Computations of elliptic units for real quadratic fields
نویسنده
چکیده
Elliptic units, which are obtained by evaluating modular units at quadratic imaginary arguments of the Poincaré upper half-plane, allow the analytic construction of abelian extensions of imaginary quadratic fields. The Kronecker limit formula relates the complex absolute values of these units to values of zeta functions, and allowed Stark to prove his rank one archimedean conjecture for abelian extensions of quadratic imaginary fields [9]. A conjectural construction of an analogous theory for real quadratic fieldsK was proposed in [2], by replacing the infinite prime of Q with a prime p that remains inert in K. The completion Kp is a quadratic unramified extension of Qp. The construction of [2] associates to a modular unit α and any τ ∈ K −Q an element u(α, τ) ∈ K× p which is conjectured to be a p-unit in a specific narrow ring class field of K depending on τ and denoted Hτ (cf. [2, Conjecture 2.14], hereafter denoted Conjecture DD). In harmony with the fact that the role of ∞ is played by that of p, the construction of u(α, τ) involves p-adic integration in a manner motivated by the definition of “Stark-Heegner points” given in [1] and generalized in [5].
منابع مشابه
On the real quadratic fields with certain continued fraction expansions and fundamental units
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