Jacksyn Bakeberg | Searcharxiv

arXiv2023

Mod $\ell$ gamma factors and a converse theorem for finite general linear groups

The local converse theorem for Rankin-Selberg gamma factors of $\mathrm{GL}_2(\mathbb{F}_q)$ proved by Piatetski-Shapiro over $\mathbb{C}$ no longer holds after reduction modulo $\ell \neq p$. To remedy this, we construct new $\mathrm{GL}_n \times \mathrm{GL}_m$ gamma factors valued in arbitrary $\mathbb{Z}[1/p, \zeta_p]$-algebras for Whittaker-type representations, show that they satisfy a functional equation, and then prove a $\mathrm{GL}_n \times \mathrm{GL}_{n-1}$ converse theorem for irreducible cuspidal representations. In the $\mathrm{GL}_2 \times \mathrm{GL}_1$ case, we define an alternative "new" gamma factor, which takes values in $k$ and satisfies a converse theorem that matches the converse theorem in characteristic $0$.

math.NT↗

arXiv2018

Revisiting the computation of cohomology classes of the Witt algebra using conformal field theory and aspects of conformal algebra

In this article, we revisit some aspects of the computation of the cohomology class of $H^2 ( \text{Witt}, \mathbb{C})$ using some methods in two-dimensional conformal field theory and conformal algebra to obtain the one-dimensional central extension of the Witt algebra to the Virasoro algebra. Even though this is well-known in the context of standard mathematical physics literature, the operator product expansion of the energy-momentum tensor in two-dimensional conformal field theory is presented almost axiomatically. In this paper, we attempt to reformulate it with the help of a suitable modification of conformal algebra (as developed by V. Kac), and apply it to compute the representative element of the cohomology class which gives the desired central extension. This paper was written in the scope of an undergraduate's exploration of conformal field theory and his attempt to gain insight on the subject from a mathematical perspective.

math-ph↗