A boundary built into the theory
A mathematical preprint presents a lattice-based framework for higher-rank Drinfeld modular forms, a class of mathematical functions, in which lower-rank lattices become part of the completed space. It also establishes correspondences for cusp forms, which vanish on lower-rank boundary strata, with the Basson–Breuer–Pink and Gekeler formulations. The work is theoretical: it develops mathematical spaces, metrics and group actions rather than reporting an empirical study.
The starting point is an equivalence between Drinfeld modules and lattices of the same rank. That equivalence gives the paper a way to describe higher-rank objects in lattice language while keeping the two mathematical descriptions linked. The paper then studies level structures, components and group actions around those lattice spaces.
To formalize when two such objects are close, the paper equips prelattices with a metric. The distance is based on the supremum, or largest value, of the differences between the exponential functions associated with the prelattices. This metric supplies the notion of convergence used to enlarge the full-rank space.
With that metric, the space of lattices of rank at most r is the metric completion of the rank-r lattice space. A completion adds the limiting points required by the distance; in this construction, those points are represented by lower-rank lattices. The inverse-level-structure space is likewise the metric completion of its full-rank level-structure subspace.
At full rank, the level-structure lattice space is realized through a stated bijection involving Ψr, finite adeles and K(N). The paper also describes the topology of its pieces: each irreducible component is completed by adjoining its boundary. Together, those statements give the framework a full-rank description and a specified way to include what lies at its edge.
Definitions built for a completed space
That completed space is not just a container for limits. Strong modular forms are defined as continuous and homogeneous functions on it that are holomorphic on the main stratum. In ordinary language, they must fit continuously across the completed space, obey the required scaling behavior, and have holomorphic behavior on the main part.
Cusp forms add a sharper boundary rule. They are strong modular forms that vanish on boundary strata where the lattice rank is below r. The lower-rank pieces therefore enter the definition directly: they are where the vanishing condition is checked.
Symmetry is built into the construction. The GLr(A/N) action on the completed level-structure space is an isometry, meaning it preserves metric distances. Fractional ideals act homeomorphically on the completed lattice space by inverse ideal scaling. In ordinary language, the action rescales the lattice in the opposite direction while preserving the space’s topological structure.
One important action remains unresolved. The paper does not establish a sufficiently well-behaved extension of the finite-adele action to the completed level-structure space. That leaves a gap between the finite-adele description used at full rank and the completed level-structure space that includes lower-rank boundary points.
Where the comparison lands
The comparison with Basson–Breuer–Pink is stated as a bijection for cusp forms: the paper’s cusp forms correspond to the corresponding Basson–Breuer–Pink cusp-form spaces. The map to Gekeler modular forms becomes a bijection after restriction to cusp forms. Both results connect the lattice-and-metric construction to those two formulations at the level where the boundary-vanishing condition is imposed.
Alongside those correspondences, the strong modular-form space with level structure is finite-dimensional for every integer weight. This gives the theory a finite-dimensional space of strong forms at each weight, despite the completed lattice spaces and their boundary strata.
The work is an arXiv version 1 preprint dated 28 August 2026. Its stated results are the construction of the lattice and metric framework, completion by lower-rank strata, the group actions that are defined, and the cusp-form comparisons. The finite-adele extension on the completed level-structure space remains the clearest unresolved part of the picture.
Paper data and sources
Original title: Drinfeld modular forms of higher rank from a lattice-oriented point of view
Authors: Liam Baker
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text