ISBN-13(EAN): 9783528064334
Издательство: Springer
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Описание: 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.

Автор: John Coates; Peter Schneider; R. Sujatha; Otmar Ve
Название: Noncommutative Iwasawa Main Conjectures over Totally Real Fields
ISBN: 3642443354 ISBN-13(EAN): 9783642443350
Издательство: Springer
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Описание: The algebraic techniques developed by Kakde will almost certainly lead eventually to major progress in the study of congruences between automorphic forms and the main conjectures of non-commutative Iwasawa theory for many motives.