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Ultrametric theta functions and abelian varieties

Published online by Cambridge University Press:  22 January 2016

Horacio Tapia-Recillas
Horacio Tapia-Recillas*
Affiliation:
Departamento de Matemáticas Centro de Investigación, I.P.N. México, D.F. México
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Let k be a field complete with respect to a non-trivial, non-archimedean valuation and let g be a positive integer. Consider the following question : if Γ is a multiplicative subgroup of Gg = (k*)g satisfying certain “Riemann conditions”, can one construct in a natural way an abelian variety defined over k having Gg/Γ as its set of k-rational points? This problem was first considered by Morikawa [3]. J. Tate provided a complete solution for g = 1 (cf. for example [6]). J. McCabe [2] gave a partial solution for g > 1. He showed how to attach to Γ a graded ring R of theta functions such that A = Proj. R is g-dimensional abelian variety over k.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 69 , February 1978 , pp. 65 - 96
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1978

References

[1] Gerritzen, L. On non-archimedean representation of abelian varieties. Math. Ann., vol. 196 (1972), pp. 323–346.CrossRefGoogle Scholar
[2] McCabe, J. P-adic theta functions. Ph.D. thesis, Harvard University (1968), unpublished.Google Scholar
[3] Morikawa, H. On theta functions and abelian varieties over valuation fields of rank one. I, II Nagoya Math. Jour. vol. 20 (1962), pp. 1–27 and pp. 231–250.CrossRefGoogle Scholar
[4] Mumford, D. An analytic construction of degenerating abelian varieties over complete local rings. Composito Math. vol. 24 (1972), pp. 129–174 and pp. 239–272.Google Scholar
[5] Raynaud, M. Variétés abéliennes et géométrie rigide. Actes Congress Intern. Math. 1 (1970), pp. 473–477.Google Scholar
[6] Roquette, P. Analytic theory of elliptic functions over local fields. Vandenhoeck and Ruprecht in Göttingen (1970).Google Scholar