ZhengTong Chern-Weil Symposium Autumn 2025: Vesselin Dimitrov (Caltech)

The Square Root Method in the Algebraization of Dirichlet Series

Abstract:

Founded upon an interplay of integral coefficients and large analytic continuations, the algebraization of formal functions is an old theme that dates back to Emile Borel’s reading of a passage from Hadamard’s Essai sur l’étude des fonctions données par leur développement de Taylor. The entire corpus of arithmetic algebraization works has so far aligned to the title of Hadamard’s essay: the case that the formal function was initially given as a power series. While this line of work connects directly to some applications—old and new—to function field arithmetic, it is by far less obvious if and how to perform an analogous Diophantine analysis in the global realm: the case that the formal function is initially given by a general (non-sparse) Dirichlet series. In this symposium talk, we explain how to devise an algebraization method in order to characterize automorphic forms as the only analytic continuation mechanism to a sufficiently large domain of the complex plane, for pairs of Dirichlet series with almost integral coefficients and connected by a functional equation of the standard GL(2) type. We will be guided by blueprint theorems of a Drinfeld—Vladut kind, except now concerning the “non almost doubling" of zeros in functional field arithmetic. In the number-theoretic realm, the key new principle proposes how to use the global Euler product in the context of the original Deuring—Heilbronn phenomenon. This ultimately leads to an effectivization of the first historical proof of the finiteness of the class number one quadratic imaginary fields.

Видео ZhengTong Chern-Weil Symposium Autumn 2025: Vesselin Dimitrov (Caltech) канала University of Chicago Department of Mathematics