'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. 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. Birch and Swinnerton-Dyer conjecture 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. 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. Elliptic curves over Q An elliptic curve is a specific type of variety. 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 p2 does not; this is written p || N), and it has additive reduction elsewhere (i.e.