Hasse–Weil zeta function
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.
Hasse–Weil L-functions form one of the two major classes of global L-functions, alongside the L-functions associated to automorphic representations. Conjecturally, these two types of global L-functions are actually two descriptions of the same type of global L-function; this would be a vast generalisation of the Taniyama-Weil conjecture, itself an important result in number theory.
For an elliptic curve over a number field K, the Hasse–Weil zeta function is conjecturally related to the group of rational points of the elliptic curve over K by the Birch and Swinnerton-Dyer conjecture.
Hasse–Weil conjecture[edit]
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.
Birch and Swinnerton-Dyer conjecture[edit]
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.[2] 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.[3]