Exceptional sets of hypergeometric series
Archinard N.
For a family of transcendental hypergeometric series, we determine explicitly the set of algebraic points at which the series takes algebraic values (the so-called exceptional set). This answers a question of Siegel in special cases. For this, we first prove identities, each one relating locally one hypergeometric series to modular functions. In some cases, the identity and the theory of complex multiplication allow the determination of an infinite subset of the exceptional set. These subsets are shown to be the whole sets in using a consequence of Wustholz's Analytic Subgroup Theorem together with mapping properties of Schwarz triangle functions. Further consequences of the identities are explicit evaluations of hypergeometric series at algebraic points. Some of them provide examples for Kroneckers Jugendtraum.
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