Points on curves in small boxes en applications

Abstract

We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x)\equiv y(\mathrm{mod}p)$ and $f(x)\equiv y^2(\mathrm{mod}p),$ with a prime $p$ and a polynomial $f$, where $(x,y)$ belongs to an arbitrary square with side length $M$. We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curves $Y^2=X^{2g+1}+a_{2g-1}{X}^{2g-1}+...+a_{1}X+a_0$ over the finite field $\F_p$ of $p$ elements, with coefficients in a $2g$-dimensional cube $(a_0,...,a_{2g-1})\in [R_0+1,R_0+M]\times ...\times [R_{2g-1}+1,R_{2g-1}+M]$ that are isomorphic to a given curve and give an almost sharp lower bound on the number of non-isomorphic hyperelliptic curves with coefficients in that cube. Furthermore, we study the size of the smallest box that contain a partial trajectory of a polynomial dynamical system over $\F_p$.

33 pages. Revised version, Theorem 2 was improved

References in corpus (1)

• On Congruences with Products of Variables from Short Intervals and Applications

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