Epiphanies, Metaphors, and Liminality: Religion and Mountains in the Seventeenth Century English Mind

This thesis studies the relationship between religion and mountains as represented in seventeenth century English thought. In particular, it seeks to discover trends of continuity in connections between divinity and mountains. It demonstrates that at least two distinct trends of continuity exist. First, between mountains and divinity as represented by metaphor and allegory, both represented in a variety of mediums, from poetry to letters and books. And secondly, it demonstrates continuity with regards to mountain experiences, which often evoke religion, either as a religious experience, experiences that use religious language, or experiences to religious places. In charting these continuities, it then seeks to argue that mountains can be seen as liminal spaces that mediate between various features of culture, such as civilization and barbarity, humans and nature, and the divine and the human. This feature of liminality has implications for the relationship between humans and mountains more broadly

On Elliptic Curves, Modular Forms, and the Distribution of Primes

In this thesis, we present four problems related to elliptic curves, modular forms, the distribution of primes, or some combination of the three. The first chapter surveys the relevant background material necessary for understanding the remainder of the thesis. The four following chapters present our problems of interest and their solutions. In the final chapter, we present our conclusions as well as a few possible directions for future research. Hurwitz class numbers are known to have connections to many areas of number theory. In particular, they are intimately connected to the theory of binary quadratic forms, the structure of imaginary quadratic number fields, the theory of elliptic curves, and the theory of modular forms. Hurwitz class number identities of a certain type are studied in Chapter 2. To prove these identities, we demonstrate three different techniques. The first method involves a relation between the Hurwitz class number and elliptic curves, while the second and third methods involve connections to modular forms. In Chapter 3, we explore the construction of finite field elements of high multiplicative order arising from modular curves. The field elements are constructed recursively using the equations that Elkies discovered to describe explicit modular towers. Using elementary techniques, we prove lower bounds for the orders of these elements. Prime distribution has been a central theme in number theory for hundreds of years. Mean square error estimates for the Chebotarëv Density Theorem are proved in Chapter 4. These estimates are related to the classical Barban-Davenport-Halberstam Theorem and will prove to be indispensable for our work in Chapter 5, where we take up the study of the Lang-Trotter Conjecture \u27on average\u27 for elliptic curves defined over number fields. We begin Chapter 4 by proving upper bounds on the mean square error in Chebotarëv\u27s theorem. It is this upper bound which features as a key ingredient in Chapter 5. As another application of this upper bound, we continue in Chapter 4 to prove an asymptotic formula for the mean square error. In Chapter 5, we turn to the discussion of the Lang-Trotter Conjecture for number fields \u27on average.\u27 The Lang-Trotter Conjecture is an important conjecture purporting to give information about the arithmetic of elliptic curves, the distribution of primes, and GL(2)-representations of the absolute Galois group. In this chapter, we present some results in support of the conjecture. In particular, we show that the conjecture holds in an average sense when one averages over all elliptic curves defined over a given number field

A generalization of the Barban–Davenport–Halberstam Theorem to number fields

AbstractFor a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarëv Density Theorem when averaging over all q⩽Q and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the Barban–Davenport–Halberstam Theorem

Development of a Time of Flight Spectrometer for Rutherford Backscattering Studies with keV ions

The solid-state silicon surface barrier (SBD) detectors used in conventional Rutherford backscattering spectroscopy necessitate the use of MeV ion beams, as lower energy ions are stopped in the detectors’ dead layer. However, the particle accelerators needed to reach such energies are expensive and often have long startup times. A time of flight spectrometer has been developed at SUNY Geneseo’s Low Energy Ion Facility to perform Rutherford backscattering studies using ions with energies in the 10-50 keV range. It has been demonstrated that this spectrometer can accurately measure the kinetic energies of ions scattered off of targets. The spectrometer can also be used to measure the composition and thickness of thin targets. Initial results show good agreement between the results obtained with low energy ions from Geneseo’s Duoplasmatron ion source and those obtained via conventional RBS with MeV ions from Geneseo’s Pelletron tandem accelerator. Although initial results are encouraging, improvements in resolution and sensitivity must be made before the spectrometer offers the expected advantages

The frequency of elliptic curve groups over prime finite fields

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $\mathbb{F}_p$, we study the frequency to which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$. It is well-known that the only permissible groups are of the form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M(G_{m,k})$ be the frequency to which the group $G_{m,k}$ arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M(G_{m,k})$, pointwise and on average. In particular, we show that $M(G_{m,k})$ is bounded above by a constant multiple of the expected quantity when $m\le k^A$ and that the conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups $G_{m,k}$ when $m\le k^{1/4-\epsilon}$. We also apply our methods to study the frequency to which a given integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.Comment: 40 pages, with an appendix by Chantal David, Greg Martin and Ethan Smith. Final version, to appear in the Canad. J. Math. Major reorganization of the paper, with the addition of a new section, where the main results are summarized and explaine