Finding Exact Values of a Character Sum

Abstract

Let $F_p$ be a field with p elements, where p is a positive prime. For x in $F_p$ the quadratic character $\chi$ is defined as follows: If x is a nonzero square, then $\chi (x)=1;$ if x is a non-square, then $\chi (x)=-1;$ $\chi (0)=0.$ Note that x is a square in $F_p$ if and only if there exists a in $F_p$ such that $x=a^2.$ Let $f(x)=x^2+bx+c$ and $g(x)=x^2+\widetilde{b}x+\widetilde{c}$ be two irreducible polynomials in $F_p\left[ x\right] .$ (That is, $\chi (b^2-4c)=\chi (\widetilde{b}^2-4\widetilde{c})=-1$). We will also assume that the resultant of f(x) and g(x) is nonzero in an algebraic closure of $F_p$. That is $\text {Re}\,s(f,\,g)=\prod \limits _{(\alpha ,\,\beta ):f(\alpha )=0\text { and } g(\beta )=0}\,(\alpha -\beta )\ne 0,$ where the product is taken over all $\alpha$ and $\beta$ in the algebraic closure for which $f(\alpha )=0$ and $g(\beta )=0.$ It is easy to show that the above no common roots condition is equivalent to $\text {Re}\,s(f(x),\,g(x))=\left( c-\widetilde{c}\right) ^2+\left( b-\widetilde{b}\right) \left( b\widetilde{c}-\widetilde{b}c\right) \ne 0.$ We now form the character sum $W_p$ given by $W_p=\sum \limits _{x\in F_p}\chi \left( f(x)g(x)\right) .$ We present a new method for computing $W_p$ when $b^2-4c\ne \widetilde{b}^2-4\widetilde{c}$ $\text {mod}\,p.$ Our method involves counting points from $F_p\times F_p$ that are on a specified elliptic curve.