Hypergeometric functions over and relations to elliptic curves and modular forms
Author:
Jenny G. Fuselier
Journal:
Proc. Amer. Math. Soc. 138 (2010), 109-123
MSC (2000):
Primary 11F30; Secondary 11T24, 11G20, 33C99
DOI:
https://doi.org/10.1090/S0002-9939-09-10068-0
Published electronically:
August 28, 2009
MathSciNet review:
2550175
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: For primes , we present an explicit relation between the traces of Frobenius on a family of elliptic curves with
-invariant
and values of a particular
-hypergeometric function over
. We also give a formula for traces of Hecke operators on spaces of cusp forms of weight
and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's
-function in terms of hypergeometric functions.
- 1. Scott Ahlgren, The points of a certain fivefold over finite fields and the twelfth power of the eta function, Finite Fields Appl. 8 (2002), no. 1, 18–33. MR 1872789, https://doi.org/10.1006/ffta.2001.0315
- 2. Scott Ahlgren and Ken Ono, Modularity of a certain Calabi-Yau threefold, Monatsh. Math. 129 (2000), no. 3, 177–190. MR 1746757, https://doi.org/10.1007/s006050050069
- 3. Scott Ahlgren and Ken Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187–212. MR 1739404, https://doi.org/10.1515/crll.2000.004
- 4. F. Beukers, Algebraic values of 𝐺-functions, J. Reine Angew. Math. 434 (1993), 45–65. MR 1195690, https://doi.org/10.1515/crll.1993.434.45
- 5. David A. Cox, Primes of the form 𝑥²+𝑛𝑦², A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322
- 6. Sharon Frechette, Ken Ono, and Matthew Papanikolas, Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not. 60 (2004), 3233–3262. MR 2096220, https://doi.org/10.1155/S1073792804132522
- 7. J.G. Fuselier, Hypergeometric functions over finite fields and relations to modular forms and elliptic curves, Ph.D. thesis, Texas A&M University, 2007.
- 8. John Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), no. 1, 77–101. MR 879564, https://doi.org/10.1090/S0002-9947-1987-0879564-8
- 9. Hiroaki Hijikata, Arnold K. Pizer, and Thomas R. Shemanske, The basis problem for modular forms on Γ₀(𝑁), Mem. Amer. Math. Soc. 82 (1989), no. 418, vi+159. MR 960090, https://doi.org/10.1090/memo/0418
- 10. Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716
- 11. Masao Koike, Hypergeometric series over finite fields and Apéry numbers, Hiroshima Math. J. 22 (1992), no. 3, 461–467. MR 1194045
- 12. Serge Lang, Cyclotomic fields I and II, 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. With an appendix by Karl Rubin. MR 1029028
- 13. Ken Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), no. 3, 1205–1223. MR 1407498, https://doi.org/10.1090/S0002-9947-98-01887-X
- 14. Matthew Papanikolas, A formula and a congruence for Ramanujan’s 𝜏-function, Proc. Amer. Math. Soc. 134 (2006), no. 2, 333–341. MR 2175999, https://doi.org/10.1090/S0002-9939-05-08029-9
- 15. John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
- 16. René Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46 (1987), no. 2, 183–211. MR 914657, https://doi.org/10.1016/0097-3165(87)90003-3
- 17. Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210
- 18. P. F. Stiller, Classical automorphic forms and hypergeometric functions, J. Number Theory 28 (1988), no. 2, 219–232. MR 927661, https://doi.org/10.1016/0022-314X(88)90067-4
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Additional Information
Jenny G. Fuselier
Affiliation:
United States Military Academy, 646 Swift Road, West Point, New York 10996
Address at time of publication:
Department of Mathematics & Computer Science, Drawer 31, High Point University, High Point, North Carolina 27262
Email:
jenny.fuselier@usma.edu, jfuselie@highpoint.edu
DOI:
https://doi.org/10.1090/S0002-9939-09-10068-0
Received by editor(s):
June 3, 2009
Published electronically:
August 28, 2009
Additional Notes:
The author thanks her advisor, Matt Papanikolas, for his advice and support during the preparation of this paper. The author also thanks the Department of Mathematics at Texas A$&$M University, where the majority of this research was conducted.
Communicated by:
Ken Ono
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.