William Stein, U.C. Berkeley

``Computing Period Lattices Associated to Newforms''

April 28, 1999

Let f=sum a_n q^n be a weight 2 newform with rational Fourier coefficients. Then the integrals Int_gamma f(z) dz as gamma varies through paths in H_1(X_0(N),Z) form a lattice Lambda_f \subset C. Conjecturally, the Weierstrass invariants c4 and c6 of Lambda_f lie in Z, and an elliptic curve E/Q with L-series the Mellin transform of f is: y^2 = x^3 - 27*c_4*x - 54*c_6. For all this and an algorithm to compute c_4 and c_6 see Cremona, Algorithms for Modular Elliptic Curves, sections 2.10--2.14. In this talk I will give a generalization of the algorithm described in Cremona which can be used to numerically compute the period lattice Lambda_f of any newform f of weight k>=2.

When the weight is greater than 2 the meaning of the "intermediate Jacobian" C^d/Lambda_f is mysterious, but when f has rational Fourier coefficient it must be an elliptic curve E_f. Can we determine E_f? Is the j-invariant of E_f transcendental or algebraic?

When k=2 this algorithm has applications to the numerical verification of the Birch and Swinnerton-Dyer conjecture for quotients of J_0(N) of dimension >= 1.