Évaluation d'une somme cubique de caractères. (Evaluation of a cubic character sum)
For a polynomial \(f(x)\in {\mathbb{Z}}[x]\), the character sum \(S_ f\) is defined to be \(\sum_{x\in {\mathbb{F}}_ p}(f(x)/p)\) where ( /p) is the Legendre symbol. The author gives a list of cubics f(x) for which \(S_ f\) is known. These are all elliptic curves with complex multiplication. The curve \(y^ 2=x^ 3-6D x^ 2-3D^ 2 x=f(x)\) has End\(\cong {\mathbb{Z}}[\sqrt{-3}]\). The author evaluates \(S_ f\) using the Frobenius automorphism on the \(\sqrt{-3}\)-division points on \(y^ 2=f(x)\); the result being \[ S_ f=(D/p)\begin{cases} 0\text{ if } p\equiv 2 (3)\\ -2c\text{ if } p\equiv 1 (3)\end{cases} \] where \(p=c^ 2+3d^ 2\) with \(c\equiv (- 1/p)\) mod 3. The useful formula for multiplication by \(\sqrt{-3}\), got by Stark's method, is given in the text.
elliptic curves with complex multiplication
formula for multiplication by \(\sqrt{-3}\)
scientific article; zbMATH DE number 4004283
scientific article