×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Y. André, On a geometric description of Gal(Qp/Qp) and a p-adic avatar of d GT , Duke Math. J. 119 (2003), no. 1, 1-39. · Zbl 1155.11356
[2] G. Faltings and C.-L. Chai, Degeneration of abelian varieties, Springer, Berlin, 1990. · Zbl 0744.14031
[3] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Springer, Berlin, 1977. · Zbl 0368.20023
[4] S. Mochizuki, Foundations of p-adic Teichmüller theory, Amer. Math. Soc., Prov- idence, RI, 1999. · Zbl 0969.14013
[5] , The absolute anabelian geometry of hyperbolic curves, in Galois theory and modular forms, 77-122, Kluwer Acad. Publ., Boston, MA, 2004. · Zbl 1062.14031
[6] , The absolute anabelian geometry of canonical curves, Doc. Math. 2003, Extra Vol., 609-640 (electronic). · Zbl 1092.14507
[7] , A survey of the Hodge-Arakelov theory of elliptic curves. I, in Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), 533-569, Proc. Sympos. Pure Math., 70, Amer. Math. Soc., Providence, RI, 2002.
[8] , A survey of the Hodge-Arakelov theory of elliptic curves. II, in Algebraic geometry 2000, Azumino (Hotaka), 81-114, Adv. Stud. Pure Math., 36, Math. Soc. Japan, Tokyo, 2002. · Zbl 1056.14032
[9] , The Hodge-Arakelov theory of elliptic curves: Global discretization of local Hodge theories, RIMS Preprint Nos. 1255, 1256, 1999.
[10] , The scheme-theoretic theta convolution, RIMS Preprint No. 1257, 1999.
[11] , Connections and related integral structures on the universal extension of an elliptic curve, RIMS Preprint No. 1279, 2000.
[12] , The Galois-theoretic Kodaira-Spencer morphism of an elliptic curve, RIMS Preprint No. 1287, 2000.
[13] , The Hodge-Arakelov theory of elliptic curves in positive characteristic, RIMS Preprint No. 1298, 2000.
[14] , The local pro-p anabelian geometry of curves, Invent. Math. 138 (1999), no. 2, 319-423. · Zbl 0935.14019
[15] , The geometry of anabelioids, Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, 819-881. · Zbl 1113.14021
[16] , Galois sections in absolute anabelian geometry, Nagoya Math. J. 179 (2005), 17-45. 349 · Zbl 1129.14042
[17] S. Mochizuki, Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci. 42 (2006), no. 1, 221-322. · Zbl 1113.14025
[18] , A combinatorial version of the Grothendieck conjecture, Tohoku Math. J. (2) 59 (2007), no. 3, 455-479. · Zbl 1129.14043
[19] , Absolute anabelian cuspidalizations of proper hyperbolic curves, J. Math. Kyoto Univ. 47 (2007), no. 3, 451-539. · Zbl 1143.14305
[20] , The geometry of Frobenioids I: The general theory, Kyushu J. Math. 62 (2008), 293-400.
[21] , The geometry of Frobenioids II: Poly-Frobenioids, Kyushu J. Math. 62 (2008), 401-460. · Zbl 1200.14008
[22] D. Mumford, An analytic construction of degenerating abelian varieties over com- plete rings, Appendix to [FC]. · Zbl 0241.14020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.