arXiv subjects

Finn McGlade

Publications and source records attributed to Finn McGlade.

4 recordsLinked to original sources

arXiv2026

The quaternionic Maass Spezialschar on split $\mathrm{SO}(8)$

The classical Maass Spezialschar is a Hecke-stable subspace of the space of holomorphic Siegel modular forms of genus two and level one cut out by certain linear relations among Fourier coefficients. We define an analogous quaternionic Maass Spezialschar, which consists of the quaternionic modular forms of level one on split $\mathrm{SO}(8)$ whose Fourier coefficients satisfy certain linear relations. We characterize this space in terms of a theta lift from the space of holomorphic Siegel modular forms on $\mathrm{Sp}(4)$, and in terms of periods. We also give a conjecture for the Dirichlet series of the standard $L$-function of quaternionic modular eigenforms on $\mathrm{SO}(8)$ and verify our conjecture on the quaternionic Maass Spezialschar.

math.NT↗

arXiv2026

On the Vanishing and Cuspidality of $D_4$ Modular Forms

We develop vanishing and cuspidality criteria for quaternionic modular forms on $G=\mathrm{Spin}(4,4)$ using a theory of scalar Fourier coefficients. By analyzing a Fourier-Jacobi expansion for these forms, we prove that a level one quaternionic modular form on $G$ vanishes if and only if its primitive Fourier coefficients are zero. Using this criterion, we characterize Pollack's quaternionic Saito-Kurokawa subspace by imposing a system of linear relation among certain primitive Fourier coefficients. This characterization strengthens earlier work of the author with Johnson-Leung, Negrini, Pollack, and Roy. We also study quaternionic modular forms in the more general setting of a group $G_J$ associated to a cubic norm structure $J$. Here we establish a new relationship between the degenerate Fourier coefficients of quaternionic modular forms, and the Fourier coefficients of the holomorphic modular forms associated to their constant terms. As a consequence, we prove that in weights $\ell\geq 5$, a level one quaternionic modular form on $G$ is cuspidal if and only if its non-degenerate Fourier coefficients satisfy a polynomial growth condition.

math.NT↗

arXiv2025

Fourier Coefficients and Algebraic Cusp Forms on $\mathrm{U}(2,n)$

We establish a theory of scalar Fourier coefficients for a class of non-holomorphic, automorphic forms on the quaternionic real Lie group $\mathrm{U}(2,n)$. By studying the theta lifts of holomorphic modular forms from $\mathrm{U}(1,1)$, we apply this theory to obtain examples of non-holomorphic cusp forms on $\mathrm{U}(2,n)$ whose Fourier coefficients are algebraic numbers.

math.NT↗

arXiv2019

Positive level, negative level and level zero

This is a survey on the combinatorics and geometry of integrable representations of quantum affine Lie algebras with a particular focus on level 0. Pictures and examples are included to illustrate the affine Weyl group orbits, crystal graphs and Macdonald polynomials that provide detailed understanding of the structure of the extremal weight modules and their characters. The final section surveys the alcove walk method of working with the positive level, negative level and level zero affine flag varieties and describes the corresponding actions of the affine Hecke algebra.

math.RT↗