On character sums related to elliptic curves with complex multiplication
In 1975 \textit{T. Hadano} [Proc. Japan Acad. 51, 92-95 (1975; Zbl 0329.14018)]\ determined the most general equation of an elliptic curve defined over \(\mathbb{Q}\) with complex multiplication by the ring of integers of \(\mathbb{Q} (\sqrt {-d})\) \((d>0)\). There are nine such equations of the form \(y^2= f_d (x)\), where \(d=1, 2, 3, 7, 11, 19, 43, 67, 163\). Let \(p\) be a prime number and \((\cdot/p)\) the Legendre symbol. The sum of characters \(S_d (p)= \sum^{p-1}_{x=0} (f_d (x)/ p)\), is useful for the implementation of the primality test ECPP. So it is interesting to know its exact value. Rajwade and others, using the theory of complex multiplication, calculated the sum of characters \(S_d (p)\) for \(d=1, 2, 3, 7, 11, 19\). In the paper under review, the authors, simplifying the method of Rajwade, give the value of the sum \(S_d (p)\) for the last three cases. They prove that for \(d\in \{43, 67, 163\}\) we have \[ S_d (p)= \begin{cases} 0 &\text{ if } (-d/p) =-1,\\ (2/p) (u/d)u &\text{ if } (-d/p) =1\text{ and } 4p= u^2+ dv^2 \end{cases} \] {}.
elliptic curves with complex multiplication
scientific article; zbMATH DE number 821867