Strong multiplicity one theorem

Statement

Let $F$ be a global field, and let $\pi$ and $\pi'$ be cuspidal automorphic representations of $\operatorname{GL}_n(\mathbb A_F)$ , where $\mathbb A_F$ is the adele ring. If

\pi_v\simeq\pi'_v

for all but finitely many places $v$ of $F$ , then $\pi\simeq\pi'$ . This is the strong multiplicity one theorem.

Thus a cuspidal representation of $\operatorname{GL}_n$ is determined by its almost-everywhere unramified Satake parameters.

Strength and scope

Ordinary multiplicity one says that a fixed cuspidal representation occurs with multiplicity one in the cuspidal spectrum. Strong multiplicity one is a separate uniqueness statement comparing two representations from their local components.

The theorem extends to isobaric automorphic representations after comparing their cuspidal constituents. For general reductive groups, nearly equivalent automorphic representations can be distinct, so the statement is not valid without modification.

Use in reciprocity

If an automorphic construction matches Frobenius or Satake data outside a finite set of places, strong multiplicity one identifies the resulting general-linear-group representation globally. On the Galois side, Chebotarev density plays the analogous uniqueness role.

References

Hervé Jacquet and Joseph A. Shalika, “On Euler products and the classification of automorphic representations I,” American Journal of Mathematics 103 (1981), 499–558. JSTOR.
I. I. Piatetski-Shapiro, “Multiplicity one theorems,” in Automorphic Forms, Representations and L-Functions, Proceedings of Symposia in Pure Mathematics 33, part 1, 1979, 209–212.