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The Étale Theta Function and Its Frobenioid-Theoretic ManifestationsShinichi Mochizuki (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