Evaluation of character sums connected with elliptic curves

Williams, Kenneth S.

doi:10.1090/S0002-9939-1979-0518507-4

Evaluation of character sums connected with elliptic curves
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Proc. Amer. Math. Soc. 73 (1979), 291-299 Request permission

Abstract:

Let p be an odd prime and let $(\tfrac { \cdot }{p})$ be the Legendre symbol. It is shown how to evaluate the character sum $\Sigma _{x = 0}^{p - 1}(\tfrac {{f(x)}}{p})$, for certain quartic polynomials $f(x)$. For example, it is shown that \[ \sum \limits _{x = 0}^{p - 1} {\left ( {\frac {{{x^4} - 8{x^3} + 12{x^2} - 16x + 4}}{p}} \right )} = \left \{ {\begin {array}{*{20}{c}} {2\left ( {\frac {2}{p}} \right ){x_1} - 1,} \hfill & {{\text {if}}\;p \equiv 1\;\pmod 4,} \hfill \\ { - 1,} \hfill & {{\text {if}}\;p \equiv 3\;\pmod 4,} \hfill \\ \end {array} } \right .\] where ${x_1}$ is defined for primes $p \equiv 1\;\pmod 4$ by \[ p = x_1^2 + y_1^2,\quad {x_1} \equiv - 1\quad \pmod 4.\]

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