Comparing Hecke coefficients of automorphic representations
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- by Liubomir Chiriac and Andrei Jorza PDF
- Trans. Amer. Math. Soc. 372 (2019), 8871-8896 Request permission
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.References
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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
- MR Author ID: 875465
- Email: chiriac@math.umass.edu
- Andrei Jorza
- Affiliation: University of Notre Dame, 275 Hurley Hall, Notre Dame, Indiana 46556
- MR Author ID: 876071
- Email: ajorza@nd.edu
- 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. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 8871-8896
- MSC (2010): Primary 11F30, 11F41
- DOI: https://doi.org/10.1090/tran/7903
- MathSciNet review: 4029715