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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.
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."
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