Supercongruences between truncated hypergeometric functions and their Gaussian analogs
Author:
Eric Mortenson
Journal:
Trans. Amer. Math. Soc. 355 (2003), 9871007
MSC (2000):
Primary 11F85, 11L10
Published electronically:
October 25, 2002
MathSciNet review:
1938742
Fulltext PDF Free Access
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References 
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Additional Information
Abstract: Fernando RodriguezVillegas has conjectured a number of supercongruences for hypergeometric CalabiYau manifolds of dimension . For manifolds of dimension , he observed four potential supercongruences. Later the author proved one of the four. Motivated by RodriguezVillegas's work, in the present paper we prove a general result on supercongruences between values of truncated hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.
 [A]
S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Dev. Math., 4, Kluwer, Dordrecht, 2001, pp. 112.
 [AO]
S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. reine angew. Math. 518 (2000), 187212. MR 2001c:11057
 [B]
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), 201210. MR 88b:11002
 [COV]
P. Candelas, X. de la Ossa, and F. RodriguezVillegas, CalabiYau manifolds over finite fields I, http://xxx.lanl.gov/abs/hepth/0012233.
 [G]
J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), 77101. MR 88e:11122
 [GrKo]
B. Gross and N. Koblitz, Gauss sums and the adic function, Ann. Math 109 (1979), 569581. MR 80g:12015
 [I]
T. Ishikawa, On Beukers' conjecture, Kobe J. Math 6 (1989), 4951. MR 90i:11001
 [IR]
K. Ireland and M. Rosen, A classical introduction to modern number theory, SpringerVerlag, New York, 1982. MR 83g:12001
 [M]
E. Mortenson, A supercongruence conjecture of RodriguezVillegas for a certain truncated hypergeometric function, J. Number Theory, to appear.
 [PWZ]
M. Petkovsek, H. Wilf, and D. Zeilberger, A=B, A. K. Peters, Ltd., Wellesley, MA, 1996. MR 97j:05001
 [RV1]
F. RodriguezVillegas, Hypergeometric families of CalabiYau manifolds, preprint.
 [RV2]
F. RodriguezVillegas, private communication.
 [A]
 S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Dev. Math., 4, Kluwer, Dordrecht, 2001, pp. 112.
 [AO]
 S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. reine angew. Math. 518 (2000), 187212. MR 2001c:11057
 [B]
 F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), 201210. MR 88b:11002
 [COV]
 P. Candelas, X. de la Ossa, and F. RodriguezVillegas, CalabiYau manifolds over finite fields I, http://xxx.lanl.gov/abs/hepth/0012233.
 [G]
 J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), 77101. MR 88e:11122
 [GrKo]
 B. Gross and N. Koblitz, Gauss sums and the adic function, Ann. Math 109 (1979), 569581. MR 80g:12015
 [I]
 T. Ishikawa, On Beukers' conjecture, Kobe J. Math 6 (1989), 4951. MR 90i:11001
 [IR]
 K. Ireland and M. Rosen, A classical introduction to modern number theory, SpringerVerlag, New York, 1982. MR 83g:12001
 [M]
 E. Mortenson, A supercongruence conjecture of RodriguezVillegas for a certain truncated hypergeometric function, J. Number Theory, to appear.
 [PWZ]
 M. Petkovsek, H. Wilf, and D. Zeilberger, A=B, A. K. Peters, Ltd., Wellesley, MA, 1996. MR 97j:05001
 [RV1]
 F. RodriguezVillegas, Hypergeometric families of CalabiYau manifolds, preprint.
 [RV2]
 F. RodriguezVillegas, private communication.
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Additional Information
Eric Mortenson
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email:
mort@math.wisc.edu
DOI:
http://dx.doi.org/10.1090/S0002994702031720
PII:
S 00029947(02)031720
Keywords:
Supercongruences
Received by editor(s):
February 27, 2002
Received by editor(s) in revised form:
July 22, 2002
Published electronically:
October 25, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
