|
 -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
Posted:
October 13, 2011
Full-text PDF
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.
References
- 1.
Alan
Adolphson, Hypergeometric functions and rings generated by
monomials, Duke Math. J. 73 (1994), no. 2,
269–290. MR 1262208
(96c:33020), http://dx.doi.org/10.1215/S0012-7094-94-07313-4
- 2.
Alan
Adolphson, Higher solutions of hypergeometric systems and Dwork
cohomology, Rend. Sem. Mat. Univ. Padova 101 (1999),
179–190. MR 1705287
(2001b:14032)
- 3.
Alan
Adolphson and Steven
Sperber, On twisted de Rham cohomology, Nagoya Math. J.
146 (1997), 55–81. MR 1460954
(98k:14027)
- 4.
Frits
Beukers, Algebraic 𝐴-hypergeometric functions, Invent.
Math. 180 (2010), no. 3, 589–610. MR 2609251
(2011k:33034), http://dx.doi.org/10.1007/s00222-010-0238-y
- 5.
Bernard
Dwork, Generalized hypergeometric functions, Oxford
Mathematical Monographs, The Clarendon Press Oxford University Press, New
York, 1990. Oxford Science Publications. MR 1085482
(92h:14006)
- 6.
B.
Dwork and F.
Loeser, 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.
Nicholas
M. Katz, On the differential equations satisfied by period
matrices, Inst. Hautes Études Sci. Publ. Math.
35 (1968), 223–258. MR 0242841
(39 #4168)
- 9.
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 0291177
(45 #271)
- 10.
Nicholas
M. Katz, Algebraic solutions of differential equations
(𝑝-curvature and the Hodge filtration), Invent. Math.
18 (1972), 1–118. MR 0337959
(49 #2728)
- 11.
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
(84h:14014a)
- 12.
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
(84h:14014b)
- 13.
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
(88h:13001)
- 14.
Mathias
Schulze and Uli
Walther, Hypergeometric D-modules and twisted Gauß-Manin
systems, J. Algebra 322 (2009), no. 9,
3392–3409. MR 2567427
(2010m:14028), http://dx.doi.org/10.1016/j.jalgebra.2008.09.010
Similar Articles
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:
http://dx.doi.org/10.1090/S0002-9939-2011-11073-6
PII:
S 0002-9939(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
Posted:
October 13, 2011
Communicated by:
Lev Borisov
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
|
