Hanson Smith

• Doctor of Philosophy
• Professor (Assistant) at California State University, San Marcos

This is the second paper in a series of two studying monogenicity of number rings from a moduli-theoretic perspective. By the results of the first paper in this series, a choice of a generator \(\theta \) for an A-algebra B is a point of the scheme \({\mathcal {M}}_{B/A}\). In this paper, we study and relate several notions of local monogenicity th...

Consider an elliptic curve E over a number field K. Suppose that E has supersingular reduction at some prime p of K lying above the rational prime p. We completely classify the valuations of the pn-torsion points of E by the valuation of a coefficient of the pth division polynomial. This classification corrects an error in earlier work of Lozano-Ro...

This is the first in a series of two papers that study monogenicity of number rings from a moduli-theoretic perspective. Given an extension of algebras B/A, when is B generated by a single element \(\theta \in B\) over A? In this paper, we show there is a scheme \({\mathcal {M}}_{B/A}\) parameterizing the choice of a generator \(\theta \in B\), a “...

Let A be an abelian variety over a finite field k with |k| = q = pm. Let π ∈Endk(A) denote the Frobenius and let v = qπ−1 denote Verschiebung. Suppose the Weil q-polynomial of A is irreducible. When Endk(A) = ℤ[π,v], we construct a matrix which describes the action of π on the prime-to-p-torsion points of A. We employ this matrix in an algorithm th...

This is the sequel paper to arXiv:2108.07185, continuing a study of monogenicity of number rings from a moduli-theoretic perspective. By the results of the first paper in this series, a choice of a generator $\theta$ for an $A$-algebra $B$ is a point of the scheme $\mathcal{M}_{B/A}$. In this paper, we study and relate several notions of local mono...

Let $A$ be an abelian variety over a finite field $k$ with $|k|=q=p^m$. Let $\pi\in \text{End}_k(A)$ denote the Frobenius and let $v=\frac{q}{\pi}$ denote Verschiebung. Suppose the Weil $q$-polynomial of $A$ is irreducible. When $\text{End}_k(A)=\mathbb{Z}[\pi,v]$, we construct a matrix which describes the action of $\pi$ on the prime-to-$p$-torsio...

Given an extension of algebras $B/A$, when is $B$ generated by a single element $\theta \in B$ over $A$? We show there is a scheme $\mathcal{M}_{B/A}$ parameterizing the choice of a generator $\theta \in B$, a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ampl...

We give a formula for divisors of modular units on X1(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1(N)$$\end{document} and use it to prove that the Q\documentcl...

For various positive integers n, we show the existence of infinite families of elliptic curves over Q with n-division fields that are not monogenic, i.e., such that the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every E/Q without CM has infinitely many non-monogen...

For various positive integers $n$, we show the existence of infinite families of elliptic curves over $\mathbb{Q}$ with $n$-division fields, $\mathbb{Q}(E[n])$, that are not monogenic, i.e., the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every $E/\mathbb{Q}$ witho...

We give a formula for divisors of modular units on $X_1(N)$ and use it to prove that the $\mathbb{Q}$-gonality of the modular curve $X_1(N)$ is bounded above by $\left[\frac{11N^2}{840}\right]$, where $[\bullet]$ denotes the nearest integer.

We give necessary and sufficient conditions for the Kummer extension $\mathbb{Q}\left(\zeta_n,\sqrt[n]{\alpha}\right)$ to be monogenic over $\mathbb{Q}(\zeta_n)$ with $\sqrt[n]{\alpha}$ as a generator, i.e., for $\mathcal{O}_K=\mathbb{Z}\left[\zeta_n\right]\left[\sqrt[n]{\alpha}\right]$. We generalize these ideas to radical extensions of an arbitra...

We call a polynomial monogenic if a root $\theta$ has the property that $\mathbb{Z}[\theta]$ is the full ring of integers in $\mathbb{Q}(\theta)$. Using the Montes algorithm, we find sufficient conditions for $x^n + ax + b$ and $x^n + cx^{n-1} + d$ to be monogenic (this was first studied by Jakhar, Khanduja, and Sangwan using other methods). Weaker...

Consider an elliptic curve $E$ over a number field $K$ and let $\mathfrak{p}$ be a prime of $\mathcal{O}_K$ lying above a prime $p$ of $\mathbb{Z}$. Suppose $E$ has supersingular reduction at $\mathfrak{p}$. Fix a positive integer $n$ and define $L$ to be a minimal extension of $K$ such that $E(L)$ has a point of exact order $p^n$. If $\mathfrak{p}...

Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\mid a$, and the integers $b$, $d$, $256d-27c^4$, and $\dfrac{256b^3-27a^4}{\gcd(256b^3,27a^4)}$ are all square-free. Using the Montes algorithm, we show that a root of $f_{a,b}(x)$ or $g_{c,d}(x)$ defines a...

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and t...

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and t...

We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial $3$-torsion fields for a certain one-parameter family of non-CM el...

We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial $3$-torsion fields for a certain one-parameter family of non-CM el...