Hasse–Weil zeta function

In mathematics, the Hasse-Weil zeta-function attached to an algebraic variety V defined over a number field K is one of the two most important types of L-function. Such L-functions are called 'global', in that they are defined as Euler products in terms of local zeta-functions. They form one of the two major classes of global L-functions, the other being the L-functions associated to automorphic representations. Conjecturally there is just one essential type of global L-function, with two descriptions (coming from an algebraic variety, coming from an automorphic representation); this would be a vast generalisation of the Taniyama-Shimura conjecture, itself a very deep and recent result (as of 2004) in number theory.

The description of the Hasse-Weil zeta-function up to finitely many factors of its Euler product is relatively simple. This follows the initial suggestions of Helmut Hasse and André Weil, motivated by the case in which V is a single point, and the Riemann zeta-function results.

Taking the case of K the rational number field Q, and V a non-singular projective variety, we can for almost all prime numbers p consider the reduction of V modulo p, an algebraic variety V_p over the finite field $F_{p}$ with p elements, just by reducing equations for V. Again for almost all p it will be non-singular. We define

$Z_{V,Q}(s)$

to be the Dirichlet series of the complex variable s, which is the infinite product of the local zeta-functions

$\zeta _{V,p}\left(p^{-s}\right).$

Then $Z(s)$, according to our definition, is well-defined only up to multiplication by rational functions in a finite number of $p^{-s}$.

Since the indeterminacy is relatively anodyne, and has meromorphic continuation everywhere, there is a sense in which the properties of $Z(s)$ do not essentially depend on it. In particular, while the exact form of the functional equation for Z(s), reflecting in a vertical line in the complex plane, will definitely depend on the 'missing' factors, the existence of some such functional equation does not.

A more refined definition became possible with the development of étale cohomology; this neatly explains what to do about the missing, 'bad reduction' factors. According to general principles visible in ramification theory, 'bad' primes carry good information (theory of the conductor). This manifests itself in the étale theory in the Ogg-Néron-Shafarevich criterion for good reduction; namely that there is good reduction, in a definite sense, at all primes p for which the Galois representation ρ on the étale cohomology groups of V is unramified. For those, the definition of local zeta-function can be recovered in terms of the characteristic polynomial of

$\rho (Frob(p)),$

$Frob(p)$ being a Frobenius element for p. What happens at the ramified p is that ρ is non-trivial on the inertia group $I(p)$ for p. At those primes the definition must be 'corrected', taking the largest quotient of the representation ρ on which the inertia group acts by the trivial representation. With this refinement, the definition of $Z(s)$ can be upgraded successfully from 'almost all' p to all p participating in the Euler product. The consequences for the functional equation were worked out by Serre and Deligne in the later 1960s; the functional equation itself has not been proved in general.

Reference

• J.-P. Serre, Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures), 1969/1970, Sém. Delange-Pisot-Poitou, exposé 19