Certain character sums and 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.

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

Finite Fields and Their Applications ◽

10.1016/j.ffa.2021.101832 ◽

2021 ◽

Vol 72 ◽

pp. 101832

Author(s):

Gautam Kalita ◽

Arijit Jana

Keyword(s):

Hypergeometric Series ◽

Algebraic Curves ◽

Character Sums ◽

Gaussian Hypergeometric Series

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The Mean Values of Character Sums and Their Applications

Mathematics ◽

10.3390/math9040318 ◽

2021 ◽

Vol 9 (4) ◽

pp. 318

Author(s):

Jiafan Zhang ◽

Yuanyuan Meng

Keyword(s):

Character Sums ◽

Gauss Sums ◽

Mean Values ◽

Special Polynomials ◽

Elementary Methods ◽

The Mean

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

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Higher depth mock theta functions and q-hypergeometric series

Forum Mathematicum ◽

10.1515/forum-2021-0013 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Joshua Males ◽

Andreas Mono ◽

Larry Rolen

Keyword(s):

Theta Function ◽

Modular Forms ◽

Hypergeometric Series ◽

Theta Functions ◽

Mock Theta Function ◽

Mock Theta Functions ◽

Maass Forms ◽

Mock Modular Forms ◽

Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

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A-hypergeometric series and a p-adic refinement of the Hasse-Witt matrix

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ◽

10.1007/s12188-021-00243-1 ◽

2021 ◽

Author(s):

Alan Adolphson ◽

Steven Sperber

Keyword(s):

Hypergeometric Series

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Equidistribution Among Cosets of Elliptic Curve Points in Intervals

Journal of Mathematical Cryptology ◽

10.1515/jmc-2019-0020 ◽

2020 ◽

Vol 14 (1) ◽

pp. 339-345

Author(s):

Taechan Kim ◽

Mehdi Tibouchi

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Character Sums ◽

Fault Analysis ◽

Signature Schemes ◽

Large Subgroup

AbstractIn a recent paper devoted to fault analysis of elliptic curve-based signature schemes, Takahashi et al. (TCHES 2018) described several attacks, one of which assumed an equidistribution property that can be informally stated as follows: given an elliptic curve E over 𝔽q in Weierstrass form and a large subgroup H ⊂ E(𝔽q) generated by G(xG, yG), the points in E(𝔽q) whose x-coordinates are obtained from xG by randomly flipping a fixed, sufficiently long substring of bits (and rejecting cases when the resulting value does not correspond to a point in E(𝔽q)) are close to uniformly distributed among the cosets modulo H. The goal of this note is to formally state, prove and quantify (a variant of) that property, and in particular establish sufficient bounds on the size of the subgroup and on the length of the substring of bits for it to hold. The proof relies on bounds for character sums on elliptic curves established by Kohel and Shparlinski (ANTS–IV).

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