Vectorial Modular Forms over Tate Algebras

In a joint work with F. Pellarin, we have continued the development of
his vectorial modular forms over Tate algebras for a certain very
specific representation. We introduce natural expansions for our forms
at the infinite cusp of Drinfeld's moduli space for rank two Drinfeld
modules. Such expansions allow us to prove that our modules of forms of
a given weight and type have finite rank, to introduce Hecke operators
on these modules, and ultimately to prove that a vectorial Hecke
eigenform specializes to infinitely many scalar valued Hecke eigenforms
(of different levels, but often with the same eigenvalues) as introduced
by Goss. Hyperderivatives also play a role in the rank two setting, and
I aim to draw some parallels with the rank one and rank two settings
coming from my recent joint work with A. Maurischat and prior work of
Angles-Pellarin.