Sums of Legendre symbols

Question

Recently, I am considering a class of character sums concerning Legendre symbols. Let $p$ be an odd prime, and $\phi$ the Legendre symbol mod $p$. It is well known that $$\sum_x\phi(x+a)\phi(x+b)=-1,\ \ (a-b,p)=1,$$where $x$ runs over the residue system mod $p$. However, I haven't found any references on the character sums $$\sum_x\phi(x+a)\phi(x+b)\phi(x+c)$$ and $$\sum_x\phi(x+a)\phi(x+b)\phi(x+c)\phi(x+d).$$

Of courese, the sums of such types are useful as studying the consecutive quadratic residues.

Answer 1

The sums that you give are related to the number of points on

$y^2 = (x+a)(x+b)(x+c)$ and $y^2=(x+a)(x+b)(x+c)(x+d)$ respectively.

Assuming that $a,b,c,d$ are distinct, the first is an elliptic curve, and the second a curve of genus 2 also a curve of genus 1. By the "Riemann Hypothesis" for curves over finite fields each has the value

$a_p$ where $a_p$ is the trace of Frobenius. In the both cases first case $|a_p| \le 2 \sqrt{p}$ and in the second $|a_p| \le 2 \lfloor 2 \sqrt{p} \rfloor$ (a result of Serre strengthening the normal RH) .

[added: the formula I gave includes the "point at infinity" so that you need to subtract 1 in the first case and 2 in the second. The first case that you give corresponds to a curve of genus 0].