In mathematics, the **Hasse–Weil zeta function** attached to an algebraic variety *V* defined over an algebraic 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 2009) 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

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

Since the indeterminacy is relatively harmless, 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

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.

# Example: elliptic curve over Q

Let *E* be an elliptic curve over **Q** of conductor *N*. Then, *E* has good reduction at all primes *p* not dividing *N*, it has multiplicative reduction at the primes *p* that *exactly* divide *N* (i.e. such that *p* divides *N*, but *p*2 does not; this is written *p* || *N*), and it has additive reduction elsewhere (i.e. at the primes where *p*2 divides *N*). The Hasse–Weil zeta function of *E* then takes the form

Here, ζ(*s*) is the usual Riemann zeta function and *L*(*s*, *E*) is called the *L*-function of *E*/**Q**, which takes the form^{[1]}

where, for a given prime *p*,

where, in the case of good reduction *a**p* is *p* + 1 − (number of points of *E* mod *p*), and in the case of multiplicative reduction *a**p* is ±1 depending on whether *E* has split or non-split multiplicative reduction at *p*.

# Hasse–Weil conjecture

The Hasse–Weil conjecture states that the Hasse–Weil zeta function should extend to a meromorphic function for all complex *s*, and should satisfy a functional equation similar to that of the Riemann zeta function. For elliptic curves over the rational numbers, the Hasse–Weil conjecture follows from the modularity theorem.