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No description available.

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 ƒ > 1 be a positive integer coprime to p. In this thesis we give a p-adic construction of special elements u(r, ??) ∈ Kxp for special pairs (r, ??) ∈ (ℤ/ƒℤ)x x Hp where Hp = ℙ¹(ℂp) ℙ¹(ℚp) is the so called p-adic upper half plane. These pairs (r, ??) can be thought of as an analogue of classical Heegner points on modular curves. The special elements u(r, ??) are conjectured to be global p-units in the narrow ray class field of K of conductor ƒ. 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 ??=ℤpxℤp pℤpxpℤp . 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 -logₚ NKp/Qp u(r, ??). Finally we also provide some evidence both theoretical and numerical for the algebraicity of u(r, ??). Namely we relate a certain norm of our p-adic invariant with Gauss sums of the cyclotomic field Q (zetaf, zetap). The norm here is taken via a conjectural Shimura reciprocity law. We also have included some numerical examples at the end of section 18.

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In this thesis I will outline the impact of Pell's equation on various branches of number theory, as well as some of the history. I will also discuss some recently discovered properties of the solutions of Pell's equation.

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No description available.

Let K be an imaginary quadratic field with class number one and let [Special characters omitted.] be a degree one prime ideal of norm p not dividing 6 d K . In this thesis we generalize an algorithm of Schoof to compute the class number of ray class fields [Special characters omitted.] heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura's reciprocity law. We have discovered a very interesting phenomena where p divides the class number of [Special characters omitted.] . This is a counterexample to the elliptic analogue of a well-known conjecture, namely the Vandiver's conjecture.

Thesis (Doctoral)--Universität Bonn, 1990. / Includes bibliographical references.

Let K be an imaginary quadratic field, with associated quadratic character α. We construct an analytic p-adic L-function interpolating the special values L(1, ad(f) ⊗ α) as f varies in a Hida family; these values are non-critical in the sense of Deligne. Our approach is based on Greenberg--Stevens' idea of Λ-adic modular symbols. By considering cohomology with values in a space of p-adic measures, we construct a Λ-adic evaluation map that interpolates Hida's integral expression as the weight varies. The p-adic L-function is obtained by applying this map to a cohomology class corresponding to the given Hida family.