Coverings of elliptic curves with complex multiplication by hyperelliptic curves and character sums
Let \(p\) be a prime number, \(x\), \(A\), \(B\) and \(n\geq 1\) integers. Denote by \(\chi(x)\) the Legendre symbol of \(x\) at \(p\). In this paper the authors study the following character sums \[ \Gamma_{p,n}(A,B)=\sum_x\chi(x(x^{2n}-Ax^n+B)) \quad\text{and}\quad \Delta_{p,n}(A,B)=\sum_x\chi(x^{2n}-Ax^n+B). \] In the case \(n=2\) they use coverings of elliptic curves with complex multiplication by hyperelliptic curves to compute \(\Gamma_{p,2}(\theta,1)\). This is done using previous results from \textit{B. W. Brewer} [Trans. Am. Math. Soc. 99, 241--245 (1961; Zbl 0103.03205), and Proc. Am. Math. Soc. 17, 502--509 (1966; Zbl 0147.29801)] where he gives a formula for \(\Gamma_{p,2}(5Q,5Q^2)\) in the case where \(p=a^2+b^2=u^2+5v^2\), where \(a,b,u,v\) are integers. Another tool is the theory of complex multiplication and Weber's functions. If \(E\) is an elliptic curve with complex multiplication by an order \(\mathcal O_z\) of conductor \(m\) in a complex quadratic field \(K=\mathbb Q(\sqrt{-p})\) of discriminant \(-D\) and equation \(y^2=f(x)\), one can compute the character sum \(S_p(f)=\sum_x\chi(f(x))\) in terms of the representation of \(p\) of the form \(4p=U^2+m^2DV^2\), where \(U\) and \(V\) are integers satisfying certain normalizing conditions. One of the goals of the paper is the search for these normalizing conditions. The previous decomposition of \(p\) means that \(p\) is decomposed in \(\mathbb Q(\sqrt{-1},\sqrt{5})\) which is the class field of \(\mathbb Q(\sqrt{-20})\). So it is natural to search for a connection between the curve of equation \(y^2=x(x^4-5Qx^2+5Q^2)\) and a curve with complex multiplication by \(\mathbb Q(\sqrt{-20})\). The authors then consider the character sum \(\Gamma_{p,2}(\theta,1)\) with \(\theta\neq\pm 2\), the curve \(C_{\theta}\) given by the equation \(y^2=x(x^4-\theta x^2+1)\) which has genus \(2\) and whose Jacobian variety is isogenous to the product of the elliptic curves \(E_{\theta}\) and \(E'_{\theta}\) given by equations \(y^2=P_{\theta}(x)=x(x^2+4x+2-\theta)\) and \(y^2=x(x^2-4x+2-\theta)\), respectively. In the case where \(\theta\) takes particular values, which are expressed as quotients of Weber functions in a complex variable \(z\), \(E_{\theta}\) has modular invariant equal to \(j(z)\), and from this they show that if \(p\) is an odd prime and \(\theta\in\mathbb F_p-\{2\}\) then \(\Gamma_{p,2}(\theta,1)=(1-\chi(-1))S_p(P_{\theta})\).
coverings of elliptic curves with complex multiplication
scientific article; zbMATH DE number 1025607
scientific article