Paola Francesca Chilla: Introduction to weak vectorial Drinfeld modular forms

Our goal in this talk is to introduce weak vectorial Drinfeld modular forms which will
have a crucial role to determine special values of Goss L-functions. We need to emphasize
that VDMFs given in [Pel12, Def. 12] are indeed seen as weak VDMFs in [PP18, Def. 3.4].
Throughout the seminar, we will borrow this terminology and call them weak VDMFs. Our
main goal for the talk is to analyze the C∞-vector spaces of a certain subclass of weak
VDMFs studied in [Pel12]. The talk will start with basic definitions. Later on we prove
[Pel12, Lem. 13] which indeed implies that one dimensional weak VDMFs corresponding to
the trivial representation 1 are nothing but weak Drinfeld modular forms tensored with T.
This will imply that the space of Drinfeld modular forms tensored with T is equal to the
space of VDMFs corresponding to 1. Thus one needs to focus on the higher dimension case
to produce non-trivial examples. For this aim, we define the functions F and F∗ given in
[Pel12, §2.2, 2.3], which are examples of weak VDMFs of dimension two constructed by using
Anderson generating functions. We also define the deformation of the Eisenstein series. We
will finalize the talk with a sketch of the proof of [Pel12, Prop. 19]. The main references are
[Pel12, §1,2] and [Pel14, §2,3]