Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves

Author:
Catherine Lennon

Journal:
Proc. Amer. Math. Soc. **139** (2011), 1931-1938

MSC (2010):
Primary 11T24, 11G20

Published electronically:
November 3, 2010

MathSciNet review:
2775369

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present here a formula for expressing the trace of the Frobenius endomorphism of an elliptic curve over satisfying and in terms of special values of Gaussian hypergeometric series. This paper uses methods introduced by Fuselier for one-parameter families of curves to express the trace of Frobenius of as a function of its -invariant and discriminant instead of a parameter, which are more intrinsic characteristics of the curve.

**1.**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**, 10.1155/S1073792804132522**2.**J. Fuselier,*Hypergeometric functions over finite fields and relations to modular forms and elliptic curves*, Ph.D. thesis, Texas A&M University, 2007.**3.**Jenny G. Fuselier,*Hypergeometric functions over 𝔽_{𝕡} and relations to elliptic curves and modular forms*, Proc. Amer. Math. Soc.**138**(2010), no. 1, 109–123. MR**2550175**, 10.1090/S0002-9939-09-10068-0**4.**John Greene,*Hypergeometric functions over finite fields*, Trans. Amer. Math. Soc.**301**(1987), no. 1, 77–101. MR**879564**, 10.1090/S0002-9947-1987-0879564-8**5.**Yasutaka Ihara,*Hecke Polynomials as congruence 𝜁 functions in elliptic modular case*, Ann. of Math. (2)**85**(1967), 267–295. MR**0207655****6.**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****7.**Nicholas M. Katz,*Exponential sums and differential equations*, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR**1081536****8.**Masao Koike,*Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields*, Hiroshima Math. J.**25**(1995), no. 1, 43–52. MR**1322601****9.**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****10.**C. Lennon,*A Trace Formula for Certain Hecke Operators and Gaussian Hypergeometric Functions*, http://arxiv.org/abs/1003.1157.**11.**Ken Ono,*The web of modularity: arithmetic of the coefficients of modular forms and 𝑞-series*, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR**2020489****12.**Ken Ono,*Values of Gaussian hypergeometric series*, Trans. Amer. Math. Soc.**350**(1998), no. 3, 1205–1223. MR**1407498**, 10.1090/S0002-9947-98-01887-X**13.**Joseph H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
11T24,
11G20

Retrieve articles in all journals with MSC (2010): 11T24, 11G20

Additional Information

**Catherine Lennon**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Email:
clennon@math.mit.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10609-3

Received by editor(s):
March 22, 2010

Received by editor(s) in revised form:
May 28, 2010

Published electronically:
November 3, 2010

Additional Notes:
This work was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.

Communicated by:
Matthew A. Papanikolas

Article copyright:
© Copyright 2010
American Mathematical Society

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