# The Étale Theta Function and Its Frobenioid-Theoretic Manifestations

### Shinichi Mochizuki

Kyoto University, Japan

## Abstract

We develop the theory of the *tempered anabelian* and *Frobenioid-theoretic* aspects of the “*étale theta function*”, i.e., the Kummer class of the classical formal algebraic e theta function associated to a Tate curve over a nonarchimedean mixed-characteristic local ﬁeld. In particular, we consider a certain natural “environment” for the study of the étale theta function, which we refer to as a “*mono-theta environment*” — essentially a *Kummer-theoretic* version of the classical *theta trivialization* — and show that this mono-theta environment satisﬁes certain *remarkable rigidity properties* involving *cyclotomes, discreteness*, and *constant multiples*, all in a fashion that is *compatible* with the *topology* of the tempered fundamental group and the *extension structure* of the associated tempered Frobenioid.