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Joachim Schwermer

Publications and source records attributed to Joachim Schwermer.

5 recordsLinked to original sources
arXiv2024

On arithmetically defined hyperbolic $5$-manifolds arising from maximal orders in definite $\mathbb{Q}$-algebras

Using the quaternionic formalism for the description of the group of isometries of hyperbolic $5$-space we consider arithmetically defined $5$-dimensional hyperbolic manifolds which are non-compact but of finite volume. They arise from maximal orders $Λ$ in the central simple algebra $M_2(D)$ of degree $4$ where $D$ denotes a definite quaternion $\mathbb{Q}$-algebra. The affine $\mathbb{Z}$-group scheme $SL_Λ$ determines an integral structure for the algebraic $\mathbb{Q}$-group $G = SL_Λ \times_{\mathbb{Z}} \mathbb{Q}$ obtained by base change. The group $G$ is an inner form of the special linear $\mathbb{Q}$-group $SL_4$. Each torsion-free subgroup $Γ\subset SL_Λ(\mathbb{Z})$ determines a hyperbolic $5$-manifold, to be denoted $X_G/Γ$. Given a principal congruence subgroup $Γ(\frak{p}^e)$, we determine the number of ends and the dimensions of the cohomology groups at infinity of the manifold $X_G/Γ(\frak{p}^e)$.

math.NT↗
arXiv2020

Eisenstein series and the top degree cohomology of arithmetic subgroups of $SL_n/\mathbb{Q}$

The cohomology $H^*(Γ, E) $ of a torsion-free arithmetic subgroup $Γ$ of the special linear $\mathbb{Q}$-group $\mathsf{G} = SL_n$ may be interpreted in terms of the automorphic spectrum of $Γ$. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes $\{\mathsf{P}\}$ of associate proper parabolic $\mathbb{Q}$-subgroups of $\mathsf{G}$. Each summand $H^*_{\mathrm{\{P\}}}(Γ, E)$ is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in $\{\mathsf{P}\}$. The cohomology $H^*(Γ, E) $ vanishes above the degree given by the cohomological dimension $\mathrm{cd}(Γ) = \frac{n(n-1)}{2}$. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes $\{\mathsf{P}\}$ for which the corresponding summand $H^{\mathrm{cd}(Γ)}_{\mathrm{\{\mathsf{P}\}}}(Γ, E)$ vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span $H^{\mathrm{cd}(Γ)}_{\mathrm{\{\mathsf{Q}\}}}(Γ, \mathbb{C})$. Finally, in the case of a principal congruence subgroup $Γ(q)$, $q = p^ν > 5$, $p\geq 3$ a prime, we give lower bounds for the size of these spaces if not even a precise formula for its dimension for certain associate classes $\{\mathsf{Q}\}$.

math.NT↗
arXiv2019

Central morphisms and Cuspidal automorphic Representations

Let $F$ be a global field. Let $G$ and $H$ be two connected reductive group defined over $F$ endowed with an $F$-morphism $f: H\rightarrow G$ such that the induced morphism $H_{der}\rightarrow G_{der}$ on the derived groups is a central isogeny. Our main results yield in particular the following theorem: Given any irreducible cuspidal representation $π$ of $G(\mathbb A_F)$ its restriction to $H(\mathbb A_F)$ contains a cuspidal representation $σ$ of $H(\mathbb A_F)$. Conversely, assuming moreover that $f$ is an injection, any irreducible cuspidal representation $σ$ of $H(\mathbb A_F)$ appears in the restriction of some cuspidal representation $π$ of $G(\mathbb A_F)$. This theorem has an obvious local analogue.

math.NT↗
arXiv2012

On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds

We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce the following theorem: Given an arithmetically defined cocompact subgroup of SL(2,C), provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence of finite index subgroups of such that the first Betti number grows at least as fast as the square root of the index.

math.NT↗