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.
Cite this article
Shinichi Mochizuki, The Étale Theta Function and Its Frobenioid-Theoretic Manifestations. Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, pp. 227–349DOI 10.2977/PRIMS/1234361159