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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Evaluation of character sums connected with elliptic curves


Author: Kenneth S. Williams
Journal: Proc. Amer. Math. Soc. 73 (1979), 291-299
MSC: Primary 10G15; Secondary 10D25
MathSciNet review: 518507
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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

$\displaystyle \sum\limits_{x = 0}^{p - 1} {\left( {\frac{{{x^4} - 8{x^3} + 12{x... ...,} \hfill & {{\text{if}}\;p \equiv 3\;\pmod 4,} \hfill \\ \end{array} } \right.$

where $ {x_1}$ is defined for primes $ p \equiv 1\;\pmod 4$ by

$\displaystyle p = x_1^2 + y_1^2,\quad {x_1} \equiv - 1\quad \pmod 4.$


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DOI: https://doi.org/10.1090/S0002-9939-1979-0518507-4
Keywords: Legendre symbol, character sums, elliptic curves, complex multiplication
Article copyright: © Copyright 1979 American Mathematical Society