A strong multiplicity one theorem for SL(2)

TL;DR

This paper establishes a strong multiplicity one theorem for SL(2) under specific local conditions, extending the known multiplicity one results by incorporating additional local data and a local converse theorem.

Contribution

It proves a strong multiplicity one property for SL(2) when representations share local components at certain places and are generic, using a new local converse theorem.

Findings

Strong multiplicity one holds under specified local conditions.

Local converse theorem for SL(2) is utilized in the proof.

Results extend the understanding of multiplicity one phenomena for SL(2).

Abstract

It is known that multiplicity one property holds for SL(2), while the strong multiplicity one property fails. However, in this paper, we show that if we require further that a pair of cuspidal representations $\pi$ and $\pi'$ of SL(2) have the same local components at archimedean places and the places above 2, and they are generic with respect to the same additive character, then they also satisfy the strong multiplicity one property. The proof is based on a local converse theorem for SL(2).