p-adic Generalized Hypergeometric Equations from the Viewpoint of Arithmetic D-modules

Abstract

We study the p-adic (generalized) hypergeometric equations by using the theory of multiplicative convolution of arithmetic D-modules. As a result, we prove that the hypergeometric isocrystals with suitable rational parameters have a structure of overconvergent F -isocrystals. 0 Introduction. Katz [Ka, 5.3.1] discovered that the hypergeometric D-modules on A 1 C \{0} can be described as the multiplicative convolution of hypergeometric D-modules of rank one. Precisely speaking, Katz proved the statement (ii) in the following theorem (the statement (i) is trivial but put to compare with another theorem later).Convolution Theorem over C. Let α = (α 1 , . . . , α m ) and β = (β 1 , . . . , β n ) be two sequences of complex numbers and assume that α i −β j is not an integer for any i , j . Let Hyp(α;β) be the D-module on G m,C defined by the hypergeometric operatorThen, Hyp π (α; β) has the following properties.(i) If m = n, then Hyp π (α; β) has a structure of an overconvergent F -isocrystal on G m,k of rank max {m, n}. If m = n, then the restriction of Hyp π (α; β) to P 1 V \{0, 1, ∞} has a structure of an overconvergent F -isocrystal on G m,k \{1} of rank m.

Slopes of indecomposable $F$-isocrystals

“…According to K. Miyatani [Mi], for any c as in §A.1.2 there exists an irreducible object M c ∈ F -Isoc † (G m \ {1}) such that for every closed point x ∈ G m \ {1} the Frobenius characteristic polynomial of (M c ) x is equal to that of (E c ) x (the word "equal" makes sense because the coefficients of the Frobenius characteristic polynomial of (E c ) x are in the number field E, which is a subfield of Q p ). The construction of M c given in [Mi,§3.2] is parallel to the one from §A.1.2.…”