Understand
In this paper, we study the action of diamond operators on Hilbert modular forms with coefficients in a general commutative ring.
- In particular, we generalize a result of Chai on the surjectivity of the constant term map for Hilbert modular forms with nebentypus to the setting of group ring valued modular forms.
- As an application, we construct certain Hilbert modular forms required for Dasgupta-Kakde's proof of the Brumer-Stark conjecture at odd primes.
- Since the forms required for the Brumer-Stark conjecture live on the non-PEL Shimura variety associated to the reductive group $G = Res_{F/Q}(GL_2)$, as opposed to the PEL Shimura variety associated to the subgroup $G^* \subset G$ studied by Chai, we give a detailed explanation of theory of algebraic diamond operators for $G$, as well as how the theory of toroidal and minimal compactifications for $G$ may be deduced from the analogous theory for $G^*$.