In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is a meromorphic function on the complex plane defined in terms of the number of points on the variety after reducing modulo each prime number p. It is a global L-function defined as an Euler product of local zeta functions.
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 Riemann zeta function, which results from the case when V is a single point.
with p elements, just by reducing equations for V. Scheme-theoretically, this reduction is just the pullback of the Néron model of V along the canonical map Spec
for finitely many primes p. 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.
A more refined definition became possible with the development of étale cohomology; this neatly explains what to do about the missing, 'bad reduction' factors.
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.
The Birch and Swinnerton-Dyer conjecture states that the rank of the abelian group E(K) of points of an elliptic curve E is the order of the zero of the Hasse–Weil L-function L(E, s) at s = 1, and that the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K.[3] The conjecture is one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof.
The Hasse–Weil zeta function of E then takes the form Here, ζ(s) is the usual Riemann zeta function and L(E, s) is called the L-function of E/Q, which takes the form[5] where, for a given prime p, where in the case of good reduction ap is p + 1 − (number of points of E mod p), and in the case of multiplicative reduction ap is ±1 depending on whether E has split (plus sign) or non-split (minus sign) multiplicative reduction at p. A multiplicative reduction of curve E by the prime p is said to be split if -c6 is a square in the finite field with p elements.