Congruences via modular forms

Authors:
Robert Osburn and Brundaban Sahu

Journal:
Proc. Amer. Math. Soc. **139** (2011), 2375-2381

MSC (2010):
Primary 11A07; Secondary 11F11

DOI:
https://doi.org/10.1090/S0002-9939-2010-10771-2

Published electronically:
December 9, 2010

MathSciNet review:
2784802

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove two congruences for the coefficients of power series expansions in of modular forms where is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apéry-like differential equations.

**1.**S. Ahlgren, K. Ono,*A Gaussian hypergeometric series evaluation and Apéry number congruences*, J. Reine Angew. Math.**518**(2000), 187-212. MR**1739404 (2001c:11057)****2.**G. Almkvist, D. van Straten and W. Zudilin,*Generalizations of Clausen's formula and algebraic transformations of Calabi-Yau differential equations*, Proc. Edinb. Math. Soc. (2), to appear.**3.**F. Beukers,*Some congruences for the Apéry numbers*, J. Number Th.**21**(1985), no. 2, 141-155. MR**808283 (87g:11032)****4.**F. Beukers,*Another congruence for the Apéry numbers*, J. Number Th.**25**(1987), no. 2, 201-210. MR**873877 (88b:11002)****5.**F. Beukers,*On Dwork's accessory parameter problem*, Math. Z.**241**(2002), no. 2, 425-444. MR**1935494 (2003i:12013)****6.**H. Chan, S. Chan and Z. Liu,*Domb's numbers and Ramanujan-Sato type series for*, Adv. Math.**186**(2004), 396-410. MR**2073912 (2005e:11170)****7.**H. Chan, S. Cooper and F. Sica,*Congruences satisfied by Apéry-like numbers*, Int. J. Number Theory**6**(2010), no. 1, 89-97. MR**2641716****8.**H. Chan, A. Kontogeorgis, C. Krattenthaler and R. Osburn,*Supercongruences satisfied by coefficients of hypergeometric series*, Ann. Sci. Math. Québec**34**(2010), no. 1, 25-36.**9.**H. Chan, H. Verrill,*The Apéry numbers, the Almkvist-Zudilin numbers and new series for*, Math. Res. Lett.**16**(2009), no. 3, 405-420. MR**2511622 (2010d:11152)****10.**M. Coster,*Supercongruences*, Ph.D. thesis, Universiteit Leiden, 1988.**11.**N. Fine,*Basic hypergeometric series and applications*, American Mathematical Society, Providence, RI, 1988. MR**956465 (91j:33011)****12.**F. Jarvis, H. Verrill,*Supercongruences for the Catalan-Larcombe-French numbers*, Ramanujan J.**22**(2010), no. 2, 171-186. MR**2643702****13.**K. Ono,*The web of modularity: Arithmetic of the coefficients of modular forms and -series*, CBMS Regional Conf. in Math., vol. 102, Amer. Math. Soc., Providence, RI, 2004. MR**2020489 (2005c:11053)****14.**C. Peters, J. Stienstra,*A pencil of -surfaces related to Apéry's recurrence for and Fermi surfaces for potential zero*. Arithmetic of complex manifolds (Erlangen, 1988), 110-127, Lecture Notes in Math., 1399, Springer, Berlin, 1989. MR**1034260 (91e:14036)****15.**W. Stein,*Modular forms, a computational approach*, Graduate Studies in Mathematics, 79, American Mathematical Society, Providence, RI, 2007. MR**2289048 (2008d:11037)****16.**J. Stienstra, F. Beukers,*On the Picard-Fuchs equation and the formal Brauer group of certain elliptic -surfaces*, Math. Ann.**271**(1985), no. 2, 269-304. MR**783555 (86j:14045)****17.**H. Verrill,*Congruences related to modular forms*, Int. J. Number Theory**6**(2010), no. 6, 1367-1390.**18.**D. Zagier,*Integral solutions of Apéry-like recurrence equations*, Groups and Symmetries: From Neolithic Scots to John McKay, 349-366, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009. MR**2500571 (2010h:11069)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
11A07,
11F11

Retrieve articles in all journals with MSC (2010): 11A07, 11F11

Additional Information

**Robert Osburn**

Affiliation:
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Email:
robert.osburn@ucd.ie

**Brundaban Sahu**

Affiliation:
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Address at time of publication:
School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar 751005, India

Email:
brundaban.sahu@niser.ac.in

DOI:
https://doi.org/10.1090/S0002-9939-2010-10771-2

Keywords:
Coefficients of power series expansions,
congruences,
modular forms

Received by editor(s):
December 1, 2009

Received by editor(s) in revised form:
June 29, 2010

Published electronically:
December 9, 2010

Additional Notes:
The authors were partially supported by Science Foundation Ireland 08/RFP/MTH1081.

Communicated by:
Ken Ono

Article copyright:
© Copyright 2010
American Mathematical Society

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