# Euler characteristic and cohomology of $\mathrm{Sp}_4(\mathbb{Z})$ with nontrivial coefficients

## TL;DR

This paper investigates the cohomology and Euler characteristic of the arithmetic group Sp_4(Z) with nontrivial coefficients, providing explicit dimension calculations and insights into cuspidal cohomology structure.

## Contribution

It offers a detailed computation of Euler characteristics and cohomology dimensions for Sp_4(Z) with various coefficients, extending previous theoretical frameworks.

## Key findings

- Explicit formulas for Euler characteristics with coefficients.
- Dimension calculations for cohomology spaces.
- Description of cuspidal cohomology structure.

## Abstract

In this article, the cohomology of the arithmetic group $\mathrm{Sp}_4(\mathbb{Z})$ with coefficients in any finite dimensional highest weight representation $\mathcal{M}_\lambda$ have been studied. Euler characteristic with coefficients in $\mathcal{M}_\lambda$ have been carried out in detail. Combining the results obtained on Euler characteristic and the work of Harder on Eisenstein cohomology, the description of the cuspidal cohomology has been achieved. At the end, we employ our study to compute the dimensions for the cohomology spaces $H^{\bullet}(\mathrm{Sp}_4(\mathbb{Z}), \mathcal{M}_\lambda)$.