3F2 HYPERGEOMETRIC SERIES AND PERIODS OF ELLIPTIC CURVES

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

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Certain transformations for hypergeometric series in the p-adic setting

International Journal of Number Theory ◽

10.1142/s1793042115500359 ◽

2015 ◽

Vol 11 (02) ◽

pp. 645-660 ◽

Cited By ~ 2

Author(s):

Rupam Barman ◽

Neelam Saikia

Keyword(s):

Elliptic Curves ◽

Finite Fields ◽

Gamma Function ◽

Hypergeometric Functions ◽

Hypergeometric Series ◽

Transformation Formulas ◽

Trace Of Frobenius

In [The trace of Frobenius of elliptic curves and the p-adic gamma function, Pacific J. Math. 261(1) (2013) 219–236], McCarthy defined a function nGn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. This function extends hypergeometric functions over finite fields to the p-adic setting. In this paper, we give certain transformation formulas for the function nGn[⋯] which are not implied from the analogous hypergeometric functions over finite fields.

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Elliptic curves and special values of Gaussian hypergeometric series

Journal of Number Theory ◽

10.1016/j.jnt.2013.03.010 ◽

2013 ◽

Vol 133 (9) ◽

pp. 3099-3111 ◽

Cited By ~ 12

Author(s):

Rupam Barman ◽

Gautam Kalita

Keyword(s):

Elliptic Curves ◽

Hypergeometric Series ◽

Special Values ◽

Gaussian Hypergeometric Series

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CERTAIN VALUES OF GAUSSIAN HYPERGEOMETRIC SERIES AND A FAMILY OF ALGEBRAIC CURVES

International Journal of Number Theory ◽

10.1142/s179304211250056x ◽

2012 ◽

Vol 08 (04) ◽

pp. 945-961 ◽

Cited By ~ 13

Author(s):

RUPAM BARMAN ◽

GAUTAM KALITA

Keyword(s):

Finite Field ◽

Elliptic Curves ◽

Algebraic Curve ◽

Hypergeometric Series ◽

Algebraic Curves ◽

Alternate Proof ◽

Special Values ◽

Gaussian Hypergeometric Series

Let λ ∈ ℚ\{0, -1} and l ≥ 2. Denote by Cl, λ the nonsingular projective algebraic curve over ℚ with affine equation given by [Formula: see text] In this paper, we give a relation between the number of points on Cl, λ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of [D. McCarthy, 3F2 Hypergeometric series and periods of elliptic curves, Int. J. Number Theory6(3) (2010) 461–470]. We find some special values of 3F2 and 2F1 Gaussian hypergeometric series. Finally we evaluate the value of 3F2(4) which extends a result of [K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc.350(3) (1998) 1205–1223].

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The number of edges on generalizations of Paley graphs

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201002071 ◽

2001 ◽

Vol 27 (2) ◽

pp. 111-123

Author(s):

Lawrence Sze

Keyword(s):

Elliptic Curves ◽

Hypergeometric Series ◽

Quadratic Residues ◽

Gaussian Hypergeometric Series ◽

Paley Graphs

Evans, Pulham, and Sheenan computed the number of complete4-subgraphs of Paley graphs by counting the number of edges of the subgraph containing only those nodesxfor whichxandx−1are quadratic residues. Here we obtain formulae for the number of edges of generalizations of these subgraphs using Gaussian hypergeometric series and elliptic curves. Such formulae are simple in several infinite families, including those studied by Evans, Pulham, and Sheenan.

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Evaluation of Gaussian hypergeometric series using Huff’s models of elliptic curves

The Ramanujan Journal ◽

10.1007/s11139-018-0075-y ◽

2018 ◽

Vol 48 (2) ◽

pp. 357-368 ◽

Cited By ~ 2

Author(s):

Mohammad Sadek ◽

Nermine El-Sissi ◽

Arman Shamsi Zargar ◽

Naser Zamani

Keyword(s):

Elliptic Curves ◽

Hypergeometric Series ◽

Gaussian Hypergeometric Series

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Special values of the hypergeometric series II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005254 ◽

2001 ◽

Vol 131 (2) ◽

pp. 309-319 ◽

Cited By ~ 5

Author(s):

I. J. ZUCKER ◽

G. S. JOYCE

Keyword(s):

Gamma Function ◽

Hypergeometric Series ◽

Elliptic Integral ◽

Singular Values ◽

Complete Elliptic Integral ◽

Special Values ◽

Gaussian Hypergeometric Series

Several authors [1, 5, 9] have investigated the algebraic and transcendental values of the Gaussian hypergeometric series(formula here)for rational parameters a, b, c and algebraic and rational values of z ∈ (0, 1). This led to several new identities such as(formula here)and(formula here)where Γ(x) denotes the gamma function. It was pointed out by the present authors [6] that these results, and others like it, could be derived simply by combining certain classical F transformation formulae with the singular values of the complete elliptic integral of the first kind K(k), where k denotes the modulus.Here, we pursue the methods used in [6] to produce further examples of the type (1·2) and (1·3). Thus, we find the following results:(formula here)The result (1·6) is of particular interest because the argument and value of the F function are both rational.

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