An Algorithm for Modular Elliptic Curves over Real Quadratic Fields

Let F be a real quadratic field with narrow class number one, and f a Hilbert newform of weight 2 and level n with rational Fourier coefficients, where n is an integral ideal of F . By the Eichler-Shimura construction, which is still a conjecture in many cases when [F : Q] > 1, there exists an elliptic curve Ef over F attached to f . In this paper, we develop an algorithm that computes the (candidate) elliptic curve Ef under the assumption that the Eichler-Shimura conjecture is true. We give several illustrative examples which explain among other things how to compute modular elliptic curves with everywhere good reduction. Such curves do not admit any parametrization by Shimura curves, and so the Eichler-Shimura construction is still conjectural in this case. Introduction Let F be a totally real number field of degree n, OF its ring of integers and n ⊆ OF an integral ideal. Let f be a Hilbert newform of weight 2 and level n. The differential form attached to f is given by ωf = (2πi)f(z1, . . . , zn)dz1 · · · dzn and, for each prime p, we let ap(f) be the Fourier coefficient of f at p. Let E be an elliptic curve defined over F . The trace of the Frobenius endomorphism acting on E at the prime p is denoted by ap(E). We recall that, for p n, ap(E) = N(p) + 1−#Ē(Fp), where Fp = OF /p is the residue field at p and Ē the reduction of E modulo p; and N(p) is the norm of p. The L-series of f is given by