Mathematics Colloquium

Borel and Dwork gave conditions on when a nice power series with rational number coefficients comes from a rational function in terms of meromorphic convergence radii at all places. Such a criterion was used in Dwork's proof of the rationality of zeta functions of varieties over finite fields. Later, the work of Andre, Bost and many others generalized the rationality criterion of Borel--Dwork and deduced many applications in the arithmetic of differential equations and elliptic curves. In this talk, we will discuss some further refinements and generalizations of the criteria of Andre and Bost and their applications to Ogus conjecture for cycles of abelian varieties, the unbounded denominators conjecture for modular forms, and irrationality of certain p-adic zeta value. The second and third applications are joint work with Frank Calegari and Vesselin Dimitrov.