Searcharxiv⌕ Search

arXiv subjects

John Cullinan

Publications and source records attributed to John Cullinan.

17 recordsLinked to original sources
arXiv2026

Hasse-Witt invariants for trace forms of Jacobi polynomials

In \cite{feit}, Feit used the Generalized Laguerre Polynomials (GLP) to prove that the groups $\widetilde{A}_{5}$ and $\widetilde{A}_{7}$ occur as Galois groups over $\Q$. Hajir, in \cite{hajir} extended these results to prove that $\widetilde{A}_{n}$ is Galois over $\Q$ whenever $n \equiv 1 \pmod{8}$. A key ingredient of both proofs is the explicit determination of the Hasse-Witt invariant of (the diagonalization of) the trace form of the root fields of the GLP, which relies on the calculation of a certain determinant, $\Delta_t$. The explicit formula for $\Delta_t$ used in \cite{feit} and \cite{hajir} was derived using properties specific to the GLP which do not generalize to other polynomials. In this paper we revisit Feit's original calculation of $\Delta_t$ and situate it in the context of Hankel determinants. We give an alternate derivation of $\Delta_t$ using standard combinatorial arguments and then apply these results to the Jacobi polynomials, a two-parameter family of orthogonal polynomials encompassing the GLP as a special case. We compute an explicit formula for the $\Delta_t$ of the Jacobi polynomials as well as the associated Hasse-Witt invariant. The techniques used in this paper are not specific to the Jacobi polynomials and are widely applicable.

math.NT↗
arXiv2026

Kodaira-Neron statistics for rational elliptic curves with $j$-invariant 0 and 1728

Elliptic curves over $\Q$ with $j$-invariant 0 or 1728 have additive reduction at all primes of bad reduction. In addition, all elliptic curves with $j$-invariant 0 have bad reduction at $p=3$ and all elliptic curves with $j$-invariant 1728 have bad reduction at $p=2$. In this paper we count elliptic curves with $j$-invariant 0 and 1728 by height and determine asymptotics for the various Kodaira-N\'eron types at 3 and 2, respectively. We also give related statistics by holding the torsion subgoup and isogeny-torsion graph constant.

math.NT↗
arXiv2026

The Gibbs phenomenon for the Krawtchouk polynomials

We study the Fourier approximation $\mathcal{F}_N$ of the sign function by the Krawtchouk polynomials. We give numerical evidence that the Gibbs phenomenon of the approximation differs from the classical Gibbs constant; this is in contrast to other families of orthogonal polynomials. We also show that the steepness $\mathcal{F}_N'(0)$ of the approximation is bounded by explicitly proving $\lim_{N \to \infty} \mathcal{F}_N'(0) = \log 4$. This is also in contrast to approximations by classical orthogonal polynomials, where the steepness has been shown to be unbounded as the degree increases.

math-ph↗
arXiv2024

Tamagawa Numbers of Elliptic Curves with an $\ell$-isogeny

Let $\ell$ be an odd prime, and suppose $E$ is an elliptic curve defined over the rational numbers $\mathbb{Q}$. If $E$ has an $\ell$-torsion point, then there has been significant work done on characterizing the $\ell$-divisibility of the global Tamagawa number of $E$. In this paper, we consider elliptic curves that are $\ell$-isogenous to elliptic curves with an $\ell$-torsion point and study the $\ell$-divisibility of their global Tamagawa numbers.

math.NT↗
arXiv2024

Unisingular Specht Modules

Let $G$ be a finite group and $ρ:G \to \GL(V)$ a finite dimensional representation of $G$. We say that $ρ$ is unisingular if $\det(1-ρ(g)) = 0$ for all $g \in G$. Building on previous work in \cite{cullinan}, we consider the symmetric groups $S_n$ and prove that certain families of Specht modules are always unisingular as well as raise new questions for future study.

math.RT↗
arXiv2024

On the modular Plesken Lie algebra

Let G be a finite group. The Plesken Lie algebra L[G] is a subalgebra of the complex group algebra C[G] and admits a direct-sum decomposition into simple Lie algebras based on the ordinary character theory of G. In this paper we review the known results on L[G] and related Lie algebras, as well as introduce a conjecture on a characteristic p analog L_p[G], with a focus on when p divides the order of G.

math.RT↗
arXiv2023

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, I

Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$. Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where $K$ is an $\ell$-power extension of $k$. In a previous work we have shown that, under mild rationality hypotheses, the case of interest is when $\ell=2$ and $K$ is the unique quadratic extension of $k$. In this paper we study the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves $E$, $E'$ over ${\mathbf{Q}}$ and asking for the proportion of primes $p$ for which $E(\mathbf{F}_p) \simeq E'(\mathbf{F}_p)$ and $E(\mathbf{F}_{p^2}) \not \simeq E'(\mathbf{F}_{p^2})$.

math.NT↗
arXiv2022

On the Irreducibility of the Krawtchouck Polynomials

The Krawtchouck polynomials arise naturally in both coding theory and probability theory and have been studied extensively from these points of view. However, very little is known about their irreducibility and Galois properties. Just like many classical families of orthogonal polynomials (e.g. the Legendre and Laguerre), the Krawtchouck polynomials can be viewed as special cases of Jacobi polynomials. In this paper we determine the Newton Polygons of certain Krawtchouck polynomials and show that they are very similar to those of the Legendre polynomials (and exhibit new cases of irreducibility). However, we also show that their Galois groups are significantly more complicated to study, due to the nature of their coefficients, versus those of other classical orthogonal families.

math.NT↗
arXiv2022

Towards a Universal Gibbs Constant

In this paper we build on the work of \cite{kaber} where it was shown that the one-parameter family of Gegenbauer Polynomials (GP) exhibit a Gibbs Phenomenon at a jump discontinuity. We show that the one-parameter family of Generalized Laguerre Polynomials (GLP) also exhibit a Gibbs Phenomenon. Among many differences, a major one is that the GLP are orthogonal on a non-compact subset of $\R$, while the GP are orthogonal on $[-1,1]$. Our strategy follows that of \cite{kaber} and we use entirely elementary methods to arrive at our result. As a special case we show that the Hermite Polynomials also possess a Gibbs Phenomenon. We conclude with a numerical example exhibiting the rate of convergence to the Gibbs constant and a conjectured identity for special values of the GLP.

math-ph↗
arXiv2021

Realizations of Unisingular Representations by Hyperelliptic Jacobians

A representation of a finite group $G$ on a finite dimensional vector space $V$ is called \textbf{unisingular} if every $g\in G$ has 1 as an eigenvalue in its action on $V$. In this paper we show that certain unisingular representations can be realized as mod 2 representations of hyperelliptic Jacobians over $\Q$. We additionally identify new unisingular representations of the symmetric and alternating groups.

math.NT↗
arXiv2021

On a probabilistic local-global principle for torsion on elliptic curves

Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid#E(\mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability that $m$ divides the the order of the torsion subgroup of $E(\mathbb{Q})$: we find it is nonzero for all $m \in \{ 1, 2, \dots, 10, 12, 16\}$ and we compute it exactly when $m \in \{ 1,2,3,4,5,7 \}$. As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve arises from the quotient by a torsion-free group of genus zero.

math.NT↗
arXiv2021

Unisingular representations in arithmetic and Lie theory

Let G be a subgroup of GL(V), where V is a finite dimensional vector space over a finite field of characteristic p >0. If det(g-1) = 0 for all g \in G then we call G a fixed-point subgroup of GL(V). Motivated in parallel by questions in arithmetic and linear group theory, we classify all irreducible fixed-point subgroups of Sp_8(2) and give new infinite series of irreducible fixed-point subgroups of symplectic groups Sp_m(2) for various m arising from certain representations of groups of Lie type.

math.NT↗
arXiv2021

Divisibility of torsion subgroups of abelian surfaces over number fields

Let $A$ be a 2-dimensional abelian variety defined over a number field $K$. Fix a prime number $\ell$ and suppose $\#A(\mathbb{F}_p) \equiv 0 \pmod{\ell^2}$ for a set of primes $\mathfrak{p} \subset \mathcal{O}_K$ of density 1. When $\ell=2$ Serre has shown that there does not necessarily exist a $K$-isogenous $A'$ such that $\#A'(K)_{\mathrm{tors}} \equiv 0 \pmod{4}$. We extend those results to all odd $\ell$ and classify the abelian varieties that fail this divisibility principle for torsion in terms of the image of the mod-$\ell^2$ representation.

math.NT↗
arXiv2020

On the arithmetic of Padé approximants to the exponential function

The $(u,v)$-Padé approximation to a function $f$ is the (unique, up to scaling) rational approximation $f(x) = P(x)/Q(x) + O(x^{u+v+1})$, where $P$ has degree $u$ and $Q$ has degree $v$. Motivated by recent work of Molin, Pazuki, and Rabarison, we study the arithmetic of the Padé approximants of the exponential polynomials. By viewing the approximants as certain Generalized Laguerre Polynomials, we determine the Galois groups of the diagonal approximants and prove some special cases of irreducibility.

math.NT↗