Hasse–Weil L-function

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^[update]) 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

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