Supercongruences between truncated hypergeometric functions and their Gaussian analogs

Author:
Eric Mortenson

Journal:
Trans. Amer. Math. Soc. **355** (2003), 987-1007

MSC (2000):
Primary 11F85, 11L10

DOI:
https://doi.org/10.1090/S0002-9947-02-03172-0

Published electronically:
October 25, 2002

MathSciNet review:
1938742

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension . For manifolds of dimension , he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas'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. 1-12.**[A-O]**S. Ahlgren and K. Ono,*A Gaussian hypergeometric series evaluation and Apéry number congruences*, J. reine angew. Math.**518**(2000), 187-212. MR**2001c:11057****[B]**F. Beukers,*Another congruence for the Apéry numbers*, J. Number Theory**25**(1987), 201-210. MR**88b:11002****[COV]**P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas,*Calabi-Yau manifolds over finite fields I*, http://xxx.lanl.gov/abs/hep-th/0012233.**[G]**J. Greene,*Hypergeometric functions over finite fields*, Trans. Amer. Math. Soc.**301**(1987), 77-101. MR**88e:11122****[Gr-Ko]**B. Gross and N. Koblitz,*Gauss sums and the**-adic**-function*, Ann. Math**109**(1979), 569-581. MR**80g:12015****[I]**T. Ishikawa,*On Beukers' conjecture*, Kobe J. Math**6**(1989), 49-51. MR**90i:11001****[I-R]**K. Ireland and M. Rosen,*A classical introduction to modern number theory*, Springer-Verlag, New York, 1982. MR**83g:12001****[M]**E. Mortenson,*A supercongruence conjecture of Rodriguez-Villegas 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. Rodriguez-Villegas,*Hypergeometric families of Calabi-Yau manifolds*, preprint.**[RV2]**F. Rodriguez-Villegas,*private communication*.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11F85,
11L10

Retrieve articles in all journals with MSC (2000): 11F85, 11L10

Additional Information

**Eric Mortenson**

Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Email:
mort@math.wisc.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03172-0

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