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Transactions of the American Mathematical Society

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Special values of symmetric hypergeometric functions

Author: Francesco Baldassarri
Journal: Trans. Amer. Math. Soc. 348 (1996), 2249-2289
MSC (1991): Primary 11T23, 11S31, 12H25, 14F30
MathSciNet review: 1361637
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Abstract: We discuss the $p$-adic formula (0.3) of P. Th. Young, in the framework of Dwork's theory of the hypergeometric equation. We show that it gives the value at 0 of the Frobenius automorphism of the unit root subcrystal of the hypergeometric crystal. The unit disk at 0 is in fact singular for the differential equation under consideration, so that it's not a priori clear that the Frobenius structure should extend to that disk. But the singularity is logarithmic, and it extends to a divisor with normal crossings relative to $\mathbf {Z}_{p}$ in $\mathbf {P}^{1}_{\mathbf {Z}_{p}}$. We show that whenever the unit root subcrystal of the hypergeometric system has generically rank 1, it actually extends as a logarithmic $F$-subcrystal to the unit disk at 0. So, in these optics, ``singular classes are not supersingular''. If, in particular, the holomorphic solution at 0 is bounded, the extended logarithmic $F$-crystal has no singularity in the residue class of 0, so that it is an $F$-crystal in the usual sense and the Frobenius operation is holomorphic. We examine in detail its analytic form.

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Additional Information

Francesco Baldassarri
Affiliation: Dipartimento di Matematica, Università di Padova, Via Belzoni 7, I-35131, Padova, Italy

Received by editor(s): November 15, 1994
Dedicated: Dedicated to Professor Bernard Dwork on his 73rd birthday
Article copyright: © Copyright 1996 American Mathematical Society

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