Special values of symmetric hypergeometric functions

Abstract

We discuss the p 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 Z p \mathbf {Z}_{p} in P Z p 1 \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 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 F -crystal has no singualrity in the residue class of 0, so that it is an F F -crystal in the usual sense and the Frobenius operation is holomorphic. We examine in detail its analytic form.