𝐴-hypergeometric systems that come from geometry

Adolphson, Alan; Sperber, Steven

doi:10.1090/S0002-9939-2011-11073-6

${A}$-hypergeometric systems that come from geometry
HTML articles powered by AMS MathViewer

by Alan Adolphson and Steven Sperber PDF

Proc. Amer. Math. Soc. 140 (2012), 2033-2042 Request permission

Abstract:

In recent work, Beukers characterized ${A}$-hypergeometric systems having a full set of algebraic solutions. He accomplished this by (1) determining which ${A}$-hypergeometric systems have a full set of polynomial solutions modulo $p$ for almost all primes $p$ and (2) showing that these systems come from geometry. He then applied a fundamental theorem of N. Katz, which says that such systems have a full set of algebraic solutions. In this paper we establish some connections between nonresonant $A$-hypergeometric systems and de Rham-type complexes, which leads to a determination of which $A$-hypergeometric systems come from geometry. We do not use the fact that the system is irreducible or find integral formulas for its solutions.

References

Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), no. 2, 269–290. MR 1262208, DOI 10.1215/S0012-7094-94-07313-4
Alan Adolphson, Higher solutions of hypergeometric systems and Dwork cohomology, Rend. Sem. Mat. Univ. Padova 101 (1999), 179–190. MR 1705287
Alan Adolphson and Steven Sperber, On twisted de Rham cohomology, Nagoya Math. J. 146 (1997), 55–81. MR 1460954, DOI 10.1017/S0027763000006218
Frits Beukers, Algebraic $A$-hypergeometric functions, Invent. Math. 180 (2010), no. 3, 589–610. MR 2609251, DOI 10.1007/s00222-010-0238-y
Bernard Dwork, Generalized hypergeometric functions, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1085482
B. Dwork and F. Loeser, Hypergeometric series, Japan. J. Math. (N.S.) 19 (1993), no. 1, 81–129. MR 1231511, DOI 10.4099/math1924.19.81
Katz, Nicholas. Thesis (1966), Princeton University.
Nicholas M. Katz, On the differential equations satisfied by period matrices, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 223–258. MR 242841
Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. MR 291177, DOI 10.1007/BF02684688
Nicholas M. Katz, Algebraic solutions of differential equations ($p$-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1–118. MR 337959, DOI 10.1007/BF01389714
Nicholas M. Katz, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France 110 (1982), no. 2, 203–239 (English, with French summary). MR 667751, DOI 10.24033/bsmf.1960
Nicholas M. Katz, Corrections to: “A conjecture in the arithmetic theory of differential equations”, Bull. Soc. Math. France 110 (1982), no. 3, 347–348. MR 688039, DOI 10.24033/bsmf.1966
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
Mathias Schulze and Uli Walther, Hypergeometric D-modules and twisted Gauß-Manin systems, J. Algebra 322 (2009), no. 9, 3392–3409. MR 2567427, DOI 10.1016/j.jalgebra.2008.09.010