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Liam May

Publications and source records attributed to Liam May.

4 recordsLinked to original sources
arXiv2026

Components of the Divisibility Graph of Finite Groups of Lie Type in Defining Characteristic $2$

We determine the connected components of the divisibility graph of several families of finite groups of Lie type in defining characteristic $2$, extending the result of Abdolghafourian, Iranmanesh, and Niemeyer (arXiv:1612.04410), who treated odd defining characteristic. We show that there is a distinguished component containing all nonidentity unipotent class sizes and that every other component is either an isolated vertex or an explicitly described two-vertex component. We also determine the isolated vertices outside the distinguished component. Additionally, we prove in Appendix A that the divisibility graph of a Frobenius group has exactly two components.

math.GR↗
arXiv2025

Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields

We present new criteria that obstruct an isogeny class of abelian varieties over a finite field with a given Weil polynomial from containing a Jacobian of a genus-3 hyperelliptic curve. Based on our analysis of the Weil polynomials of three-dimensional abelian varieties over finite fields up to $\mathbb{F}_{25}$ using the data in the L-functions and Modular Forms Database, we conjecture a collection of apparent obstructions. We provide a survey of known and conjectured results related to this problem, and a detailed statistical analysis of these findings. We conjecture that two of these obstructions classify all isogeny classes asymptotically as $q \to \infty$.

math.NT↗
arXiv2025

On Weil Polynomials of Hyperelliptic Curves over Finite Fields of Characteristic 2

We present new conditions which obstruct the existence of hyperelliptic Jacobians in isogeny classes of abelian varieties over finite fields of characteristic 2. We show that Weil polynomials of Jacobians cannot have coefficients in certain residue classes modulo 2, extending the approach of Costa et al. in arXiv:2002.02067. We prove that for 3- and 4-dimensional abelian varieties over $\mathbb{F}_{2^n}$, as $n \rightarrow\infty$, the parities of the Weil coefficients asymptotically equidistribute. Further, we show that these obstructions disqualify $\frac12$ of all 3-dimensional isogeny classes and $\frac 58$ of all 4-dimensional isogeny classes from containing a hyperelliptic Jacobian. Additionally, we present a practical enumeration algorithm which generates all isomorphism classes of hyperelliptic curves of arbitrary genus over almost any finite field of characteristic 2 based on existing algorithms over $\mathbb{F}_2$. Our analysis shows the runtime to be $\tilde{{O}}(2^{n(2g-1)})$ expected, and $\tilde{{O}}(2^{n(2g+2)})$ worst case. This runtime improvement renders the algorithm practical for fields other than $\mathbb{F}_2$.

math.NT↗