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

1 Scopus citations

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)
Journal	Mathematische Zeitschrift
DOIs	https://doi.org/10.1007/s00209-022-03071-y
State	Published - Oct 2022

All Science Journal Classification (ASJC) codes

General Mathematics

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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",

note = "Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2022",

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AU - Chiriac, Liubomir

AU - Yang, Liyang

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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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UR - https://www.scopus.com/pages/publications/85134308383#tab=citedBy

U2 - 10.1007/s00209-022-03071-y

DO - 10.1007/s00209-022-03071-y

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AN - SCOPUS:85134308383

SN - 0025-5874

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

ER -

Summing Hecke eigenvalues over polynomials

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