A note on the cancellations of sums involving Hecke eigenvalues

Guodong Hua ()
Additional contact information
Guodong Hua: Weinan Normal University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 2, 619-629

Abstract: Abstract Let f and g be distinct primitive holomorphic cusp forms of even integral weights $$k_{1}$$ k 1 and $$k_{2}$$ k 2 for the full modular group $$\Gamma =SL(2,{\mathbb {Z}})$$ Γ = S L ( 2 , Z ) , respectively. Denote by $$\lambda _{f}(n)$$ λ f ( n ) and $$\lambda _{g}(n)$$ λ g ( n ) the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we consider short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by the multiplicative function $$\lambda _{f}(n^{i})\lambda _{f}(n^{j})$$ λ f ( n i ) λ f ( n j ) and $$\lambda _{f}(n^{i})\lambda _{g}(n^{j})$$ λ f ( n i ) λ g ( n j ) for any integers $$i,j\ge 1$$ i , j ≥ 1 . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least $$q^{\frac{1}{2}+\varepsilon }$$ q 1 2 + ε for arbitrarily small $$\varepsilon >0$$ ε > 0 . In the similar manner, We also establish the nontrivial bounds for short sums of isotypic trace functions twisted by the coefficients $$\lambda _{f\times f\times f}(n)$$ λ f × f × f ( n ) and $$\lambda _{f\times f\times g}(n)$$ λ f × f × g ( n ) of triple product L-functions $$L(f\times f\times f,s)$$ L ( f × f × f , s ) and $$L(f\times f\times g,s)$$ L ( f × f × g , s ) , respectively.

Keywords: Fourier coefficients; Automorphic L-functions; Trace functions; 11N3F; 11F11; 11F30 (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-022-00280-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:54:y:2023:i:2:d:10.1007_s13226-022-00280-3

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-022-00280-3

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().