Some consequences of Arthur's conjectures for special orthogonal even groups

TL;DR

This paper explicitly constructs a special automorphic representation of the even special orthogonal group, analyzes its properties under Arthur's conjectures, and explores implications for eigenvalues and Arthur packets.

Contribution

It provides an explicit construction of a residual automorphic representation for $SO_{2n}$ and investigates the bounds on Hecke eigenvalues and the structure of Arthur packets.

Findings

Constructed a square integrable residual automorphic representation of $SO_{2n}$.

Established bounds for Hecke eigenvalues based on Arthur's conjectures.

Determined the composition of global and local Arthur's packets for the constructed parameter.

Abstract

In this paper we construct explicitly a square integrable residual automorphic representation of the special orthogonal group $SO_{2n}$, through Eisenstein series. We show that this representation comes from an elliptic Arthur parameter $\psi$ and appears in the space $L^2(SO_{2n}(\mathbb{Q})\backslash SO_{2n}(\mathbb{A}_{\mathbb{Q}}))$ with multiplicity one. Next, we consider parameters whose Hecke matrices, at the unramified places, have eigenvalues bigger (in absolute value), than those of the parameter constructed before. The main result is, that these parameters cannot be cuspidal. We establish bounds for the eigenvalues of Hecke operators, as consequences of Arthur's conjectures for $SO_{2n}$. Next, we calculate the character and the twisted characters for the representations that we constructed. Finally, we find the composition of the global and local Arthur's packets associated to our parameter $\psi$. All the results in this paper are true if we replace $\mathbb{Q}$ by any number field $F$.