UMD Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/3
New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a given thesis/dissertation in DRUM.
More information is available at Theses and Dissertations at University of Maryland Libraries.
Browse
6 results
Search Results
Item CYCLOTOMIC Z2-EXTENSION OF REAL QUADRATIC FIELDS WITH CYCLIC IWASAWA MODULE(2024) Avila Artavia, Josue David; Ramachandran, Niranjan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)For a number field K and a prime p, let K∞ denote the cyclotomic Zp-extension of K, andAn denote the p-primary part of the class group of its n-th layer Kn. Greenberg conjectured that for a totally real field, the order of An becomes constant for sufficiently large n. Motivated by the work of Mouhib and Movahhedi, we focus on the case where p = 2 and K is a real quadratic field such that the Iwasawa module X∞ = lim←An is cyclic. They determined all such fields and proved that Greenberg’s conjecture holds for some cases. In this dissertation, we provide new examples of infinite families of real quadratic fields satisfying Greenberg’s conjecture which were not covered completely in the work of Mouhib and Movahhedi. To achieve this, we use the theory of binary quadratic forms and biquadratic extensions to determine a fundamental system of units and the class number of the first few layers of the cyclotomic Z2-extension. Additionally, in certain cases, we can determine the size of the module X∞ and the level of the cyclotomic tower where the size of An becomes constant.Item DISTRIBUTION OF PRIME ORDER IDEAL CLASSES OF QUADRATIC CLASS GROUPS(2023) Wedige, Melanka Saroad; Ramachandran, Niranjan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The Cohen-Lenstra heuristics predicts that for any odd prime k, the k-part of quadraticclass groups occur randomly on the space of k-groups with respect to a natural probability measure. However, apart from the first moments of the 3-torsion part of quadratic class groups, consequences of these heuristics still remain highly conjectural. The quadratic ideal classes have geometric representations on the modular curve as CM-points in the case of negative discriminants and as closed primitive geodesics in the case of positive discriminants. We mainly focus on the asymptotic distribution of these geometrical objects. As motivation, it is seen that in the case of imaginary quadratic fields, knowledge on the (joint) distribution of k-order CM-points leads in-general to the resolution of the Cohen-Lenstra conjectures on moments of the k-part of class groups. As a first step, inspired by the works of Duke, Hough conjectures that the k-order CM-points are equidistributed on the modular curve. Although the case with k = 3 was resolved by Hough himself, k > 3 remains unresolved. In this dissertation, we revisit Hough’s conjectures, with empirical evidence. We were able to reprove the conjecture for k = 3, and even stronger to show that the result holds along certain subfamilies of imaginary quadratic fields defined by local behaviors of their discriminants. In addition, we study the case for k > 3. We introduce a heuristics model, and show that this model agrees with Hough’s conjectures. We also show that the difference between the actual asymptotics and the heuristic model reduces down to the distribution of solutions to certain quadratic congruences. We, then again inspired by Duke’s work, investigate an analog for the real quadratic fields. Backed by empirical evidence, we go on to conjecture the asymptotic behavior of the length of k-order geodesics on the modular curve. In addition, based on a theorem and its proof by Siegel, we prove certain results that may shed light on a probable proof direction of these conjectures.Item Collatz Conjecture: Generalizing the Odd Part(2013) Zavislak, Ryan Michael; Koralov, Leonid; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Our aim is to investigate the Collatz conjecture. Because the chaotic mixing from iterating the piecewise Collatz function takes place in the odd case, we restrict attention to the odd integers in the orbits to identify some regularities. The parity sequence is reinterpreted and then used to show that if a counterexample exists then there are infinitely many counterexamples with any given initial behavior. When replacing the subfunction 3x+1 in the odd case with other affine functions, our results generalize. We show that the prime factorizations of the coefficients can be used to put a lower bound on the number of weak components in the digraph generated. Furthermore, we identify pairs of functions in this class such that the graph generated by one is isomorphic to a subgraph of the graph generated by the other. In the end, the Collatz conjecture is generalized and several new conjectures are raised.Item Weil-etale Cohomology over Local Fields(2012) Karpuk, David Anton; Ramachandran, Niranjan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In a recent article, Lichtenbaum established the arithmetic utility of the Weil group of a finite field, by demonstrating a connection between certain Euler characteristics in Weil-etale cohomology and special values of zeta functions. In particular, the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field have a Weil-etale cohomological interpretation. These results rely on a duality theorem stated in terms of cup-product in Weil-etale cohomology. With Lichtenbaum's paradigm in mind, we establish results for the cohomology of the Weil group of a local field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil group modules, which implies the main theorem of Local Class Field Theory. We define Weil- smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of G_m on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.Item The Cohomological Equation for Horocycle Maps and Quantitative Equidistribution(2011) Tanis, James Holloway; Forni, Giovanni; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)There are infinitely many distributional obstructions to the existence of smooth solutions for the cohomological equation u o φ1 - u = f in each irreducible component of L2(Γ\PSL(2,R)), where φ1 is the time-one map of the horocycle flow. We study the regularity of these obstructions, determine which ones also obstruct the existence of L2 solutions and prove a Sobolev estimate of the solution in terms of f. As an application, we estimate the rate of equidistribution of horocycle maps on compact, finite volume manifolds Γ\PSL(2,R)) using an auxiliary result from Flaminio-Forni (2003) and one from Venkatesh (2010) concerning the horocycle flow and the twisted horocycle flow, respectively.Item Sigma-Delta Quantization: Number Theoretic Aspects of Refining Quantization Error(2006-07-18) Tangboondouangjit, Aram; Benedetto, John J.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The linear reconstruction phase of analog-to-digital (A/D) conversion in signal processing is analyzed in quantizing finite frame expansions for R^d. The specific setting is a K-level first order Sigma-Delta quantization with step size delta. Based on basic analysis, the d-dimensional Euclidean 2-norm of quantization error of Sigma-Delta quantization with input of elements in R^d decays like O(1/N) as the frame size N approaches infinity; while the L-infinity norm of quantization error of Sigma-Delta quantization with input of bandlimited functions decays like O(T) as the sampling ratio T approaches zero. It has been, however, observed via numerical simulation that, with input of bandlimited functions, the mean square error norm of quantization error seems to decay like O(T^(3/2)) as T approaches zero. Since the frame size N can be taken to correspond to the reciprocal of the sampling ratio T, this belief suggests that the corresponding behavior of quantization error, namely O(1/N^(3/2)), holds in the setting of finite frame expansions in R^d as well. A number theoretic technique involving uniform distribution of sequences of real numbers and approximation of exponential sums is introduced to derive a better quantization error than O(1/N) as N tends to infinity. This estimate is signal dependent.