arXiv subjects
Publications and source records attributed to Michael Spieß.
We prove the equality of three conjectural formulas for the Brumer--Stark units. The first formula has essentially been proven, so the present paper also verifies the validity of the other two formulas.
We prove a variant of Emerton's conjecture concerning the right derived functors of the ordinary parts functor $\operatorname{Ord}_P^G$. This functor plays an important role in the theory of mod $p$ representations of $p$-adic reductive groups. A key ingredient for our proof is a comparison between certain small and parabolic inductions. Additionally, our method yields an explicit description of Vignéras' right adjoint to parabolic induction. In the appendix (joint with Heyer) we apply our results to obtain a mod $p$ variant of Bernstein's Second Adjointness, i.e. we show that the right and left adjoint of derived parabolic induction are isomorphic (on complexes with admissible cohomology) up to a cohomological shift and twist by a character.
Nori's Eisenstein cohomology classes and their integral refinements due to Beilinson, Kings and Levin can be used to obtain simple proofs of the rationality and integrality properties of special values of abelian $L$-functions of totally real fields. Here we introduce an adelic refinement of these constructions. This will be used to establish new divisibility properties of Stickelberger elements associated to abelian extensions of totally real fields.
For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is split at $\mathfrak{p}$ and a $p$-adically admissible representation of $\mbox{PGL}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic representation $π$ of $D^*(\mathbb{A})$ this yields a natural construction of infinitesimal deformations of the Galois representation attached to $π$.
We introduce an integral version of the Eisenstein cocycle. As applications we prove a conjecture of Gross regarding the "order of vanishing" of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross--Stark units.