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Kit-Ho Mak

Publications and source records attributed to Kit-Ho Mak.

9 recordsLinked to original sources
arXiv2014

The distribution of values of short hybrid exponential sums on curves over finite fields II

Let $p$ be a prime number, $C$ be any absolutely irreducible affine plane curve over $\mathbb{F}_p$, and $g,f\in\mathbb{F}_p(x,y)$ be rational functions. We continue the study of the distribution of the values of short hybrid exponential sums of the form $S_{H}(x;C) = \sum_{P\in C, x<x(P)\leq x+H}χ(g(P))ψ(f(P))$ on $x\in\mathcal{I}$ for some short interval $\mathcal{I}$. We show that under some natural conditions, the limiting distribution of the sum $S_{H}(x;C)$ is Gaussian for all curve $C$. This largely generalizes a previous result of the author and Zaharescu.

math.NT↗
arXiv2013

The distribution of rational points and polynomial maps on an affine variety over a finite field on average

Let $p$ be a prime, let $V/\mathbb{F}_p$ be an absolutely irreducible affine variety inside the affine $r$-space. In this paper, we consider the problem of how often a box $B$ will contain the expected number of points. In particular, we give a lower bound on the volume of $B$ that guarantees almost all translations of $B$ in the $r$-space contain the expected number of points. This shows that the Weil estimate holds in smaller regions in an "almost all" sense.

math.NT↗
arXiv2013

The distribution of values of short hybrid exponential sums on curves over finite fields

Let $p$ be a prime number, $X$ be an absolutely irreducible affine plane curve over $\mathbb{F}_p$, and $g,f\in\mathbb{F}_p(x,y)$. We study the distribution of the values of the hybrid exponential sums S_n on $n\in\mathcal{I}$ for some short interval $\mathcal{I}$. We show that under some natural conditions the limiting distribution of the projections of the sum $S_n$, $n\in\mathcal{I}$ on any straight line through the origin is Gaussian as $p$ tends to infinity.

math.NT↗
arXiv2013

Lehmer points and visible points on affine varieties over finite fields

Let $V$ be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on $V$ is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on $V$ are obtained, and the distribution of visible points into different congruence classes is investigated.

math.NT↗
arXiv2012

On lower bounds for the Ihara constants A(2) and A(3)

Let X be a curve over the finite field of q elements and let N(X), g(X) be its number of rational points and genus respectively. The Ihara constant A(q) is defined by the limit superior of N(X)/g(X) as the genus of X goes to infinity. In this paper, we employ a variant of Serre's class field tower method to obtain an improvement of the best known lower bounds on A(2) and A(3).

math.NT↗
arXiv2011

Poisson Type Phenomena for Points on Hyperelliptic Curves modulo p

Let $p$ be a large prime, and let $C$ be a hyperelliptic curve over $\mathbb{F}_p$. We study the distribution of the $x$-coordinates in short intervals when the $y$-coordinates lie in a prescribed interval, and the distribution of the distance between consecutive $x$-coordinates with the same property. Next, let $g(P,P_0)$ be a rational function of two points on $C$. We study the distribution of the above distances with an extra condition that $g(P_i,P_{i+1})$ lies in a prescribed interval, for any consecutive points $P_i,P_{i+1}$.

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arXiv2011

On the distribution of the number of points on a family of curves over finite fields

Let $p$ be a large prime, $\ell\geq 2$ be a positive integer, $m\geq 2$ be an integer relatively prime to $\ell$ and $P(x)\in\mathbb{F}_p[x]$ be a polynomial which is not a complete $\ell'$-th power for any $\ell'$ for which $GCD(\ell',\ell)=1$. Let $\mathcal{C}$ be the curve defined by the equation $y^{\ell}=P(x)$, and take the points on $\mathcal{C}$ to lie in the rectangle $[0,p-1]^2$. In this paper, we study the distribution of the number of points on $\mathcal{C}$ inside a small rectangle among residue classes modulo $m$ when we move the rectangle around in $[0,p-1]^2$.

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