The Étale Theta Function and Its Frobenioid-theoretic Manifestations

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 theta function associated to a Tate curve over a nonarchimedean 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”, 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.