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 Dworkâs 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 the unbounded denominators conjecture for modular forms, and irrationality of 2-adic zeta value at 5 and some other linear independence problems. This is joint work with Frank Calegari and Vesselin Dimitrov.