Arithmetic distribution of tempered components of cuspidal representations of GL(3)

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Abstract

Let π=⊗v′πv be an arbitrary unitary cuspidal representation of GL (3) over a number field F. We show, for certain ray classes C, that ∑p2∈C|aπ(p)|<1logN(p)N(p)1/2=+∞,which implies that there are infinitely many unramified places v in C such that πv’s are tempered with Hecke eigenvalues lying inside the open unit disk. Furthermore, when F= Q, we consider the problem on the least prime p in an arithmetic progression such that πp satisfies the Ramanujan conjecture. An effective upper bound of Linnik type for such a prime p is proved.

Original languageEnglish (US)
Article number66
JournalMathematische Zeitschrift
Volume303
Issue number3
DOIs
StatePublished - Mar 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics

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