Principle of Riemann Zeta Functions and Their Strategies |
Zeta or L - functions are modelled on the Riemann'szeta function originally defined by the seriesand then extended to the whole complexplane. The zeta function has an "Euler product", a "functionalequation" and though very much studied still keeps secret many of itsproperties, the greatest mystery being the so-called Hiemann Hypothesis. Manysimilar (or thought to be similar) serieshave been introduced in arithmetic,algebraic geometry and even topology, dynamics (we won't discuss the latter).We plan basically to discuss zeta functions attached to algebraic varietiesover finite fields and global fields. The first applications of zeta functions haw been thearithmetic progession theorem (Dirichlet. 1837) "there exists one (hence infinitely) prime congruent to a modulo b.whenever a and b are coprime" and the prime number theorem(Kiemann 1859, with an incomplete proof: Hadamard and de la Vallee Poussin,1896) "the number of primes lessthan x is asymptotic to Butfurther applications were not restricted to the study of prime numbers, theyinclude the study of the ring of algebraic integers, class field theory, theestimation of the size of solutions of (some) diophantine equations, etc.Moreoverhave provided or suggested fundamentallinks between algebraic varieties (motives over Q). Galois representations,modular or au- tomorphic forms: for example, though they do not appearexplicitly in Wiles work, it seems fair to say they played an important role inthe theory that finally led to the solution of Shimura-Taniyama-Weil conjectureand thus of Fermat's Last Theorem. The first four lectures develop results and definitionswhich though all classical are perhaps not too often gathered together. Thefirst lecture introduces Riemann's zeta function, Dirichletassociated to a character. Dedekind zetafunctions and describes some applications of zeta functions: the secondintroduces the Hasse-Weil zeta functions associated to algebraic varietiesdefined over a finite field, a number field or a function field as well as L-functions associated to Galois representations and modular forms; the thirdreviews techniques from complex analysis and estimates for zeta functions: thefourth touches the theory of special values of zeta functions, some known likethe class number formula and some conjectured like the Birch and Swinnerton-Dyer formula. The fifth and final lecture is an exposition of recent work ofBoris Kunyavskif, Micha Tsfasman. Alexei Zykin. Amflcar Facheco and the authoraround versions and analogues of the Brauer-Siegel theorem. Frerequisite will be kept minimal whenever possible : acourse in complex variable and algebraic number t heory, a bit of Galois theoryplus some exposure to algebraic geometry should suffice.