A Short Character Sum in $\mathbb{F}_{p^3}$

TL;DR

This paper derives new bounds for short character sums over finite fields, extending previous results to higher dimensions and applying them to character sums on cubic forms, with implications for number theory.

Contribution

It introduces novel bounds for short character sums in $ ext{F}_{p^d}$, generalizing earlier work and applying to cubic form sums, advancing understanding of character sum cancellations.

Findings

Nontrivial cancellation for sums over intervals of size $p^{3/8+ ext{epsilon}}$

Extension of bounds to higher-dimensional lattices in $ ext{F}_{p^d}$

Application to character sums on binary cubic forms

Abstract

We establish a new bound for short character sums in finite fields, particularly over two-dimensional grids in $\mathbb{F}_{p^3}$ and higher-dimensional lattices in $\mathbb{F}_{p^d}$, extending an earlier work of Mei-Chu Chang on Burgess inequality in $\mathbb{F}_{p^2}$. In particular, we show that for intervals of size $p^{3/8+\varepsilon}$, the sum $\sum_{x, y} \chi(x + \omega y)$, with $\omega \in \mathbb{F}_{p^3} \setminus \mathbb{F}_p$, exhibits nontrivial cancellation uniformly in $\omega$. This is further generalized to codimension-one sublattices in $\mathbb{F}_{p^d}$, and applied to obtain an alternative estimate for character sums on binary cubic forms.