An Introduction to P -adic and Motivic Zeta Functions and the Monodromy Conjecture

Introduced by Weil, the p-adic zeta function associated to a polynomial f over Zp was systematically studied by Igusa in the non-archimedean wing of his theory of local zeta functions, which also includes archimedean (real and complex) zeta functions [18][19]. The p-adic zeta function is a meromorphic function on the complex plane, and contains information about the number of solutions of the congruences f ≡ 0 mod p for m > 0. Igusa formulated an intriguing conjecture, the monodromy conjecture, stating that, if f is defined over Z, the poles of its p-adic zeta function are closely related to the structure of the singularities of the complex hypersurface defined by f (see Conjecture 3.1 for the precise statement). Special cases of the conjecture have been solved (in particular the case where f is a polynomial in two variables) but the general case remains quite mysterious.