A conjectural refinement of strong multiplicity one for GL(n)

REFINEMENT OF STRONG MULTIPLICITY ONE FOR AUTOMORPHIC REPRESENTATIONS OF GL(n)

We state a qualitative form of strong multiplicity one for GL1. We derive refinements of strong multiplicity one for automorphic representations arising from Eisenstein series associated to a Borel subgroup on GL(n), and for the cuspidal representations on GL(n) induced from idele class characters of cyclic extensions of prime degree. These results are in accordance with a conjecture of D. Rama...

SPECTRAL MULTIPLICITY FOR Gln(R)

We study the behavior of the cuspidal spectrum of T\^, where %? is associated to Gln(i?) and V is cofinite but not compact. By a technique that modifies the Lax-Phillips technique and uses ideas from wave equation techniques, if r is the dimension of JP, Na{k) is the counting function for the Laplacian attached to a Hilbert space Ha, Ma(X) is the multiplicity function, and Ho is the space of cu...

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 ...