# Publications of the Research Institute for Mathematical Sciences

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**Volume 45, Issue 1, 2009, pp. 227–349**

**DOI: 10.2977/prims/1234361159**

The Étale Theta Function and Its Frobenioid-Theoretic Manifestations

Shinichi Mochizuki^{[1]}(1) Research Institute for Mathematical Sciences, Kyoto University, 606-8502, KYOTO, JAPAN

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.

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Mochizuki S. The Étale Theta Function and Its Frobenioid-Theoretic Manifestations. *Publ. Res. Inst. Math. Sci.* 45 (2009), 227-349. doi: 10.2977/prims/1234361159