Computations of elliptic units for real quadratic fields
Let K be a real quadratic field, and p a rational prime which is inert in K. Let a be a modular unit on Γ0(N). In an earlier joint article with Henri Darmon, we presented the definition of an element u(α, τ) ε Kpx attached to a and each τ ε K. We conjectured that the p-adic number u(α, τ) lies in a specific ring class extension of K depending on τ, and proposed a "Shimura reciprocity law" describing the permutation action of Galois on the set of u(α, τ), This article provides computational evidence for these conjectures. We present an efficient algorithm for computing u(α, τ), and implement this algorithm with the modular unit α(z) = Δ(z) 2 Δ(4z)3,Δ(2z)3. Using p = 3, 5, 7, and 11, and all real quadratic fields K with discriminant D < 500 such that 2 splits in K and K contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define w(α, τ) is shown to be Z-valued rather than only Zp ∩ Q-valued; this is an improvement over our previous result and allows for a precise definition of u(α, τ), instead of only up to a root of unity. © Canadian Mathematical Society 2007.
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