arXiv subjects

Jesse Franklin

Publications and source records attributed to Jesse Franklin.

3 recordsLinked to original sources

arXiv2024

The Geometry of Drinfeld Modular Forms

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.

math.NT↗

arXiv2024

On Drinfeld modular curves for SL(2)

We study the Drinfeld modular curves arising from the Hecke congruence subgroups of $\mathrm{SL}_2(\mathbb{F}_q[T])$. Using a combinatorial method of Gekeler and Nonnengardt, we obtain a genus formula for these curves. In cases when the genus is one, we compute the Weierstrass equation of the corresponding curve.

math.NT↗

arXiv2024

Section Rings of $\mathbb{Q}$-Divisors on Genus $1$ Curves

We compute generators and relations for the section ring of a rational divisor on an elliptic curve. Our technique generalizes the work of O'Dorney (in genus zero) and Voight--Zureick-Brown (for specific divisors arising from the study of stacky curves). For effective divisors supported on at most two points, we give explicit descriptions of the generators and the leading terms of the relations for a minimal presentation. As in the genus zero case, the generators are parametrized by best lower approximations to the coefficients, but there are added wrinkles. Following Landesman, Ruhm and Zhang we can bound the degrees of generators for the section ring of an effective divisor supported at any finite number of points.

math.NT↗