Hanson Smith

activity

20172021

collaborators

Showing math.NTShow all

6 papers · 1 filter

math.NT2021

Frobenius Finds Non-monogenic Division Fields of Abelian Varieties

Hanson Smith

Let $A$ be an abelian variety over a finite field $k$ with $∣ k ∣ = q = p^{m}$ . Let $π \in End_{k} (A)$ denote the Frobenius and let $v = \frac{q}{π}$ denote Verschiebung. Suppose the Weil…

math.NT2020

Non-monogenic Division Fields of Elliptic Curves

Hanson Smith

For various positive integers $n$ , we show the existence of infinite families of elliptic curves over $Q$ with $n$ -division fields, $Q (E [n])$ , that are not monoge…

math.NT2020

A Divisor Formula and a Bound on the $Q$ -gonality of the Modular Curve $X_{1} (N)$

Mark van Hoeij, Hanson Smith

We give a formula for divisors of modular units on $X_{1} (N)$ and use it to prove that the $Q$ -gonality of the modular curve $X_{1} (N)$ is bounded above by $\left[\frac{11N^2}…

math.NT2019

The Monogeneity of Kummer Extensions and Radical Extensions

Hanson Smith

We give necessary and sufficient conditions for the Kummer extension $K := Q (ζ_{n}, n α)$ to be monogenic over $Q (ζ_{n})$ with $n α$ as a generato…

math.NT2018

Two Families of Monogenic $S_{4}$ Quartic Number Fields

Hanson Smith

Consider the integral polynomials $f_{a, b} (x) = x^{4} + a x + b$ and $g_{c, d} (x) = x^{4} + c x^{3} + d$ . Suppose $f_{a, b} (x)$ and $g_{c, d} (x)$ are irreducible, $b ∣ a$ , and the integers $b$ , $d$ , $2…

math.NT2017

A family of monogenic $S_{4}$ quartic fields arising from elliptic curves

T. Alden Gassert, Hanson Smith, Katherine E. Stange

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…

Frobenius Finds Non-monogenic Division Fields of Abelian Varieties

Non-monogenic Division Fields of Elliptic Curves

A Divisor Formula and a Bound on the Q-gonality of the Modular Curve X1​(N)

The Monogeneity of Kummer Extensions and Radical Extensions

Two Families of Monogenic S4​ Quartic Number Fields

A family of monogenic S4​ quartic fields arising from elliptic curves

A Divisor Formula and a Bound on the $Q$ -gonality of the Modular Curve $X_{1} (N)$

Two Families of Monogenic $S_{4}$ Quartic Number Fields

A family of monogenic $S_{4}$ quartic fields arising from elliptic curves