Gamma Structures and Gauss’s Contiguity v. Golyshev and A. Mellit

We introduce gamma structures on regular hypergeometric D–modules in dimension 1 as special one–parametric systems of solutions on the compact subtorus. We note that a balanced gamma product is in the Paley–Wiener class and show that the monodromy with respect to the gamma structure is expressed algebraically in terms of the hypergeometric exponents. We compute the hypergeometric monodromy explicitly in terms of certain diagonal matrices, Vandermonde matrices and their inverses (or generalizations of those in the resonant case). A hypergeometric D–module with rational indices is motivic, i. e. may be realized as a constituent of the pushforward of the constant D–module O in a pencil of varieties over Gm defined over Q. The de Rham to Betti comparison arises in each fiber; as a result, the vector space of solutions is endowed with two K–rational structures for a number field K. On the other hand, no rational structure exists in the case of irrational exponents, and yet one still wishes to have the benefits of the Dwork/Boyarsky method of parametric exponents. A substitute is the gamma structure on a hypergeometric D–module which manifests itself as a rational structure in the case of rational exponents and gives rise to an extension of the Betti to de Rham comparison in the non-motivic direction. Operating in this framework, one might try to study period matrices of traditional motives by representing them as limiting cases of hypergeometric ones, or even degenerate the hypergeometric period matrix into a resonant singularity. Reverting this process yields a perturbation of a Tate type period to an expression in gamma–values, cf [Gol08]. F. Baldassarri has emphasized that the key to hypergeometric monodromy is Gauss’s contiguity principle: with a translation of the set of indices 1 by a vector in an integral lattice is associated an explicit isomorphism of the respective D–modules, whose shape leads one to an a priori guess on the shape of the monodromy. Y. Andre has remarked that the situation is even better with p–adic hypergeometrics, as the translation lattice is dense in the space of indices. We introduce the gamma structure and replication as a means to make up for the lack of density of the translations in the complex case by interpolating the shifts to non–integral ones. —— We follow Katz’s treatment [Kat90] of hypergeometrics in order to fix our basics . Let Gm = Spec C[z, z ] be a one–dimensional torus. By D denote the algebra of differential operators on Gm, by D denote the differential operator z ∂ ∂z . One has D = C[z, z, D]. 1We adopt the terminology in which, in xα, α is the index, and exp(2πiα), the exponent. 1 2 V. GOLYSHEV AND A. MELLIT Definition. Let n and m be a pair of nonnegative integers. Let P and Q be polynomials of degrees n andm respectively. Define a hypergeometric differential operator of type (n,m): H(P,Q) = P (D)− zQ(D), and the hypergeometric D-module: H(P,Q) = D/DH(P,Q). If P (t) = p ∏ i(t− ai), Q(t) = q ∏ j(t− bj), λ = p/q, we shall write Hλ(ai, bj) = D/D 