A strong multiplicity one theorem for SL2

A Theorem on Analytic Strong Multiplicity One

Let K be an algebraic number field, and π = ⊗πv an irreducible, automorphic, cuspidal representation of GLm(AK) with analytic conductor C(π). The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant c depending on ε > 0,m, and K only, such that π can be decided completely by its local components πv with norm N(v) < c · C...

On a Spectral Analogue of the Strong Multiplicity One Theorem

We prove spectral analogues of the classical strong multiplicity one theorem for newforms. Let Γ1 and Γ2 be uniform lattices in a semisimple group. Suppose all but finitely many irreducible unitary representations (resp. spherical) of G occur with equal multiplicities in L(Γ1\G) and L(Γ2\G). Then L(Γ1\G) ∼= L(Γ2\G) as G modules (resp. the spherical spectra of L(Γ1\G) and L(Γ2\G) are equal).

Effective Strong Multiplicity One for Gl M

On Strong Multiplicity One for Automorphic Representations

We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let π be a unitary, cuspidal, automorphic representation of GLn(AK). Let S be a set of finite places of K, such that the sum ∑ v∈S Nv −2/(n+1) is convergent. Then π is uniquely determined by the collection of the local components {πv | v 6∈ S, v finite} of π. Combining this theorem with base change, it is p...

Strong Multiplicity One for the Selberg Class

In [7] A. Selberg axiomatized properties expected of L-functions and introduced the “Selberg class” which is expected to coincide with the class of all arithmetically interesting L-functions. We recall that an element F of the Selberg class S satisfies the following axioms. • In the half-plane σ > 1 the function F (s) is given by a Dirichlet series ∑∞n=1 aF (n)n with aF (1) = 1 and aF (n) ≪ǫ n ...