Elliptic curves with complex multiplication and a character sum
For a polynomial \(f(n)\in\mathbb Z[n]\) let \(S_p(f)\) denote the character sum \(\sum_{n\pmod p}(f(n)/p)\), where \(p\) is prime and \((\cdot/p)\) the Legendre symbol. If \(y^2=f(n)\) is an elliptic curve with complex multiplication, it is well-known that its full ring of CMs is the ring of integers of an imaginary quadratic field of class number 1; i.e. the fields \(\mathbb Q(\sqrt{-D})\), where \(D=1,2,3,7,11,19,43,67,163\). Then by Deuring's theorem \[ S_p(f)= \begin{cases} 0\quad\text{if }p\text{ is inert in }\mathbb Q(\sqrt{-D}),\\ \pm C\quad\text{where }4p= C^2+Dd^2.\end{cases} \] To know the sum exactly, we need to fix the sign of \(C\). The cases \(D=1,2,3,7,11,19\) have already been dealt with by Rajwade and others using explicit division points on the corresponding elliptic curves. However, these division points get increasingly more complicated; even for the case \(D=19\), the formulae are much too long. In the present paper, the authors give a very neat method of dealing with the cases 19 onwards (19 inclusive) almost simultaneously, the sign of \(C\) being determined in each case by the condition \((C/D)=(2/p)\).
elliptic curve with complex multiplication
scientific article; zbMATH DE number 980269