Published Articles
(Select articles for more information)
Barake, D., & Franc, C. (2025). An Improved Lower Bound on the Image of the 2-adic Character Map for the Heisenberg Algebra via Modular Linear Differential Equations, International Journal of Number Theory 22 (2026), no. 01, 49–71.
Link: https://doi.org/10.1142/S1793042126500041
Summary: The aim is to provide strong evidence that the character map on the 2-adic rank-one Heisenberg VOA surjects onto the space of 2-adic modular forms. We provide sequences of pre-images of states yielding sequences of modular forms via the character map, which converge to powers of a hauptmodul generating the space of 2-adic modular forms. The 2-adic properties of these images determine how much of the entire space is attained, and we improve an earlier result of Franc and Mason by demonstrating that the space of (1/2)-overconvergent 2-adic modular forms lies in the image of the character map on the 2-adic Heisenberg. A major technical aspect of this computation was to compute an eisenstein basis for the sequence of pre-images appearing in the construction, and this was done by realizing these pre-images as solutions of certain hypergeometric modular linear differential equations (MLDEs). Surprisingly these MLDEs are bounded in degree despite the weight of their solutions growing exponentially, and this fact serves as the second main theorem in this paper.
Barake, D., & Franc, C. (2024). Characters in p-adic Heisenberg and lattice vertex operator algebras. Communications in Algebra, 52(7), 2903–2916.
Link: https://doi.org/10.1080/00927872.2024.2311232
Summary: This paper generalizes some of the results appearing in an earlier work of Franc and Mason as part of an ongoing program of unifying aspects of VOA theory with p-adic modular forms. For a rank-one p-adic Heisenberg VOA and a p-adic lattice VOA associated to an even unimodular lattice, we describe families of states which have characters in the space of Serre and Katz p-adic modular forms, respectively. The construction is algebro-combinatorial in nature and relies on an explicit formulation of the character map on the Heisenberg VOA, due to Mason and Tuite.
Pre-Prints
Daniel Barake, Owen Chuchman, Cameron Franc, Geoffrey Mason, and Brett Nasserden, Vertex operator algebra bundles on modular curves and their associated modular forms, 2026.
Link: https://arxiv.org/abs/2601.10686
Summary: We provide here an explicit construction of the vertex operator algebra bundle over the level-one elliptical modular curve, and describe its sections as “VOA-valued modular forms” which transform according to an action of sl2(Z).
Travel & Talks
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Denton, Texas, USA
Characters in p-adic VOAs via MLDEs
Vancouver, Canada
Pittsburgh, USA
Toronto, Canada
