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.

Related Topics:
Mathematics - Algebraic variety - Number field - L-function - Euler product - Local zeta function - Automorphic representation - Taniyama-Shimura conjecture - As of 2004 - 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.

Related Topics:
Helmut Hasse - André Weil - Riemann zeta function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 Vp 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

Related Topics:
Rational number - Non-singular - Projective variety - Almost all - Prime number - Finite field

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:Z_{V,Q}(s)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

Related Topics:
Dirichlet series - Complex variable - Infinite product

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:zeta_{V,p}left(p^{-s} ight).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

Related Topics:
Well-defined - Rational function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 functional equation for Z, 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.

Related Topics:
Meromorphic continuation - Functional equation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

Related Topics:
étale cohomology - Ramification theory - Ogg-Néron-Shafarevich criterion - Good reduction - Galois representation - Characteristic polynomial

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: ho(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.

Related Topics:
Frobenius element - Inertia group - Trivial representation - Serre - Deligne - 1960s

~ ~ ~ ~ ~ ~ ~ ~ ~ ~