arXiv subjects

Jackson Hopper

Publications and source records attributed to Jackson Hopper.

3 recordsLinked to original sources

arXiv2022

The twining character formula for reductive groups

Let $\widehat{G}$ be a connected reductive group over an algebraically closed field with a pinning-preserving outer automorphism $\sigma$. Jantzen's twining character formula relates the trace of the action of $\sigma$ on a highest-weight representation $V_{\mu}$ of $\widehat{G}$ to the character of a corresponding highest-weight representation $(V_{\sigma})_{\mu}$ of a related group $\widehat{G^{\sigma, \circ}}$. This paper extends the methods of Hong's geometric proof for the case $\widehat{G}$ is adjoint, to prove that the formula holds for all connected reductive groups, and examines the role of additional hypotheses. In the final section, it is explained how these results can be used to draw conclusions about quasi-split groups over a non-Archimedean local field. This paper thus provides a more general geometric proof of the Jantzen twining character formula and provides some apparently new results of independent interest along the way.

math.RT↗

arXiv2017

On covering systems of integers

A covering system of the integers is a finite collection of modular residue classes $\{a_m \bmod{m}\}_{m \in S}$ whose union is all integers. Given a finite set $S$ of moduli, it is often difficult to tell whether there is a choice of residues modulo elements of $S$ covering the integers. Hough has shown that if the smallest modulus in $S$ is at least $10^{16}$, then there is none. However, the question of whether there is a covering of the integers with all odd moduli remains open. We consider multiplicative restrictions on the set of moduli to generalize Hough's negative solution to the minimum modulus problem. In particular, we find that every covering system of the integers has a modulus divisible by a prime number less than or equal to $19$. Hough and Nielsen have shown that every covering system has a modulus divisible by either $2$ or $3$.

math.NT↗

arXiv2015

Digitally delicate primes

Tao has shown that in any fixed base, a positive proportion of prime numbers cannot have any digit changed and remain prime. In other words, most primes are "digitally delicate". We strengthen this result in a manner suggested by Tao: A positive proportion of primes become composite under any change of a single digit and any insertion a fixed number of arbitrary digits at the beginning or end.

math.NT↗