-hypergeometric systems that come from geometry

Authors:
Alan Adolphson and Steven Sperber

Journal:
Proc. Amer. Math. Soc. **140** (2012), 2033-2042

MSC (2010):
Primary 33C70, 14F40; Secondary 52B20

DOI:
https://doi.org/10.1090/S0002-9939-2011-11073-6

Published electronically:
October 13, 2011

MathSciNet review:
2888191

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In recent work, Beukers characterized -hypergeometric systems having a full set of algebraic solutions. He accomplished this by (1) determining which -hypergeometric systems have a full set of polynomial solutions modulo for almost all primes 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 -hypergeometric systems and de Rham-type complexes, which leads to a determination of which -hypergeometric systems come from geometry. We do not use the fact that the system is irreducible or find integral formulas for its solutions.

**1.**Adolphson, Alan. Hypergeometric functions and rings generated by monomials. Duke Math. J.**73**(1994), no. 2, 269-290. MR**1262208 (96c:33020)****2.**Adolphson, Alan. Higher solutions of hypergeometric systems and Dwork cohomology. Rend. Sem. Mat. Univ. Padova**101**(1999), 179-190. MR**1705287 (2001b:14032)****3.**Adolphson, Alan; Sperber, Steven. On twisted de Rham cohomology. Nagoya Math. J.**146**(1997), 55-81. MR**1460954 (98k:14027)****4.**Beukers, Frits. Algebraic -hypergeometric functions. Invent. Math.**180**(2010), no. 3, 589-610. MR**2609251****5.**Dwork, Bernard. Generalized hypergeometric functions. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. MR**1085482 (92h:14006)****6.**Dwork, B.; Loeser, F. Hypergeometric series. Japan. J. Math. (N.S.)**19**(1993), no. 1, 81-129. MR**1231511 (95f:33013)****7.**Katz, Nicholas. Thesis (1966), Princeton University.**8.**Katz, Nicholas. On the differential equations satisfied by period matrices. Inst. Hautes Études Sci. Publ. Math. No. 35 (1968), 223-258. MR**0242841 (39:4168)****9.**Katz, Nicholas. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Inst. Hautes Études Sci. Publ. Math. No. 39 (1970), 175-232. MR**0291177 (45:271)****10.**Katz, Nicholas. Algebraic solutions of differential equations (-curvature and the Hodge filtration). Invent. Math.**18**(1972), 1-118. MR**0337959 (49:2728)****11.**Katz, Nicholas. A conjecture in the arithmetic theory of differential equations. Bull. Soc. Math. France**110**(1982), no. 2, 203-239. MR**667751 (84h:14014a)****12.**Katz, Nicholas M. Corrections to: ``A conjecture in the arithmetic theory of differential equations''. Bull. Soc. Math. France**110**(1982), no. 3, 347-348. MR**688039 (84h:14014b)****13.**Matsumura, Hideyuki. Commutative ring theory. Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986. MR**879273 (88h:13001)****14.**Schulze, Mathias; Walther, Uli. Hypergeometric -modules and twisted Gauss-Manin systems. J. Algebra**322**(2009), no. 9, 3392-3409. MR**2567427 (2010m:14028)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
33C70,
14F40,
52B20

Retrieve articles in all journals with MSC (2010): 33C70, 14F40, 52B20

Additional Information

**Alan Adolphson**

Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Email:
adolphs@math.okstate.edu

**Steven Sperber**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
sperber@math.umn.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-11073-6

Keywords:
$A$-hypergeometric system,
de Rham cohomology

Received by editor(s):
December 9, 2010

Received by editor(s) in revised form:
January 24, 2011, and February 9, 2011

Published electronically:
October 13, 2011

Communicated by:
Lev Borisov

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.