A remark on the failure of multiplicity one for GSp(4)

Abstract

We revisit a classical result of Howe and Pitatski-Shapiro on the failure of strong multiplicity one for GSp(4).

Multiplicity One for Certain Paramodular Forms of Genus Two

“…We show the multiplicity one theorem and the strong multiplicity one theorem for paramodular cuspidal automorphic representations of GSp(4, A Q ) under certain restrictions. It is well-known that strong multiplicity one fails without the paramodularity assumption [HP83], [FT13].…”

A relation between multiplicity one and Böcherer’s conjecture

“…Let d > 0 be an integer such that −d is a fundamental discriminant. 2 Put K = Q( √ −d) and let Cl K denote the ideal class group of K. It is a fact going back to Gauss that the SL 2 (Z)−equivalence classes of binary quadratic forms of discriminant −d are in natural bijective correspondence with the elements of Cl K . In view of the comments above, it follows that for any f ∈ S k (Γ 2 ) and any c ∈ Cl K the notation a(f, c) makes sense.…”

“…Indeed, there are many examples of cuspidal representations on GSp 4 that are nearly equivalent but not equivalent, e.g. those provided by the various Saito-Kurokawa lifts [11], the various Yoshida lifts [16], and the CAP representations of Borel type [6,2].…”

“…The remarkable conjecture below was first made by Böcherer [1]. 2 Recall that an integer n is a fundamental discriminant if either n is a squarefree integer congruent to 1 modulo 4 or n = 4m where m is a squarefree integer congruent to 2 or 3 modulo 4.…”

“…a(f, t ASA) = a(f, S) for all A ∈ SL 2 (Z), thus showing that a(f, S) only depends on the SL 2 (Z)-equivalence class of the matrix S, or equivalently, only on the proper equivalence class of the associated binary quadratic form.Let d > 0 be an integer such that −d is a fundamental discriminant 2. Put K = Q( √ −d) and let Cl K denote the ideal class group of K. It is a fact going back to Gauss that the SL 2 (Z)−equivalence classes of binary quadratic forms of discriminant −d are in natural bijective correspondence with the elements of Cl K .…”