Elliptic units in ray class fields of real quadratic number fields

Let K be a real quadratic number field. Let p be a prime which is inert in K. We denote the completion of K at the place p by Kp. Let f > 1 be a positive integer coprime to p. In this thesis we give a p-adic construction of special elements u(r, τ) ∈ K× p for special pairs (r, τ) ∈ (Z/fZ)× × Hp where Hp = P(Cp)\P(Qp) is the so called p-adic upper half plane. These pairs (r, τ) can be thought of as an analogue of classical Heegner points on modular curves. The special elements u(r, τ) are conjectured to be global p-units in the narrow ray class field of K of conductor f . The construction of these elements that we propose is a generalization of a previous construction obtained in [DD06]. The method consists in doing p-adic integration of certain Z-valued measures on X = (Zp ×Zp)\(pZp × pZp). The construction of those measures relies on the existence of a family of Eisenstein series (twisted by additive characters) of varying weight. Their moments are used to define those measures. We also construct p-adic zeta functions for which we prove an analogue of the so called Kronecker’s limit formula. More precisely we relate the first derivative at s = 0 of a certain p-adic zeta function with − logpNKp/Qpu(r, τ). Finally we also provide some evidence both theoretical and numerical for the algebraicity of u(r, τ). Namely we relate a certain norm of our p-adic invariant with Gauss sums of the cyclotomic field Q(ζf , ζp). The norm here is taken via a conjectural Shimura reciprocity law. We also have included some numerical examples at the end of section 18.