The Geometry of Drinfeld Modular Forms

Jesse Franklin

Home/Archive/arXiv

arXiv · 2310.19623

The Geometry of Drinfeld Modular Forms

Jesse Franklin

View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a geometric perspective on the algebra of Drinfeld modular forms for congruence subgroups $Γ\leq \GL_2(\bbF_q[T]).$ In particular, we describe an isomorphism between the section ring of a line bundle on the stacky modular curve for $Γ_2$ and the algebra of Drinfeld modular forms for $Γ_2,$ where $Γ_2$ is the subgroup of square-determinant matrices in $Γ.$ This allows one to compute the latter ring by geometric invariants using the techniques of Voight, Zureick-Brown and O'Dorney. We also show how to decompose the algebra of modular forms for $Γ_2$ into a direct sum of two algebras of modular forms for $Γ$ and generalize this result to a larger class of congruence subgroups.

Explore related subjects

math.NT math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jesse Franklin. 2024-10-11. The Geometry of Drinfeld Modular Forms. https://arxiv.org/abs/2310.19623

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The degrees of irreducible factors of binomials and multiplicativity of the n-Hartley condition

We prove that the degrees of irreducible factors of a binomial $t^n-α$ are divisible by the least such degree. As a consequence, we deduce that for relatively prime natural numbers $m,n$, a polynomial has the $mn$-Hartley condition if and only if it has the $m$-Hartley and $n$-Hartley conditions.

math.NT↗

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗

arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗