The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of $f$, its local monodromy. We will discuss in this survey talk rationality issues for these zeta functions and the origins of the conjecture.[-]

We are interested in the behaviour of Frobenius roots when the base field is fixed and the genus of the curve or the dimension of the abelian variety tends to infinity. I shall explain how to put the question and what are the answers. This happens to be a question in algebraic number theory and harmonic analysis. For curves (and for number fields) these are my old results with Serge Vladuts, for abelian varieties those of J.-P. Serre (séminaire Bourbaki, 2018) and my work in progress with Nicolas Nadirashvili.[-]