Divisibility of zeta functions of curves in a covering

摘要

We prove, as an analogy of a conjecture of Artin, that if Y → X is a finite flat morphism between two singular reduced absolutely irreducible projective algebraic curves defined over a finite field, then the numerator of the zeta function of X divides that of Y in Z[T]. Then, we give some interpretations of this result in terms of semi-abelian varieties.