In mathematics, the Hasse–Weil zeta function attached to an algebraic varietyV defined over an algebraic number fieldK 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 ; this would be a vast generalisation of the Taniyama–Shimura conjecture, itself a very deep and recent result 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 fieldQ, and V a non-singularprojective variety, we can for almost allprime numbersp consider the reduction of V modulo p, an algebraic variety Vp over the finite fieldFp with p elements, just by reducing equations for V. Again for almost all p it will be non-singular. We define to be the Dirichlet series of the complex variables, which is the infinite product of the localzeta functions Then Z, according to our definition, is well-defined only up to multiplication by rational functions in a finite number of. Since the indeterminacy is relatively harmless, and has meromorphic continuation everywhere, there is a sense in which the properties of Z do not essentially depend on it. In particular, while the exact form of the 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. 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. 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 Frob being a Frobenius element for p. What happens at the ramified p is that ρ is non-trivial on the inertia groupI 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 can be upgraded successfully from 'almost all' p to allp 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.
Let E be an elliptic curve over Q of conductorN. Then, E has good reduction at all primes p not dividing N, it has multiplicative reduction at the primes p that exactly divide N, and it has additive reduction elsewhere. The Hasse–Weil zeta function of E then takes the form Here, ζ is the usual Riemann zeta function and L is called the L-function of E/Q, which takes the form where, for a given prime p, where, in the case of good reduction ap is p + 1 − , and in the case of multiplicative reduction ap is ±1 depending on whether E has split or non-split multiplicative reduction at p.
