|
Supercongruences for truncated  hypergeometric series with applications to certain weight three newforms
Author(s):
Eric
Mortenson
Journal:
Proc. Amer. Math. Soc.
133
(2005),
321-330.
MSC (2000):
Primary 11F85, 11L10
Posted:
September 20, 2004
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
We prove general results on supercongruences between values of truncated  hypergeometric functions and their character analogs. As a consequence of the main results of this paper, we prove Beukers-type supercongruences for certain weight three newforms.
References:
-
- [A]
- S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic computation, number theory, special functions, physics and combinatorics. Dev. Math., 4, Kluwer, Dordrecht, 2001. MR 1880076 (2003i:33025)
- [AO]
- S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. reine angew. Math. 518 (2000), 187-212. MR 1739404 (2001c:11057)
- [B]
- F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), 201-210. MR 0873877 (88b:11002)
- [BE]
- B. Berndt and R. Evans, Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer, Illinois J. Math. 23 (1979), 374-437. MR 0537798 (81j:10055)
- [BEW]
- B. Berndt, R. Evans, and K. Williams, Gauss and Jacobi Sums, J. Wiley & Sons, Inc., New York, 1998. MR 1625181 (99d:11092)
- [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 0879564 (88e:11122)
- [GK]
- B. Gross and N. Koblitz, Gauss sums and the
 -adic  -function, Ann. Math 109 (1979), 569-581. MR 0534763 (80g:12015) - [I]
- T. Ishikawa, On Beukers' congruence, Kobe J. Math 6 (1989), 49-52. MR 1023525 (90i:11001)
- [M1]
- E. Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory 99 (2003), 139-147. MR 1957248 (2004e:11089)
- [M2]
- E. Mortenson, Supercongruences between truncated
 hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc. 355 (2003), 987-1007. MR 1938742 (2003i:11119) - [O]
- K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), 1205-1223. MR 1407498 (98e:11141)
- [RV1]
- F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, preprint. MR 2019156
- [RV2]
- F. Rodriguez-Villegas, private communication.
- [SB]
- J. Stienstra and F. Beukers, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic
 -surfaces., Math. Ann. 271 (1985), 269-304. MR 0783555 (86j:14045) - [vH]
- L. van Hamme, Proof of a conjecture of Beukers on Apéry numbers, Proceedings of the conference on
 -adic analysis (Houthalen, 1987), 189-195. MR 0921871 (89b:11007)
Similar Articles:
Retrieve articles in Proceedings 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
Address at time of publication:
Max-Planck-Institut für Mathematik, Bonn, Germany
Email:
mort@mpim-bonn.mpg.de
DOI:
10.1090/S0002-9939-04-07697-X
PII:
S 0002-9939(04)07697-X
Keywords:
Supercongruences
Received by editor(s):
April 16, 2003
Posted:
September 20, 2004
Communicated by:
David E. Rohrlich
Copyright of article:
Copyright
2004,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org