**Title:** Arithmetic algebraization theorems and height gap for holonomic functions

**Speaker:** Vesselin Dimitrov

**Speaker Info:** University of Toronto

**Brief Description:**

**Special Note**:

**Abstract:**

An analytic continuation of a function germ past its disk of convergence is a rare and miraculous property, usually linked to either of two basic mechanisms: (a) a differential equation (holonomic functions propagate to any simply connected region avoiding the singularities); or (b) a modular form (having as major example the analytic continuation of an L-function by its functional equation).In this talk I will present some new uses of analytic continuation, coming out of "gluing" theorems to the effect that a formal function or subscheme germ is algebraic as soon as it propagates analytically on a sufficiently large domain. Chow's theorem on projective space is the fundamental example in the analytic category, whereas in the adelic category such results go by the name of arithmetic algebraization theorems. I will survey the basic ones and their proofs (due to Carlson-Polya-Bertrandias, and AndrĂ©), and present a new one as well.

As applications, focusing primarily on (a) for the present talk, I will prove a height gap theorem on holonomic functions that includes as a special case the Schinzel-Zassenhaus conjecture (the case of the Lehmer problem where the Galois orbit is maximally equalized around the unit circle), and discover a new constraint on the critical value set of a rational function. When the holonomic function is irrational, our height gap theorem includes a general Arakelovian version of the height, such as the Call-Silverman dynamical heights and Rumely's capacitary heights.

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