Special values of symmetric hypergeometric functions
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- by Francesco Baldassarri PDF
- Trans. Amer. Math. Soc. 348 (1996), 2249-2289 Request permission
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 singualrity 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.References
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Additional Information
- Francesco Baldassarri
- Affiliation: Dipartimento di Matematica, Università di Padova, Via Belzoni 7, I-35131, Padova, Italy
- Email: baldassarri@pdmat1.math.unipd.it
- Received by editor(s): November 15, 1994
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 2249-2289
- MSC (1991): Primary 11T23, 11S31, 12H25, 14F30
- DOI: https://doi.org/10.1090/S0002-9947-96-01676-5
- MathSciNet review: 1361637
Dedicated: Dedicated to Professor Bernard Dwork on his 73rd birthday