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Transcendental methods and theta-functions. (English) Zbl 0679.10026
Theta functions, Proc. 35th Summer Res. Inst. Bowdoin Coll., Brunswick/ME 1987, Proc. Symp. Pure Math. 49, Pt. 2, 167-232 (1989).
[For the entire collection see Zbl 0672.00004.]
This densely written article presents a survey of the deep results obtained by the authors in recent years on the arithmetic theory of differential equations. Padé approximation methods, yielding global informations on the differential equation, are discussed in the first part of the paper [cf., e.g., Proc. Natl. Acad. Sci. USA 82, 2212-2216 (1985; Zbl 0577.14034)]. An interesting notion of p-adic spectrum is here introduced, and investigated in the case of Lamé equations.
The second part of the paper deals with special values of solutions of hypergeometric equations and their connection with period relations. Here again, Padé approximations play a fundamental role (for a recent approach of a different nature, see J. Wolfart, Invent. Math. 92, No.1, 187-216 (1988; Zbl 0649.10022)).
Reviewer: D.Bertrand

11J81 Transcendence (general theory)
12H25 \(p\)-adic differential equations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
34A30 Linear ordinary differential equations and systems, general
41A21 Padé approximation
11J85 Algebraic independence; Gel’fond’s method