Gebhard Böckle: Vectorial Drinfeld modular forms

Our fifth talk is to start investigating VDMFs as well as discussing their several properties
which will be later used to reveal some applications for Drinfeld modular forms. They are
weak VDMFs corresponding to a certain character with a regularity condition introduced in
[PP18, Def. 3.4(2)]. After defining VDMFs, we revisit the deformation of Eisenstein series
and prove [PP18, Prop. 3.7] which gives the Fourier expansion of their each entry (see also
[Pel12, Lem. 21]). An equivalent condition for the regularity [PP18, Cor. 2.6] should also
be analyzed. Later on, we introduce the function F discussed in the previous talk and show
that it is indeed not a VDMF in the sense of [PP18]. The final goal is to prove [PP18, Thm.
3.9] which allows one to decompose a certain space of VDMFs into components generated
by an Eisenstein series E1 and its twist, the image of E1 under the q-th power Frobenius
automorphism of T.