Special values of symmetric hypergeometric functions

Baldassarri, Francesco

doi:10.1090/S0002-9947-96-01676-5

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.

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