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.

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

Quelques résultats sur les équations axp + byp = cz2

2006 ◽  
Vol 58 (1) ◽  
pp. 115-153 ◽  
Author(s):  
W. Ivorra ◽  
A. Kraus
Keyword(s):  
Prime Number ◽  
Diophantine Equation ◽  
Hyperelliptic Curves ◽  
Rational Points ◽  
Modular Representations ◽  
Natural Numbers ◽  
Prime Divisors

AbstractLet p be a prime number ≥ 5 and a, b, c be non zero natural numbers. Using the works of K. Ribet and A. Wiles on the modular representations, we get new results about the description of the primitive solutions of the diophantine equation axp + byp = cz2, in case the product of the prime divisors of abc divides 2ℓ, with ℓ an odd prime number. For instance, under some conditions on a, b, c, we provide a constant f (a, b, c) such that there are no such solutions if p > f (a, b, c). In application, we obtain information concerning the ℚ-rational points of hyperelliptic curves given by the equation y2 = xp + d with d ∈ ℤ.

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
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