Vectorial Drinfeld modular forms over Tate algebras
June 6 at 13:30 – 15:00 CEST
Oğuz Gezmiş: Hecke operators
Our goal in the last talk is to study Hecke operators acting on the space of VDMFs and consequences of such an action for Drinfeld modular forms. In particular, applying hyperderivatives and evaluating the coordinates of vectorial modular forms at roots of unity, we will obtain Hecke eigenforms for certain congruence subgroups of GL2(A). We start with defining the slash operator and the Hecke operator Tp for each monic irreducible polynomial p in A. Later on, we give the necessary ideas for the proof of [PP18, Prop. 5.12, Prop. 5.18] and briefly mention why the regularity condition is required for the stability of the space of VDMFs under Hecke operators [PP18, Rem. 5.13]. We continue with giving some examples of Hecke eigenforms in our setting and analyze the behavior of vectorial Eisenstein series under Tp as well as provide a sketch of a proof for [PP18, Cor. 5.23-5.24]. The main reference for the talk is [PP18, §5–5.3].