Hanson Smith

Sunrise pano from the summit of the volcano Pico de Orizaba, the highest mountain in Mexico.

Generally, I am interested in algebraic number theory and arithmetic geometry.

Here is a talk I gave at the Charles University Number Theory Seminar about my thesis work. It includes a good deal of background.

A current motivation of my research is understanding the relationship between monogeneity and other arithmetic properties of number fields.
[click here for an Infinite Jest-esque footnote on etymology from my thesis.]

It appears that 'monogeneity' is more commonly used to indicate "the quality of being monogenic" in mathematics. However, in other fields, 'monogeneity' indicates "the quality of being monogeneous," while 'monogenicity' refers to "the quality of being monogenic." Some sources in mathematics refer to number fields as 'monogeneous,' rather than 'monogenic.' (We note that at least one paper refers to 'monogenesis,' which is often used to indicate "the theory that all humankind originated with a single ancestor or ancestral couple." For the sake of brevity, but at the expense of allegory, we will not delve deeper into this possible terminology here.) The author has chosen to use 'monogeneity' simply because it returns more relevant results on MATHSCINET. Though, in an effort to find some solice, the author decided to investigate the etymology. According to Wikitionary, which proved a much more satisfactory resource than many more well-established dictionaries, 'monogeneous' and 'monogenic' are derived from the Ancient Greek words μόνος (mόnos), meaning "alone," "only," "sole," or "single," and γενής (genḗs), meaning "offspring" or "kind." (The interested reader should note that multiple diacritics, e.g. 'ḗ,' on one character is a difficult feat to achieve in LaTeX. The package covington yields a solution that the author finds adequate; however, linguists may want to delve deeper down the StackExchange rabbit hole.) Thus the difference lies in the suffixes '-ic' and '-ous.' The origin of '-ic' is the Latin '-icus,' meaning "belonging to" or "derived from." Conversely, '-ous' is derived from the Latin '-ōsus,' indicating "full," or "full of." The modern usages of '-ic' and '-ous' are more similar, but retain a connotation coming from their Latin roots. As such, 'monogenic' seems the more appropriate term to describe the number fields we will study. Frustratingly, it appears 'monogenicity' should be the more canonical way to turn our preferred adjective into a noun. It may also be worthwhile to note that both 'monogeneous' and 'monogenic' include mathematical definitions in their Wiktionary entries and neither definition is at all related to our current study.

[Click for an enumerated list of my papers.]

Frobenius Finds Non-monogenic Division Fields of Abelian Varieties
The Scheme of Monogenic Generators and its Twists
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 Radical Extensions
Monogenic Fields Arising from Trinomials
Ramification in Division Fields and Sporadic Points on Modular Curves
Two Families of Monogenic S₄ Quartic Number Fields
A Family of Monogenic Quartic Fields Arising from Elliptic Curves
Optimal Packings of Two to Four Equal Circles on any Flat Torus

This paper uses a new representation of the Frobenius endomorphism to investigate the monogeneity of division fields of abelian varieties of dim > 1.

Frobenius Finds Non-monogenic Division Fields of Abelian Varieties
[ show abstract | arXiv: 2109.04262 ] Let A be an abelian variety over a finite field k with |k| = q = p^m. Let π ∈ End_k(A) denote the Frobenius and let v = q/π denote Verschiebung. Suppose the Weil q-polynomial of A is irreducible. When End_k(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 that detects when p is an obstruction to the monogeneity of division fields of certain abelian varieties.

Here are slides from a talk I gave at YRANT about this work.

Here is a paper with my friends Sarah Arpin, Sebastian Bozlee, and Leo Herr recasting monogeneity in a more geometric way.

The Scheme of Monogenic Generators and its Twists
[ show abstract | arXiv: 2108.07185 ] Given an extension of algebras B/A, when is B generated by a single element θ∈B over A? We show there is a scheme M_B/A parameterizing the choice of a generator θ∈B, a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples. A choice of a generator θ is a point of the scheme M_B/A. This inspires a local-to-global study of monogeneity, piecing together monogenerators over points, completions, open sets, and so on. Local generators may not come from global ones, but they often glue to twisted monogenerators that we define. We show a number ring has class number one if and only if each twisted monogenerator is in fact a global generator θ. The moduli spaces of various twisted monogenerators are either a Proj or stack quotient of M_B/A by natural symmetries. The various moduli spaces defined can be used to apply cohomological tools and other geometric methods for finding rational points to the classical problem of monogenic algebra extensions.

Here is a paper investigating the monogeneity and non-monogeneity of division fields of elliptic curves.

Non-monogenic Division Fields of Elliptic Curves
[ show abstract | video abstract | Journal of Number Theory | arXiv: 2007.12781 ] 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 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/ℚ without CM has infinitely many non-monogenic division fields. Our main technique combines a global description of the Frobenius obtained by Duke and Tóth with a simple algorithm based on ideas of Dedekind.

Here is a video introducing this work that I recorded for the Junior Mathematician Research Archive.

Here is a paper I wrote with Mark van Hoeij investigating divisors of modular units and bounding the ℚ-gonality of X₁(N).

A Divisor Formula and a Bound on the ℚ-gonality of the Modular Curve X₁(N)
[ show abstract |Research in Number Theory, volume 7, article 22 | arXiv: 2004.13644 ] We give a formula for divisors of modular units on X₁(N) and use it to prove that the ℚ-gonality of the modular curve X₁(N) is bounded above by [11N²/840], where [•] denotes the nearest integer.

Below is a paper investigating monogeneity in Kummer extensions and radical extensions.

The Monogeneity of Radical Extensions
[ show abstract | Acta Arithmetica, volume 198, pages 313-327 | arXiv: 1909.07184 ] We give necessary and sufficient conditions for the Kummer extension K:=ℚ(ζ_n,α^1/n) to be monogenic over ℚ(ζ_n) with α^1/n as a generator, i.e., for O_K=ℤ[ζ_n][α^1/n]. We generalize these ideas to radical extensions of an arbitrary number field L and provide necessary and sufficient conditions for α^1/n to generate a power O_L-basis for O_L(α^1/n). We also give sufficient conditions for K to be non-monogenic over ℚ and establish a general criterion relating ramification and relative monogeneity. Using this criterion, we find a necessary and sufficient condition for a relative cyclotomic extension of degree φ(n) to have ζ_n as a monogenic generator.

Here are my slides from a presentation about this work.

Below is the paper that came out of the REU Katherine Stange and I ran in the summer of 2018. The REU students (Ryan Ibarra, Henry Lembeck, Mohammad Ozaslan) were excellent and we were able to generalize some of my results on the monogeneity of quartic fields to trinomials of arbitrary degree.

Monogenic Fields Arising from Trinomials
[ show abstract | accepted for publication in Involve | arXiv: 1908.09793 ] We call a polynomial monogenic if a root θ has the property that ℤ[θ] is the full ring of integers in ℚ(θ). Using the Montes algorithm, we find sufficient conditions for xⁿ +ax+b and xⁿ+cx^n-1+d to be monogenic (this was first studied by Jakhar, Khanduja, and Sangwan using other methods). Weaker conditions are given for n=5 and n=6. We also show that each of the families xⁿ+bx+b and xⁿ+ cx^n-1+cd are monogenic infinitely often and give some positive densities in terms of the coefficients.

This is a paper that uses ramification in division fields to preclude certain supersingular elliptic curves from corresponding to sporadic points on modular curves.

Ramification in Division Fields and Sporadic Points on Modular Curves
[ show abstract | arXiv: 1810.04809 ] 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 and that E(K) has a point of exact order pⁿ. To describe the minimum necessary ramification at p, we completely classify the valuations of the pⁿ-torsion points of E by the valuation of a coefficient of the p-th division polynomial. In particular, if E does not have a canonical subgroup at p, we show that p has ramification index at least p²ⁿ-p^2n-2 over p.
We apply this bound to show that sporadic points on the modular curve X₁(pⁿ) cannot correspond to supersingular elliptic curves without a canonical subgroup. Our methods are generalized to X₁(N) with N composite.

Here are my slides from a presentation at JMM 2021 about this work.

This is a paper classifying two infinite families of monogenic S₄ Quartic fields.

Two Families of Monogenic S₄ Quartic Number Fields
[ show abstract | Acta Arithmetica, volume 186, pages 257-271 | arXiv: 1802.09599 ] Consider the integral polynomials f_a,b(x)=x⁴+ax+b and g_c,d(x)=x⁴+cx³+d. Suppose f_a,b(x) and g_c,d(x) are irreducible, b|a, and the integers b, d, 256d-27c⁴, and (256b³-27a⁴)/gcd(256b³,27a⁴) 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 monogenic extension of Q and serves as a generator for a power integral basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generating monogenic S₄ fields within the families f_b,b(x) and g_1,d(x).

Below is a paper I wrote with Alden Gassert and Katherine Stange.

A Family of Monogenic Quartic Fields Arising from Elliptic Curves
[ show abstract | Journal of Number Theory, volume 197, pages 361-382 | arXiv: 1708.03953 ] 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 elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic S₄ fields given by T⁴ − 6T² − αT − 3 for α ϵ Z such that α ± 8 are squarefree, are monogenic.

Here are my slides from a presentation about this work.

Here is a paper I helped with while at an REU at Grand Valley State University. William Dickinson was our project advisor.

Optimal Packings of Two to Four Equal Circles on any Flat Torus
[ show abstract | Discrete Mathematics, volume 342 | arXiv: 1708.05395 ] 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 topological graph theory.