| Summary: | 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) + (∞).
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