An upper bound on the denominator of Eisenstein classes in Bianchi manifolds

Abstract

A general conjecture of Harder relates the denominator of the Eisenstein cohomology of certain locally symmetric spaces to special values of $L$-functions. In this paper we consider the locally symmetric space $\operatorname{SL}_2(\mathcal{O}) \backslash \mathbb{H}_3$ where $\mathcal{O}$ is the ring of integers of an imaginary quadratic field $K$ and $\mathbb{H}_3$ is the hyperbolic $3$-space. Tobias Berger proves a lower bound on the denominator of the Eisenstein cohomology in certain cases. The goal of this paper is to show how results of Ito and Sczech can be used to prove an upper bound on the denominator in terms of a special value of a Hecke $L$-function. When the class number of $K$ is one, we combine this result with Berger's result to obtain the exact denominator.

Figures (3)

• Figure 1: The embedding of $\mathcal{H}_r$ by $\iota_{r,v}$ is a horosphere in $\mathbb{H}_3$, tangent to the plane $v=0$ at the cusp $r$. As $v$ increases, the radius of the sphere decreases. Hence we can see the boundary components as horospheres at infinity.
• Figure 2: The equivalence $[a,\gamma a] \sim [b , \gamma b ]$.
• Figure 3: The equivalence $[a,b] \sim [\gamma a , \gamma b ]$.

Theorems & Definitions (51)

• Theorem
• Corollary
• Remark 1.1
• Proposition 2.1
• Remark 2.1
• Proposition 2.2: Sczech
• Proposition 2.3
• proof
• Proposition 2.4
• proof