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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Comparing Hecke coefficients of automorphic representations


Authors: Liubomir Chiriac and Andrei Jorza
Journal: Trans. Amer. Math. Soc. 372 (2019), 8871-8896
MSC (2010): Primary 11F30, 11F41
DOI: https://doi.org/10.1090/tran/7903
Published electronically: August 15, 2019
MathSciNet review: 4029715
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Abstract: We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of $ \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 $ L$-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 $ \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.


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Additional Information

Liubomir Chiriac
Affiliation: Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, P.O. Box 751, Portland, Oregon 97207
Address at time of publication: Department of Mathematics and Statistics, University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, Massachusetts 01003
Email: chiriac@math.umass.edu

Andrei Jorza
Affiliation: University of Notre Dame, 275 Hurley Hall, Notre Dame, Indiana 46556
Email: ajorza@nd.edu

DOI: https://doi.org/10.1090/tran/7903
Received by editor(s): December 4, 2018
Received by editor(s) in revised form: May 3, 2019
Published electronically: August 15, 2019
Additional Notes: The first author was partially supported by an AMS-Simons travel grant.
The second author was partially supported by NSA Grant H98230-16-1-0302.
Article copyright: © Copyright 2019 American Mathematical Society