# **CANCELED** Schinzel-Zassenhaus, height gap and overconvergence

NUMBER THEORY

##### Speaker:

Vesselin Dimitrov *- University of Toronto*

We explain a new height gap result on holonomic power series

with rational coefficients, and prove the Schinzel-Zassenhaus conjecture

as its consequence: a monic irreducible non-cyclotomic integer polynomial

of degree $n > 1$ has at least one complex root of modulus exceeding

$2^{1/(4n)}$. The method, an arithmetic algebraization argument with input

from two fixed places of the global field $\mathbb{Q}$, takes a

combination of a $p$-adic overconvergence for a certain algebraic

function, at a fixed prime $p$ on the one hand (this is the place where

the cyclotomic examples get recognized and excluded); and, on the other

hand, of the automatic Archimedean analytic continuation for solutions of

linear ODE. The input at the Archimedean place has a potential-theoretic

flavor and comes out of Dubinin's solution of an extremal problem of

Gonchar about harmonic measure. A possible connection to the $p$-curvature

conjecture is indicated, as an analogous height gap hypothesis for

G-operators with infinite monodromy.

Repeating the same pattern, we shall follow up by a sharp arithmetic

criterion on a formal power series to satisfy a linear differential

equation with polynomial coefficients. It upgrades the classical

Polya-Bertrandias rationality criterion, and we shall conclude by

explaining how this new criterion serves to amplify Calegari's p-adic

counterpart of Apery's theorem, yielding thus a proof of irrationality of

the Kubota-Leopoldt 2-adic zeta value $\zeta_2(5)$. The latter is joint

work with Frank Calegari and Yunqing Tang.