Values of Gaussian hypergeometric series
Author:
Ken Ono
Journal:
Trans. Amer. Math. Soc. 350 (1998), 12051223
MSC (1991):
Primary 11T24
MathSciNet review:
1407498
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Abstract: Let be prime and let be the finite field with elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions where and respectively are the quadratic and trivial characters of For all but finitely many rational numbers there exist two elliptic curves and for which these values are expressed in terms of the trace of the Frobenius endomorphism. We obtain bounds and congruence properties for these values. We also show, using a theorem of Elkies, that there are infinitely many primes for which is zero; however if or , then the set of such primes has density zero. In contrast, if or , then there are only finitely many primes for which Greene and Stanton proved a conjecture of Evans on the value of a certain character sum which from this point of view follows from the fact that is an elliptic curve with complex multiplication. We completely classify all such CM curves and give their corresponding character sums in the sense of Evans using special Jacobsthal sums. As a consequence of this classification, we obtain new proofs of congruences for generalized Apéry numbers, as well as a few new ones, and we answer a question of Koike by evaluating over every
 1.
Bruce
C. Berndt and Ronald
J. Evans, Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and
Brewer, Illinois J. Math. 23 (1979), no. 3,
374–437. MR
537798 (81j:10055)
 2.
B.C. Berndt, R.J. Evans, and K.S. Williams, Gauss and Jacobi sums, Wiley Publ., to appear.
 3.
F.
Beukers, Some congruences for the Apéry numbers, J.
Number Theory 21 (1985), no. 2, 141–155. MR 808283
(87g:11032), http://dx.doi.org/10.1016/0022314X(85)900472
 4.
F.
Beukers, Another congruence for the Apéry numbers, J.
Number Theory 25 (1987), no. 2, 201–210. MR 873877
(88b:11002), http://dx.doi.org/10.1016/0022314X(87)900254
 5.
Jan
Stienstra and Frits
Beukers, On the PicardFuchs equation and the formal Brauer group
of certain elliptic 𝐾3surfaces, Math. Ann.
271 (1985), no. 2, 269–304. MR 783555
(86j:14045), http://dx.doi.org/10.1007/BF01455990
 6.
J.
E. Cremona, Algorithms for modular elliptic curves, Cambridge
University Press, Cambridge, 1992. MR 1201151
(93m:11053)
 7.
Noam
D. Elkies, Distribution of supersingular primes,
Astérisque 198200 (1991), 127–132 (1992).
Journées Arithmétiques, 1989 (Luminy, 1989). MR 1144318
(93b:11070)
 8.
Arthur
Erdélyi, Wilhelm
Magnus, Fritz
Oberhettinger, and Francesco
G. Tricomi, Higher transcendental functions. Vols. I, II,
McGrawHill Book Company, Inc., New YorkTorontoLondon, 1953. Based, in
part, on notes left by Harry Bateman. MR 0058756
(15,419i)
 9.
R.
J. Evans, Character sums over finite fields, Finite fields,
coding theory, and advances in communications and computing (Las Vegas, NV,
1991) Lecture Notes in Pure and Appl. Math., vol. 141, Dekker, New
York, 1993, pp. 57–73. MR 1199822
(94c:11118)
 10.
Ronald
J. Evans, Identities for products of Gauss sums over finite
fields, Enseign. Math. (2) 27 (1981), no. 34,
197–209 (1982). MR 659148
(83i:10050)
 11.
R.
J. Evans, J.
R. Pulham, and J.
Sheehan, On the number of complete subgraphs contained in certain
graphs, J. Combin. Theory Ser. B 30 (1981),
no. 3, 364–371. MR 624553
(83c:05075), http://dx.doi.org/10.1016/00958956(81)90054X
 12.
John
Greene, Hypergeometric functions over finite
fields, Trans. Amer. Math. Soc.
301 (1987), no. 1,
77–101. MR
879564 (88e:11122), http://dx.doi.org/10.1090/S00029947198708795648
 13.
J.
Greene and D.
Stanton, A character sum evaluation and Gaussian hypergeometric
series, J. Number Theory 23 (1986), no. 1,
136–148. MR
840021 (88a:11076), http://dx.doi.org/10.1016/0022314X(86)900090
 14.
A.
V. Yagzhev, Finiteness of the set of conservative polynomials of a
given degree, Mat. Zametki 41 (1987), no. 2,
148–151, 285 (Russian). MR 888954
(88g:12001)
 15.
Anthony
W. Knapp, Elliptic curves, Mathematical Notes, vol. 40,
Princeton University Press, Princeton, NJ, 1992. MR 1193029
(93j:11032)
 16.
Neal
Koblitz, Introduction to elliptic curves and modular forms,
Graduate Texts in Mathematics, vol. 97, SpringerVerlag, New York,
1984. MR
766911 (86c:11040)
 17.
Masao
Koike, Hypergeometric series over finite fields and Apéry
numbers, Hiroshima Math. J. 22 (1992), no. 3,
461–467. MR 1194045
(93i:11146)
 18.
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
(96b:11079)
 19.
Y. Martin and K. Ono, Etaquotients and elliptic curves, Proc. Amer. Math. Soc., to appear.
 20.
K. Ono, Euler's concordant forms, Acta Arith. 78 (1996), 101123. CMP 97:05
 21.
A.
R. Rajwade, The Diophantine equation
𝑦²=𝑥(𝑥²+21𝐷𝑥+112𝐷²)
and the conjectures of Birch and SwinnertonDyer, J. Austral. Math.
Soc. Ser. A 24 (1977), no. 3, 286–295. MR 0472828
(57 #12518)
 22.
Joseph
H. Silverman, The arithmetic of elliptic curves, Graduate
Texts in Mathematics, vol. 106, SpringerVerlag, New York, 1986. MR 817210
(87g:11070)
 1.
 B. Berndt and R. Evans, Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer, Illinois J. Math 23 (1979), 374437. MR 81j:10055
 2.
 B.C. Berndt, R.J. Evans, and K.S. Williams, Gauss and Jacobi sums, Wiley Publ., to appear.
 3.
 F. Beukers, Some congruences for Apéry numbers, J. Number Theory 21 (1985), 141155. MR 87g:11032
 4.
 , Another congruence for Apéry numbers, J. Number Theory 25 (1987), 201210. MR 88b:11002
 5.
 F. Beukers and J. Stienstra, On the PicardFuchs equation and the formal Brauer group of certain surfaces and elliptic curves, Math. Ann. 271 (1985), 269304. MR 86j:14045
 6.
 J. E. Cremona, Algorithms for modular elliptic curves, Cambridge Univ. Press, 1992. MR 93m:11053
 7.
 N. Elkies, Distribution of supersingular primes, J. Arithmétiques, Astérisque 198200 (1991), 127132. MR 93b:11070
 8.
 A. Erdélyi et al., Higher transcendental functions, Vol.1, McGrawHill, New York, 1953. MR 15:419i
 9.
 R. Evans, Character sums over finite fields, Finite fields, coding theory, and advances in communications and and computing, Eds. G. Mullen and P. Shiue, Lecture Notes in Pure and Appl. Math., vol. 141, Marcel Dekker, 1993, pp. 5773. MR 94c:11118
 10.
 , Identities for products of Gauss sums over finite fields, Enseign. Math. (2) 27 (1981), 197209. MR 83i:10050
 11.
 R. Evans, J. Pulham, and J. Sheehan, On the number of complete subgraphs contained in certain graphs, J. Combin. Theory Ser. B 30 (1981), 364371. MR 83c:05075
 12.
 J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), 77101. MR 88e:11122
 13.
 J. Greene and D. Stanton, A character sum evaluation and Gaussian hypergeometric series, J. Number Theory 23 (1986), 136148. MR 88a:11076
 14.
 K. Ireland and M. Rosen, A classical introduction to modern number theory, SpringerVerlag, 1982. MR 88g:12001
 15.
 A. Knapp, Elliptic curves, Princeton Univ. Press, 1992. MR 93j:11032
 16.
 N. Koblitz, Introduction to elliptic curves and modular forms, SpringerVerlag, 1984. MR 86c:11040
 17.
 M. Koike, Hypergeometric series over finite fields and Apéry numbers, Hiroshima Math. J. 22 (1992), 461467. MR 93i:11146
 18.
 , Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields, Hiroshima Math. J. 25 (1995), 4352. MR 96b:11079
 19.
 Y. Martin and K. Ono, Etaquotients and elliptic curves, Proc. Amer. Math. Soc., to appear.
 20.
 K. Ono, Euler's concordant forms, Acta Arith. 78 (1996), 101123. CMP 97:05
 21.
 A.R. Rajwade, The Diophantine equation and the conjectures of Birch and SwinnertonDyer, J. Austral. Math. Soc. Ser. A 24 (1977), 286295. MR 57:12518
 22.
 J. Silverman, The arithmetic of elliptic curves, SpringerVerlag, 1986. MR 87g:11070
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Additional Information
Ken Ono
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540;
Department of Mathematics, Penn State University, University Park, Pennsylvania 16802
Email:
ono@math.ias.edu, ono@math.psu.edu
DOI:
http://dx.doi.org/10.1090/S000299479801887X
PII:
S 00029947(98)01887X
Keywords:
Gaussian hypergeometric series,
elliptic curves,
Ap\'{e}ry numbers,
character sums
Received by editor(s):
September 8, 1995
Received by editor(s) in revised form:
July 3, 1996
Additional Notes:
The author is supported by NSF grants DMS9304580 and DMS9508976.
Article copyright:
© Copyright 1998
American Mathematical Society
