Elliptic curves attached to Jacobians with imaginary multiplication

Source abstract

Let M be an imaginary quadratic field and A be an abelian variety of odd dimension g defined over Q for which there is a unitary algebra embedding of M in End^0(A_M) of signature ((g+1)/2,(g-1)/2). Extending a result by Fité and Goodman, we show the existence of an elliptic curve E over Q with complex multiplication by M such that H_et^1(E_Qbar, Q_l) is a G_Q-subrepresentation of H_et^g ( A_Qbar, Q_l((g-1)/2) ). The main results of the article consist on explicit descriptions of E for A in two different families. First, when A is the Jacobian of the curve y^2 = xh(x^2), where h in Z[x] is a degree g monic separable polynomial with nonzero h(0), we show that E is given by y^2 = x^3 + h(0)x. Second, when A is the Jacobian of y^3=f(x), where h in Z[x] is a degree 4 monic separable polynomial, we show that E is given by y^2 = x^3 + 16 disc(f). This latter result builds on work by Laga and Shnidman in the bielliptic case, of which we give an alternative proof by means of a new p-adic method. As an application, we derive a practical average polynomial time algorithm to compute the Zeta functions of the reductions of a Picard curve.