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The étale theta function and its Frobenioid-theoretic manifestations. (English) Zbl 1170.14023
Author’s 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 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.

##### MSC:
 14H42 Theta functions and curves; Schottky problem 14H30 Coverings of curves, fundamental group
##### Keywords:
theta functions; anabelian geometry; fundamental group
Full Text:
##### References:
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