Modularity of elliptic curves defined over function fields
Date
2020
DOI
Authors
de Frutos Fernández, María Inés
Version
OA Version
Citation
Abstract
We provide explicit equations for moduli spaces of Drinfeld shtukas over the
projective line with Γ(N), Γ_1(N) and Γ_0(N) level structures, where N is an effective
divisor on P^1 . If the degree of N is big enough, these moduli spaces are relative
surfaces.
We study how the moduli space of shtukas over P^1 with Γ_0(N) level structure,
Sht^{2,tr}(Γ_0(N)), can be used to provide a notion of motivic modularity for elliptic
curves defined over function fields. Elliptic curves over function fields are known to
be modular in the sense of admitting a parametrization from a Drinfeld modular curve,
provided that they have split multiplicative reduction at one place. We conjecture a
different notion of modularity that should cover the curves excluded by the reduction
hypothesis.
We use our explicit equations for Sht^{2,tr}(Γ_0(N)) to verify our modularity conjecture
in the cases where N = 2(0) + (1) + (∞) and N = 3(0) + (∞).