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This is probably a silly question, as I don't really know a whole lot about modular symbols over arbitrary rings.

How do modular symbols over a finite field square with Katz modular forms? If they are the same thing (or otherwise), I wonder if there is some interest in examining the deformation theory of such modular symbols.

Is there any reason to believe that Serre's conjecture should be false in this setting, i.e a Hecke stable modular symbol over a finite field shall not necessarily lift to a Hecke stable modular symbol in characteristic zero (perhaps one can ask the same question without the Hecke stability condition)?

In general is the functor of deformations (on the category of coefficient rings) representable and what can be said about the deformation ring if this is the case? One fixes the weight, congruence group $\Gamma_0(N)$ and of course the modular symbol $\Phi:\text{Hom}_{\Gamma_0(N)}(\text{Div}^0(\mathbb{P}^1(\mathbb{Q})), \text{Sym}^{k-2}(\mathbb{F}_q))$, the deformation functor on a coefficient ring $R$ consists of all lifts $\tilde{\Phi}$ of $\Phi$ where $\tilde{\Phi}:\text{Hom}_{\Gamma_0(N)}(\text{Div}^0(\mathbb{P}^1(\mathbb{Q})), \text{Sym}^{k-2}(R))$. One may also impose a condition requiring that these symbols be Hecke stable. Given that modular symbols come with such a nice algebraic description, it seems like the deformation ring could be made explicit for some examples.

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