Distribution properties of certain exponential sums
type: Item
instance of: scholarly article
title: Distribution properties of certain exponential sums
zbMATH Open document ID: 0728.11032
DOI: 10.1215/S0012-7094-91-06209-5
author: Giuliana Davidoff
published in: Duke Mathematical Journal
publication date: 1991-01-01T00:00:00Z
review text: Let \(f(x,y)=0\) be a nonsingular affine curve of genus g. For an appropriate prime p, one can associate the number - which we denote by \(N_ p\)- of solutions in \({\mathbb{F}}_ p\times {\mathbb{F}}_ p\) as well as the ``error term'' \(a_ p=N_ p-p\). By the Riemann hypothesis for curves, \(| a_ p| \geq 2g\sqrt{p}\). We call \(d\mu_{g,f}(t)\) the asymptotic density function if \[ \lim_{x\to \infty}\frac{n\{p\geq x| \quad p\in U,\quad a_ p/\sqrt{p}\in (\alpha,\beta)\}}{n\{p\geq x| p\in U\}}=\int^{\beta}_{\alpha}d\mu_{g,f}(t) \] holds. The determination of the asymptotic density function is in general a difficult problem. However, we can get the following result in the case of a certain family of hyperelliptic curves. Let \(f_{q^ k}(t)\) be the asymptotic density function associated to the curve \(y^ 2=x^{q^ k}+1\), with genus \(g=(q^ k-1)/2\). Then we have \[ \lim_{g\to \infty}\sqrt{2q}\int^{y}_{-\infty}f_{q^ k}(\sqrt{2g}t)dt=(1/\sqrt{2\pi})\int^{y}_{-\infty}\exp (-t^ 2/2)dt. \]
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 4202378
zbMATH Keywords: affine curve
zbMATH Keywords: hyperelliptic curves
zbMATH Keywords: asymptotic density function
reviewed by: Masakazu Muro
MaRDI profile type: Publication
cites work: Q5606304
cites work: How the Number of Points of An Elliptic Curve Over a Fixed Prime Field Varies
cites work: Sums of Gauss, Jacobi, and Jacobsthal
cites work: Q3208805
cites work: Q5565773
cites work: A Proof of the Generalized Second-Limit Theorem in the Theory of Probability
cites work: Q5528098
cites work: Q3939864
cites work: Q5612629
cites work: Q5580133
cites work: Q3943047
cites work: Numbers of solutions of equations in finite fields
cites work: Jacobi Sums as "Grossencharaktere"
cites work: On an analogue of the Sato conjecture
description: scientific article; zbMATH DE number 4202378