Caltech-Tsinghua Joint Colloquium

The Arithmetic of Power Series and Applications to Period

Yunqing Tang, Department of Mathematics, Caltech,

https://caltech.zoom.us/j/83227207916?pwd=R8hqeEwfn7jb9ZVLt4px16a2LTAU00

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 André, Bost, Charles and many others generalized the rationality criterion of Dwork and deduced many applications in the arithmetic of differential equations and elliptic curves. In this talk, we will briefly review the history and then discuss some further refinements and generalizations of the criteria of André, Bost, and Charles and their application to irrationality of a special value of a certain Dirichlet L-function using rational approximations constructed by Zagier. This is joint work with Frank Calegari and Vesselin Dimitrov.

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