Research and Interests

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.

For a number field K/Q, the minimal index is the least positive integer m for which there exists a monogenic order with index m in the maximal order. For any n>2 and N>1, we construct infinitely many number fields of degree n with minimal index greater than N.

Following work of Vélez, we explicitly describe the splitting of the integral prime 2 in the radical extension Q(a^1/n), where xⁿ - a is an irreducible polynomial in Z[x]. With previous work of the second author, this fully describes the splitting of any prime in Q(a^1/n). Using this description, we classify common index divisors (the primes whose splitting prevents the existence of a power integral basis for the ring of integers). Using work of Pleasants, we extend this to describe common N-index divisors (primes that divide the index of any order generated over Z by N elements). We also present two novel constructions of non-monogenic fields with no common index divisors as well as constructions of number rings requiring N ring generators for any N. Examples are provided throughout.

The classical Boyd–Lawton theorem concerning Mahler measures has recently been extended to multivariable limits by Brunault, Guilloux, Mehrabdollahei, and Pengo. In another direction, the single-variable Boyd–Lawton theorem has been generalized to various extensions of Mahler measure by Issa and Lalín. The goal of this paper is to present a cohesive framework for extending single-variable Boyd–Lawton theorems to multivariable Boyd–Lawton theorems. With this, we broaden the single-variable Boyd–Lawton theorems of Issa and Lalín to multivariable versions in the direction of Brunault, Guilloux, Mehrabdollahei, and Pengo, providing a generalization of both works.

Let K be a number field with ring of integers O_K, and let f(x)∈O_K[x] be a monic, irreducible polynomial. We establish necessary and sufficient conditions in terms of the critical points of f(x) for the iterates of f(x) to be monogenic polynomials. More generally, we give necessary and sufficient conditions for the backwards orbits of elements of O_K under f(x) to be monogenerators. We apply our criteria to construct novel examples of dynamically monogenic polynomials, yielding infinite towers of monogenic number fields with the backward orbit of one monogenerator giving a monogenerator at the next level.

We explicitly describe the splitting of odd integral primes in the radical extension Q(a^1/n), where xⁿ - a is an irreducible polynomial in Z[x]. Our motivation is to classify common index divisors, the primes whose splitting prevents the existence of a power integral basis for the ring of integers of Q(a^1/n). Among other results, we show that if p is such a prime, even or otherwise, then p divides n.

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 fⁿ(x) for the nth iterate. We obtain necessary and sufficient conditions for fⁿ(x) to be monogenic for each n. We use this to construct multiple families where fⁿ(x) is monogenic for every n>0.

Let a be an integer and p a prime so that f(x) = x^p-a is irreducible. Write fⁿ(x) to indicate the n-fold composition of f(x) with itself. We study the monogenicity of number fields defined by roots of fⁿ(x) and give necessary and sufficient conditions for a root of fⁿ(x) to yield a power integral basis for each n≥1.

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.

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.

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 Aff¹), 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.

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.

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.

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.

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.

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.

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).

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.

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.

Research and publications

Number Fields with Arbitrarily Large Minimal Index

Prime Splitting and Common N-Index Divisors in Radical Extensions: Part p=2

General Boyd–Lawton Theorems with Multivariable Limits

Critical Point Criteria and Dynamically Monogenic Polynomials

Prime Splitting and Common Index Divisors in Radical Extensions

Iterates of Quadratics and Monogenicity

Radical Dynamical Monogenicity

Frobenius Finds Non-monogenic Division Fields of Abelian Varieties

The Scheme of Monogenic Generators I: Representability

The Scheme of Monogenic Generators II: Local Monogenicity and 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