<pre>In the first part of this thesis we show that, for a given non-arithmetic closed hyperbolic <i>$</i><i>n</i><i>$</i> manifold <i>$</i><i>M</i><i>$</i>, there exist for each positive integer <i>$</i><i>j</i><i>$</i>, a set <i>$</i><i>M_</i><i>1</i><i>,...,M_j</i><i>$</i> of pairwise nonisometric, strongly isospectral, finite covers of <i>$</i><i>M</i><i>$</i>, and such that for each <i>$</i><i>i,i'</i><i>$</i> one has isomorphisms of cohomology groups <i>$</i><i>H^*(M_i,</i><i>\Zbb</i><i>)=H^*(M_{i'},</i><i>\Zbb</i><i>)</i><i>$</i> which are compatible with respect to the natural maps induced by the cover. In the second part, we prove that hyperbolic <i>$</i><i>2</i><i>$</i>- and <i>$</i><i>3</i><i>$</i>-manifolds which arise from principal congruence subgroups of a maixmal order in a quaternion algebra having type number <i>$</i><i>1</i><i>$</i> are absolutely spectrally rigid. One consequence of this is a partial answer to an outstanding question of Alan Reid, concerning the spectral rigidity of Hurwitz surfaces.</pre>

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The Gauss circle problem in classical number theory concerns the estimation of N(x) = { (m1;m2) in ZxZ : m1^2 + m2^2 <= x }, the number of integer lattice points inside a circle of radius sqrt(x). Gauss showed that P(x) = N(x)- pi * x satisfi es P(x) = O(sqrt(x)). Later Hardy and Landau independently proved that P(x) = Omega_(x1=4(log x)1=4). It is conjectured that inf{e in R : P(x) = O(x^e )}= 1/4. I. K atai showed that the integral from 0 to X of |P(x)|^2 dx = X^(3/2) + O(X(logX)^2). Similar results to those of the circle have been obtained for regions D in R^2 which contain the origin and whose boundary dD satis fies suff cient smoothness conditions. Denote by P_D(x) the similar error term to P(x) only for the domain D. W. G. Nowak showed that, under appropriate conditions on dD, P_D(x) = Omega_(x1=4(log x)1=4) and that the integral from 0 to X of |P_D(x)|^2 dx = O(X^(3/2)). A result similar to Nowak's mean square estimate is given in the case where only "primitive" lattice points, {(m1;m2) in Z^2 : gcd(m1;m2) = 1 }, are counted in a region D, on assumption of the Riemann Hypothesis.

Duke and Jenkins defined a family of linear maps from spaces of weakly holomorphic modular forms of negative integral weight and level 1 into spaces of weakly holomorphic modular forms of half integral weight and level 4 and showed that these lifts preserve the integrality of Fourier coefficients. We show that the generalization of these lifts to modular forms of genus 0 odd prime level also preserves the integrality of Fourier coefficients.

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<p>The main focus of this dissertation is to find and explain the periodic points of certain algebraic functions that are related to some modular functions, which themselves can be represented by continued fractions. Some of these continued fractions are first explored by Srinivasa Ramanujan in early 20th century. Later on, much work has been done in terms of studying the continued fractions, and proving several relations, identities, and giving different representations for them.</p> <p><br></p> <p>The layout of this report is as follows. Chapter 1 has all the basic background knowledge and ingredients about algebraic number theory, class field theory, Ramanujan’s theta functions, etc. In Chapter 2, we look at the Ramanujan-Göllnitz-Gordon continued fraction that we call v(τ) and evaluate it at certain arguments in the field K = Q(√−d), with −d ≡ 1 (mod 8), in which the ideal (2) = ℘₂℘′₂  is a product of two prime ideals. We prove several identities related to itself and with other modular functions. Some of these are new, while some of them are known but with different proofs. These values of v(τ) are shown to generate the inertia field of ℘₂ or ℘′₂ in an extended ring class field over the field K. The conjugates over Q of these same values, together with 0, −1 ± √2, are shown to form the exact set of periodic points of a fixed algebraic function ˆF(x), independent of d. These are analogues of similar results for the Rogers-Ramanujan continued fraction. See [1] and [2]. This joint work with my advisor Dr. Morton, is submitted for publication to the New York Journal.</p> <p><br> In Chapters 3 and 4, we take a similar approach in studying two more continued fractions c(τ) and u(τ), the first of which is more commonly known as the Ramanujan’s cubic continued fraction. We show what fields a value of this continued fraction generates over Q, and we describe how the periodic points for described functions arise as values of these continued fractions. Then in the last chapter, we summarise all these results, give some possible directions for future research as well as mentioning some conjectures.</p>

<p dir="ltr">Let k be a perfect field of characteristic p > 2. We extend the equivalence of categories between Fontaine-Laffaille modules and Z_p lattices inside crystalline representations with Hodge-Tate weights at most p-2 of Fontaine to the situation where the base ring is the power series ring in d variables over the ring of Witt vectors of k. </p>

Given an elliptic curve E over Q, we can then consider E over the finite field Fp. If Np is the number of points on the curve over Fp, then we define ap(E) = p+1-Np. We say primes p for which ap(E) = 1 are anomalous. In this paper, we search for curves E so that this happens for only a finite number of primes. We call such curves finitely anomalous. This thesis deals with the frequency of their occurrence and finds several examples.

Using the representation theory of positive definite sequences some propositions in additive number theory are obtained and H. Bohr's approximation theorem is deduced. A unified approach to theorems by S. Bochner, S, N, Bernstein and H. Hamburger is discussed and some operator versions of numerical moment problems are studied. The Appendix contains comments to J. von Neumann's spectral theorem for self-adjoint operators in Hilbert space. / Thesis / Doctor of Philosophy (PhD)

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number m generates a complete or incomplete Fourier basis for a Cantor-type measure with scale g.