Modular Forms, Elliptic Curves and Drinfeld Modules

In this thesis we explore three different subfields in the area of number theory. The first topic we investigate involves modular forms, specifically nearly holomorphic eigenforms. In Chapter 3, we show the product of two nearly holomorphic eigenforms is an eigenform for only a finite list of examples. The second type of problem we analyze is related to the rank of elliptic curves. Specifically in Chapter 5 we give a graph theoretical approach to calculating the size of 3-Selmer groups for a given family of elliptic curves. By calculating the size of the 3-Selmer groups, we give an upper bound for the rank of an elliptic curve. Finally, in Chapter 7, we conclude with an exposition of work from Goss, Thakur and Diaz-Vargas related to Drinfeld modules. We discuss how to build a zeta function for Drinfeld modules and introduce a symmetric group discovered by Thakur and Diaz-Vargas. An element in the symmetric group is essentially a set permutation of the p-adic integers. It is suspected that there is a relationship between this group and the zeros of certain special zeta functions. We give a specific example of this suspected connection and make a conjecture about this action

Construction of a Dimension Two Rank One Drinfeld Module

Consider Fr[t] where r = pm for some prime p and m in the natural numbers. Let f(t) be an irreducible square-free polynomial with even degree in Fr[t] so that the leading coeffcient is not a square mod Fr. Let A = L = Fr[t][\sqrt{f(t)}]. We will examine the basic set-up required for a dimension two rank one Drinfeld module over L along with an explanation of our choice of f(t). In addition we will show the construction for the exponential function

Products of Nearly Holomorphic Eigenforms

We prove that the product of two nearly holomorphic Hecke eigenforms is again a Hecke eigenform for only finitely many choices of factors.Comment: 8 page

Serious Games with Serious Players: Game Play with International Decision-makers

In 2013 the Skoll Global Threats Fund asked CNA Corporation to design and develop a game exploring information-sharing, conflict, and cooperation on the Indian subcontinent. The goal of the game was twofold: to understand information-sharing, its impediments and effects on water sharing and decision-making, as well as understand how gaming could be a tool for social change. The game was executed in two instances, one in the Washington, DC area with U.S. subject matter experts, and the other in Kuala Lumpur, Malaysia with senior leaders from each of the countries involved. This gives us a unique opportunity to explore how games compare acrosscultures, as well as how well this game allowed senior leaders to address controversial issues. We find that the cross-cultural effects occurred mostly in how particular countries implemented their policies, but that strategic issues and attitudes remained similar across the two instances of the game. From player feedback as well as game observations we conclude that games with senior officials from countries who have a history of tension between them are possible, and mayprovide a more engaging way for them to discuss controversial issues than a traditional meeting format

Bone Dry and Flooding Soon

CNA Corporation, sponsored by the Skoll Global Threats Fund, executed two instances of a political decision-making game designed to explore informationsharing and cooperation over water on the Indian subcontinent. The game explored how Bangladesh, China, India, and Pakistan manage water resources between the Brahmaputra, Indus, and Ganges rivers. The first instance of the game took place in January 2014 in the Washington, DC area, and was played primarily by American subject matter experts. The second instance of the game was held in June 2014 in Kuala Lumpur, Malaysia, and was played by retired senior officials with policy and military backgrounds, and water experts from all four South Asian countries. This document summarizes the second (regional) instance of the game, identifies strategic insights from the regional instance, and compares the two instances deriving further insights based on that comparison

The Clinical Impact of the Covid-19 Pandemic First Wave on Patients With Cystic Fibrosis in New York

BACKGROUND: People with cystic fibrosis (pwCF) may be at risk of complications from COVID-19 but the impact of COVID-19 on pwCF remains unknown. METHODS: We conducted a multicenter retrospective cohort study to assess the impact of the COVID-19 pandemic first wave on pwCF in the New York metropolitan area (NY) from March 1, 2020 to August 31, 2020. Objectives were to determine (1) the prevalence of COVID-19 by PCR and IgG antibody testing, (2) the clinical characteristics of COVID-19, (3) delay in routine outpatient care, and (4) the effect on anxiety and depression in pwCF. RESULTS: There were 26 COVID-19 cases diagnosed by PCR or antibody testing among the study cohort of 810 pwCF. The prevalence of COVID-19 by PCR (1.6%) and IgG antibody (12.2%) testing was low. 58% of cases were asymptomatic and 82% were managed at home. 8% were hospitalized and 1 person died. 89% of pwCF experienced delay in care. The prevalence of anxiety increased from 43% baseline to 58% during the pandemic (P\u3c0.01). In post-hoc analysis, the proportion of patients with diabetes (38% versus 16%, P\u3c0.01) and pancreatic insufficiency (96% versus 66%, P\u3c0.01) were higher while CFTR modulator use was lower (46% versus 65%, P = 0.05) in pwCF who tested positive for COVID-19. CONCLUSIONS: The prevalence of COVID-19 among pwCF in NY during the pandemic first wave was low and most cases were managed at home. CFTR modulators may be protective. PwCF experienced delay in routine care and increased anxiety