Certain character sums, Gaussian hypergeometric series, and their connections to algebraic curves

2021 ◽  
Vol 72 ◽  
pp. 101832
Author(s):  
Gautam Kalita ◽  
Arijit Jana
Keyword(s):  
Hypergeometric Series ◽  
Algebraic Curves ◽  
Character Sums ◽  
Gaussian Hypergeometric Series

Values of Gaussian hypergeometric series and their connections to algebraic curves

2017 ◽  
Vol 14 (01) ◽  
pp. 1-18 ◽  
Author(s):  
Gautam Kalita
Keyword(s):  
Hypergeometric Series ◽  
Algebraic Curves ◽  
Vital Role ◽  
Special Values ◽  
Transformation Formulas ◽  
Gaussian Hypergeometric Series

In this paper, we explicitly evaluate certain special values of [Formula: see text] hypergeometric series. These evaluations are based on some summation transformation formulas of Gaussian hypergeometric series. We find expressions of the number of points on certain algebraic curves over [Formula: see text] in terms of Gaussian hypergeometric series, which play the vital role in deducing the transformation results.

scholarly journals Character sums, Gaussian hypergeometric series, and a family of hyperelliptic curves

2016 ◽  
Vol 12 (08) ◽  
pp. 2173-2187 ◽  
Author(s):  
Mohammad Sadek
Keyword(s):  
Hypergeometric Series ◽  
Hyperelliptic Curves ◽  
Character Sums ◽  
Rational Points ◽  
Quadratic Character ◽  
Jacobian Varieties ◽  
Gaussian Hypergeometric Series

We study the character sums [Formula: see text] [Formula: see text] where [Formula: see text] is the quadratic character defined over [Formula: see text]. These sums are expressed in terms of Gaussian hypergeometric series over [Formula: see text]. Then we use these expressions to exhibit the number of [Formula: see text]-rational points on families of hyperelliptic curves and their Jacobian varieties.

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

2012 ◽  
Vol 08 (04) ◽  
pp. 945-961 ◽  
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].

Special values of the hypergeometric series II

2001 ◽  
Vol 131 (2) ◽  
pp. 309-319 ◽  
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.

scholarly journals Certain character sums and hypergeometric series

2018 ◽  
Vol 295 (2) ◽  
pp. 271-289
Author(s):  
Rupam Barman ◽  
Neelam Saikia
Keyword(s):  
Hypergeometric Series ◽  
Character Sums

scholarly journals Gaussian Hypergeometric series and supercongruences

2009 ◽  
Vol 78 (265) ◽  
pp. 275-275 ◽  
Author(s):  
Robert Osburn ◽  
Carsten Schneider
Keyword(s):  
Hypergeometric Series ◽  
Gaussian Hypergeometric Series

Hyperelliptic curves and values of Gaussian hypergeometric series

2014 ◽  
Vol 102 (4) ◽  
pp. 345-355 ◽  
Author(s):  
Rupam Barman ◽  
Gautam Kalita ◽  
Neelam Saikia
Keyword(s):  
Hypergeometric Series ◽  
Hyperelliptic Curves ◽  
Gaussian Hypergeometric Series

scholarly journals 3F2 HYPERGEOMETRIC SERIES AND PERIODS OF ELLIPTIC CURVES

2010 ◽  
Vol 06 (03) ◽  
pp. 461-470 ◽  
Author(s):  
DERMOT McCARTHY
Keyword(s):  
Elliptic Curves ◽  
Hypergeometric Series ◽  
The Real ◽  
Gaussian Hypergeometric Series ◽  
Trace Of Frobenius

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.

Sign in / Sign up

Export Citation Format

Share Document