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Elliptic units for real quadratic fields. (English) Zbl 1130.11030

Let \(N \geq 1\) be a natural number, \(\Gamma_0(N)\) the standard Hecke congruence group acting on the upper half plane \({\mathcal H}\), and \(X_0(N)\) the associated modular curve. A modular unit is a nowhere vanishing function on \({\mathcal H}/\Gamma_0(N)\) which extends to a meromorphic function on the compact Riemann surface \(X_0(N)(\mathbb C)\).
Fix such a unit \(\alpha(\tau)\), and let \({\mathcal O}\) be a quadratic order of discriminant \(D < 0\) coprime to \(N\). For an element \(\tau \in {\mathcal H}\) “belonging” to such an order \({\mathcal O}\), let \(u(\alpha,\tau) = \alpha(\tau)\). By class field theory, the class group \(\text{Cl}({\mathcal O})\) is isomorphic to the Galois group of a certain ring class field \(H\) over the complex quadratic base field \(K\); let \({\mathcal O}_H\) denote its ring of integers. Then the theory of complex multiplication implies that \(u(\alpha,\tau)\) belongs to \({\mathcal O}_H[\frac1N]^\times\), and that \((\sigma-1)u(\alpha,\tau) \in {\mathcal O}_H^\times\) for all \(\sigma \in \text{Gal}(H/K)\). It is possible to let integral ideals \({\mathfrak a}\) in \({\mathcal O}\) act on the upper half plane; using this action, the reciprocity law can be made fully explicit: Shimura’s reciprocity law states \(u(\alpha, {\mathfrak a} \cdot \tau) = \text{rec}({\mathfrak a})^{-1} u(\alpha, \tau). \)
In this article, Darmon and Dasgupta construct a theory of “complex multiplication” for real quadratic base fields by replacing the complex upper half-plane with the \(p\)-adic upper half-plane \({\mathcal H}_p = P_1(\mathbb C_p) \setminus P_1(\mathbb Q_p)\). Starting with a real quadratic \(\mathbb Z[\frac1p]\)-order \({\mathcal O}\) of positive discriminant \(D\), the authors then construct elements \(u(\alpha,\tau)\) for certain elements \(\tau\) in the \(p\)-adic upper half-plane that behave as much as possible like the elliptic units in the classical theory. The actual construction is rather involved; for some background motivating this construction see M. Bertolini, H. Darmon and P. Green [Math. Sci. Res. Inst. Publ. 49, 323–367 (2004; Zbl 1173.11328)].
The main conjecture formulated by the authors is a direct analogue of Shimura’s reciprocity law whose proof would yield an explicit class field theory for real quadratic number fields. It seems that the construction yields only trivial results if the ring class field of the real quadratic base field is real; this makes it necessary to work with the class group in the strict sense, and why nontrivial results are only obtained if the fundamental unit has positive norm.
The authors next investigate special values of zeta functions of the binary quadratic form attached to certain elements \(\tau\) in the \(p\)-adic upper half plane. Using these special values they construct a “Brumer-Stark element” \(BS_\tau\) in the integral group ring of the class group of \({\mathcal O}\) in the strict sense, and observe that the Brumer-Stark conjecture on annihilators of the class group of \(H\), which was shown to hold for real quadratic base fields by A. Wiles [Ann. Math. (2) 131, 555–565 (1990; Zbl 0719.11082)], also follows from their conjectural analogue of Shimura’s reciprocity law.
In the last section, the authors prove a \(p\)-adic analogue of Kronecker’s limit formula and then deduce the Gross-Stark conjecture for \(H/K\) from the conjectural reciprocity law.

MSC:

11G16 Elliptic and modular units
11F85 \(p\)-adic theory, local fields
11R37 Class field theory
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