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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.

Let f be an analytic and one-to-one function on the unit disk such that f(0)=0. Let Q(w)dw^2 be a quadratic differential. Suppose that f maps into the complex plane or the unit disk minus analytic arcs w(t) satisfying Q(w(t))(dw/dt)^2<0. We are interested in the question: if Q is unknown but of a specified form, does f determine the quadratic differential Q uniquely? Our main result is that for functions mapping into the unit disk and quadratic differentials with a pole of order 4 at the origin, the quadratic differential is uniquely determined up to exceptional cases. This question arises in the study of extremal functions for functionals over classes of analytic one-to-one maps.

Let f be an analytic and one-to-one function on the unit disk such that f(0)=0. Let Q(w)dw^2 be a quadratic differential. Suppose that f maps into the complex plane or the unit disk minus analytic arcs w(t) satisfying Q(w(t))(dw/dt)^2<0. We are interested in the question: if Q is unknown but of a specified form, does f determine the quadratic differential Q uniquely? Our main result is that for functions mapping into the unit disk and quadratic differentials with a pole of order 4 at the origin, the quadratic differential is uniquely determined up to exceptional cases. This question arises in the study of extremal functions for functionals over classes of analytic one-to-one maps.

Thesis (Ph. D.)--University of New South Wales, 1999. / Also available online.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2008. / Includes bibliographical references (leaves 35). Also available in electronic version.

Academisch proefschrift--Amsterdam. / "Stellingen" ([3] p.) inserted. Includes bibliographical references.

Thesis (Ph. D.)--University of Missouri, 1934. / Vita. "Photo-lithoprint reproduction of author's manuscript." Includes bibliographical references (p. 30).

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Jeffrey D. Vaaler. Vita. Includes bibliographical references. Also available from UMI.

Thesis (Ph. D.)--University of Chicago, 1936. / Vita. Lithoprinted. "Private editions, distributed by the University of Chicago libraries Chicago, Illinois."