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 monogenicity and other arithmetic properties of number fields.

[Click here for a ridiculous footnote on 'monogeneity' vs. 'monogenicity' 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.

This is a paper I wrote with Zack Wolske that gives necessary and sufficient conditions for the dynamical monogenicity of a quadratic polynomial.

**Iterates of Quadratics and Monogenicity**

[ show abstract | arXiv: 2406.03629 ] We investigate monogenicity and prime splitting in extensions generated by roots of iterated quadratic polynomials. Let f(x)∈

**Z**[x] be an irreducible, monic, quadratic polynomial, and write f

^{n}(x) for the nth iterate. We obtain necessary and sufficient conditions for f

^{n}(x) to be monogenic for each n. We use this to construct multiple families where f

^{n}(x) is monogenic for every n>0.

This paper extends a result of Ruofan Li on the monogenicity of iterated quadratic radical polynomials to radical polynomials of any prime degree.

**Radical Dynamical Monogenicity**

[ show abstract | arXiv: 2306.11815 ] Let a be an integer and p a prime so that f(x) = x

^{p}-a is irreducible. Write f

^{n}(x) to indicate the n-fold composition of f(x) with itself. We study the monogenicity of number fields defined by roots of f

^{n}(x) and give necessary and sufficient conditions for a root of f

^{n}(x) to yield a power integral basis for each n≥1.

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 | International Journal of Number Theory, vol. 18, no. 10 | 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 two papers that I wrote with my friends Sarah Arpin, Sebastian Bozlee, and Leo Herr recasting monogenicity in a more geometric way.

**The Scheme of Monogenic Generators I: Representability**

[ show abstract | Research in Number Theory, volume 9, article 14 | arXiv: 2108.07185 ] 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 θ∈B over A? In this paper, 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.

**The Scheme of Monogenic Generators II: Local Monogenicity and Twists**

[ show abstract | Research in Number Theory, volume 9, article 43 | arXiv: 2205.04620 ] This is the sequel paper to

*The Scheme of Monogenic Generators I*. It continues 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 θ for an A-algebra B is a point of the scheme M

_{B/A}. In this paper, we study and relate several notions of local monogenicity that emerge from this perspective. We first consider the conditions under which the extension B/A admits monogenerators locally in the Zariski and finer topologies, recovering a theorem of Pleasants as a special case. We next consider the case in which B/A is étale, where the local structure of étale maps allows us to construct a universal monogenicity space and relate it to an unordered configuration space. Finally, we consider when B/A admits local monogenerators that differ only by the action of some group (usually

**G**

_{m}or Aff

^{1}), giving rise to a notion of twisted monogenerators. In particular, we show a number ring A has class number one if and only if each twisted monogenerator is in fact a global monogenerator θ.

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, volume 228, pages 174-187 | 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 X

_{1}(N).

**A Divisor Formula and a Bound on the ℚ-gonality of the Modular Curve X**

_{1}(N)[ show abstract |Research in Number Theory, volume 7, article 22 | arXiv: 2004.13644 ] 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 [11N

^{2}/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(α1/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 | Involve, volume 15, number 2, pages 299-317 | 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 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 conditions are given for n=5 and n=6. We also show that each of the families x

^{n}+bx+b and x

^{n}+ 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 | Research in Number Theory, volume 9, article 17 | arXiv: 1810.04809 ] Consider an elliptic curve E over a number field K. Suppose that E has supersingular reduction at some prime

**of K lying above the rational prime p and that E(K) has a point of exact order p**

*p*^{n}. To describe the minimum necessary ramification at

**, we completely classify the valuations of the p**

*p*^{n}-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

**, we show that**

*p***has ramification index at least p**

*p*^{2n}-p

^{2n-2}over p.

We apply this bound to show that sporadic points on the modular curve X

_{1}(p

^{n}) cannot correspond to supersingular elliptic curves without a canonical subgroup. Our methods are generalized to X

_{1}(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 S

_{4}Quartic fields.

**Two Families of Monogenic S**

_{4}Quartic Number Fields[ show abstract | Acta Arithmetica, volume 186, pages 257-271 | arXiv: 1802.09599 ] 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|a, and the integers b, d, 256d-27c

^{4}, and (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 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 S

_{4}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 S

_{4}fields given by T

^{4}− 6T

^{2}− 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.

Some math related website I like:

Álvaro Lozano-Robledo's Website

Keith Conrad's Expository Papers

Andrew Snowden's Course on Mazur's Theorem

David Mumford's Algebraic Geometry Work

J.S. Milne's Course Notes

Katherine Stange's Website

Terry Tao's Blog

**L**-functions and

**M**odular

**F**orms

**D**ata

**B**ase

Kiran Kedlaya's Conferences Page