Summing Hecke eigenvalues over polynomials

Liubomir Chiriac; Liyang Yang

doi:10.1007/s00209-022-03071-y

Summing Hecke eigenvalues over polynomials

Liubomir Chiriac, Liyang Yang

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we estimate sums of the form ∑n≤X|aSymmπ(|f(n)|)|, for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f(x) ∈ Z[x] of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when deg (f) ≤ 4. Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves.

Original language	English (US)
Pages (from-to)	643-662
Number of pages	20
Journal	Mathematische Zeitschrift
Volume	302
Issue number	2
DOIs	https://doi.org/10.1007/s00209-022-03071-y
State	Published - Oct 2022

All Science Journal Classification (ASJC) codes

Mathematics(all)

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10.1007/s00209-022-03071-y

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title = "Summing Hecke eigenvalues over polynomials",

abstract = "In this paper we estimate sums of the form ∑n≤X|aSymmπ(|f(n)|)|, for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f(x) ∈ Z[x] of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when deg (f) ≤ 4. Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves.",

author = "Liubomir Chiriac and Liyang Yang",

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N1 - Funding Information: We thank Andrei Jorza and Peter Sarnak for many conversations and suggestions concerning this project, and also for their encouragement. Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/10

Y1 - 2022/10

N2 - In this paper we estimate sums of the form ∑n≤X|aSymmπ(|f(n)|)|, for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f(x) ∈ Z[x] of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when deg (f) ≤ 4. Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves.

AB - In this paper we estimate sums of the form ∑n≤X|aSymmπ(|f(n)|)|, for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f(x) ∈ Z[x] of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when deg (f) ≤ 4. Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves.

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Summing Hecke eigenvalues over polynomials

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