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Computations of elliptic units for real quadratic fields. (English) Zbl 1118.11045

This article is about a recent contribution towards the solution of Hilbert’s 12th problem, the explicit construction of class fields using values of analytic functions. Classically, such solutions are known if the base field is the field of rationals (cyclotomy) or a complex quadratic number field (complex multiplication). The approach presented here works for real quadratic number fields, and goes back to work by H. Darmon [Ann. Math. (2) 154, No. 3, 589–639 (2001; Zbl 1035.11027)].
To briefly describe the relevant background, let \(K\) be a real quadratic number field, and \(p\) a prime number inert in \(K\). For each modular unit \(\alpha\) on \(\Gamma_0(N)\) and each \(\tau \in K\), H. Darmon and the author [Ann. Math. (2) 163, No. 1, 301–346 (2006; Zbl 1130.11030)] constructed an element \(u(\alpha,\tau)\) in the \(p\)-adic completion \(K_p\) of \(K\), which is conjectured to lie in a certain ring class field of \(K\) depending on \(\tau\), and formulated a conjectural reciprocity law similar to Shimura’s.
In this article, the author describes an efficient algorithm for computing \(u(\alpha,\tau)\), and presents computational evidence for these conjectures.

MSC:

11R37 Class field theory
11R11 Quadratic extensions
11Y40 Algebraic number theory computations
11G16 Elliptic and modular units
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