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.