Comparing Hecke coefficients of automorphic representations

We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of GL(2)\text {GL}(2) over number fields. Using partial bounds on the size of the Hecke coefficients, instances of Langlands functoriality, and properties of Rankin–Selberg LL-functi...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2019-12, Vol.372 (12), p.8871-8896
Main Authors: Chiriac, Liubomir, Jorza, Andrei
Format: Article
Language:English
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Research article
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Summary:We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of GL(2)\text {GL}(2) over number fields. Using partial bounds on the size of the Hecke coefficients, instances of Langlands functoriality, and properties of Rankin–Selberg LL-functions, we obtain bounds on the set of places where linear combinations of Hecke coefficients are negative. Under a mild functoriality assumption we extend these methods to GL(n)\text {GL}(n). As an application, we obtain a result related to a question of Serre about the occurrence of large Hecke eigenvalues of Maass forms. Furthermore, in the cases where the Ramanujan conjecture is satisfied, we obtain distributional results of the Hecke coefficients at places varying in certain congruence or Galois classes.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/7903