Further refinement of strong multiplicity one for GL(2)

Abstract

Abstract. We obtain a sharp refinement of the strong multiplicity one theorem for the case of unitary non-dihedral cuspidal automorphic representations for GL(2). Given two unitary cuspidal automorphic representations for GL(2) that are not twist-equivalent, we also find sharp lower bounds for the number of places where the Hecke eigenvalues are not equal, for both the general and non-dihedral cases. We then construct examples to demonstrate that these results are sharp.

Distinguishing finite group characters and refined local-global phenomena

“…(2) If n = 3 and c > 5 7 , then ρ ≃ ρ ′ . When n = 2, if ρ and ρ ′ are automorphic, i.e., satisfy the strong Artin conjecture, then the above result already follows by [Wal14]. When n = 2, the strong Artin conjecture for ρ is known in many cases-for instance, if ρ has solvable image by Langlands [Lan80] and Tunnell [Tun81], or if F = Q and ρ is "odd" via Serre's conjecture by Khare-Wintenberger [KW09].…”

Matching densities for Galois representations

“…Examples. We now delineate the construction of two examples of matching densities already known, the first being an example of Serre that provides matching densities of the form 1 − 1/2k 2 and the second being a matching density of 17/32 from [Wa14a].…”

“…We now briefly outline an example with matching density 17/32 and refer the reader to [Wa14a] for further details. This will differ from the one above in that it does not rely on twisting the first representation with a character in order to obtain the second representation.…”

“…There also exist examples that give values of 7/8 [Se77] and 3/4, 3/5, 17/32 [Wa14a]. In any case, one knows that for irreducible 2-dimensional complex Galois representations over any number field, the size of non-trivial matching densities is bounded above by 7/8.…”

“…In this section, we outline the construction of two cases of matching densities that were known earlier so as to provide some context. The first case is actually a family of examples due to Serre; the second is a tetrahedral example arising in [Wa14a], and is one of the few examples that does not use twisting or Galois conjugacy. We also outline a conjecture of Ramakrishnan on the highest non-trivial matching density for cuspidal automorphic representations for GL(n) over any number field.…”