Evaluation of character sums connected with elliptic curves
Kenneth S. Williams · Proceedings of the American Mathematical Society · 1979
Let p be an odd prime and let ( ⋅ p ) (\tfrac { \cdot }{p}) be the Legendre symbol. It is shown how to evaluate the character sum Σ x = 0 p − 1 ( f ( x ) p ) \Sigma _{x = 0}^{p - 1}(\tfrac {{f(x)}}{p}) , for certain quartic polynomials f ( x ) f(x) . For example, it is shown that \[ ∑ x = 0 p − 1 ( x 4 − 8 x 3 + 12 x 2 − 16 x + 4 p ) = { 2 ( 2 p ) x