Sums of Legendre symbols - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:47:11Z http://mathoverflow.net/feeds/question/44497 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44497/sums-of-legendre-symbols Sums of Legendre symbols arithboy 2010-11-02T00:51:01Z 2010-11-02T02:22:16Z <p>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 $$\sum_x\phi(x+a)\phi(x+b)\phi(x+c)\phi(x+d).$$</p> <p>Of courese, the sums of such types are useful as studying the consecutive quadratic residues.</p> http://mathoverflow.net/questions/44497/sums-of-legendre-symbols/44502#44502 Answer by Victor Miller for Sums of Legendre symbols Victor Miller 2010-11-02T01:28:22Z 2010-11-02T02:22:16Z <p>The sums that you give are related to the number of points on</p> <p>$y^2 = (x+a)(x+b)(x+c)$ and $y^2=(x+a)(x+b)(x+c)(x+d)$ respectively.</p> <p>Assuming that $a,b,c,d$ are distinct, the first is an elliptic curve, and the second <strike>a curve of genus 2 </strike> also a curve of genus 1. By the "Riemann Hypothesis" for curves over finite fields each has the value</p> <p>$a_p$ where $a_p$ is the trace of Frobenius. In the both cases <strike>first case </strike> $|a_p| \le 2 \sqrt{p}$ <strike> and in the second $|a_p| \le 2 \lfloor 2 \sqrt{p} \rfloor$ (a result of Serre strengthening the normal RH) </strike>.</p> <p>[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].</p>