Conjectures in Arithmetic Algebraic Geometry : A Survey

Wilfred W. J. Hulsbergen

ISBN 10: 366309507X / ISBN 13: 9783663095071

Editore: Vieweg+Teubner Verlag, Vieweg+Teubner Verlag, 2013

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Druck auf Anfrage Neuware - Printed after ordering - In this expository text we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued math ematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to introduce L functions, the main, motivation being the calculation of class numbers. In partic ular, Kummer showed that the class numbers of cyclotomic fields play a decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirichlet 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 properties of L-functions. Twentieth century number theory, class field theory and algebraic geome try only strengthen the nineteenth century number theorists's view. We just mention the work of E. H~cke, 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 generalization of Dirichlet's L-functions with a generalization of class field theory to non-abelian Galois extensions of number fields in mind.

Codice articolo 9783663095071

Dati bibliografici

• Titolo: Conjectures in Arithmetic Algebraic Geometry : A Survey
• Editore: Vieweg+Teubner Verlag, Vieweg+Teubner Verlag
• Data di pubblicazione: 2013
• Lingua: Inglese
• ISBN 10: 366309507X
• ISBN 13: 9783663095071
• Rilegatura: Taschenbuch
• Edizione: seconda edizione
• Condizione: Neu
• Dimensioni: 244 L x 170 A x 15 P (millimetri) 244 millimetri di larghezza per 170 millimetri di altezza per 15 millimetri di profondità
• Peso: 448 grammi

Descrizione articolo

Riassunto

In the early 1980's, stimulated by work of Bloch and Deligne, Beilinson stated some intriguing conjectures on special values of L-functions of algebraic varieties defined over number fields. Roughly speaking these special values are determinants of higher regulator maps relating the higher algebraic K-groups of the variety to its cohomology. In this respect, higher algebraic K-theory is believed to provide a universal, motivic cohomology theory and the regulator maps are determined by Chern characters from higher algebraic K-theory to any other suitable cohomology theory. Also, Beilinson stated a generalized Hodge conjecture. This book provides an introduction to and a survey of Beilinson's conjectures and an introduction to Jannsen's work with respect to the Hodge and Tate conjectures. It addresses mathematicians with some knowledge of algebraic number theory, elliptic curves and algebraic K-theory.

Informazioni sull?autore

Dr. Wilfried Hulsbergen is teaching at the KMA, Breda,Niederlande.