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

NUMBER THEORY

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March 11, 2020 3:00 pm - 4:00 pm
Science Center 507
Address: 1 Oxford Street, Cambridge, MA 02138 USA
##### 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.