ON FUNDAMENTAL FOURIER COEFFICIENTS of SIEGEL CUSP FORMS of DEGREE 2

Jesse Jääsaari; Stephen Lester; Abhishek Saha

doi:10.1017/S1474748021000542

ON FUNDAMENTAL FOURIER COEFFICIENTS of SIEGEL CUSP FORMS of DEGREE 2

Jesse Jääsaari, Stephen Lester, Abhishek Saha

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

7 Citations (Scopus)

Abstract

Let F be a Siegel cusp form of degree <2>, even weight k ≥ 2>, and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients <a(F,S)> of F at fundamental matrices S (i.e., with <-4 (S)> equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with <(S) X>, the sequence <a(F,S)> has at least <X 1-> sign changes and takes at least <X 1-> 'large values'. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan-Gross-Prasad conjecture, we prove the bound < a(F,S) F, (S) k2 - {1}{2} (S) 18 - > for fundamental matrices S.

Original language	English
Journal	Journal Of The Institute Of Mathematics Of Jussieu
DOIs	https://doi.org/10.1017/S1474748021000542
Publication status	Accepted/In press - 2021

Keywords

Fourier coefficients
L-function
Siegel modular form
sign changes

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10.1017/S1474748021000542

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title = "ON FUNDAMENTAL FOURIER COEFFICIENTS of SIEGEL CUSP FORMS of DEGREE 2",

abstract = "Let F be a Siegel cusp form of degree <2>, even weight k ≥ 2>, and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients of F at fundamental matrices S (i.e., with <-4 (S)> equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with <(S) X>, the sequence has at least sign changes and takes at least 'large values'. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan-Gross-Prasad conjecture, we prove the bound < a(F,S) F, (S) k2 - {1}{2} (S) 18 - > for fundamental matrices S. ",

keywords = "Fourier coefficients, L-function, Siegel modular form, sign changes",

author = "Jesse J{\"a}{\"a}saari and Stephen Lester and Abhishek Saha",

note = "Funding Information: We thank Ralf Schmidt for helpful discussions concerning the material in §. We thank Valentin Blomer and Farrell Brumley for forwarding us their preprint []. We thank the anonymous referee for useful comments and corrections which improved this paper. This work was supported by the Engineering and Physical Sciences Research Council (grant EP/T028343/1). Publisher Copyright: {\textcopyright} The Author(s), 2021. Published by Cambridge University Press.",

year = "2021",

doi = "10.1017/S1474748021000542",

language = "English",

journal = "Journal Of The Institute Of Mathematics Of Jussieu",

issn = "1474-7480",

publisher = "Cambridge University Press",

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T1 - ON FUNDAMENTAL FOURIER COEFFICIENTS of SIEGEL CUSP FORMS of DEGREE 2

AU - Jääsaari, Jesse

AU - Lester, Stephen

AU - Saha, Abhishek

N1 - Funding Information: We thank Ralf Schmidt for helpful discussions concerning the material in §. We thank Valentin Blomer and Farrell Brumley for forwarding us their preprint []. We thank the anonymous referee for useful comments and corrections which improved this paper. This work was supported by the Engineering and Physical Sciences Research Council (grant EP/T028343/1). Publisher Copyright: © The Author(s), 2021. Published by Cambridge University Press.

PY - 2021

Y1 - 2021

N2 - Let F be a Siegel cusp form of degree <2>, even weight k ≥ 2>, and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients of F at fundamental matrices S (i.e., with <-4 (S)> equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with <(S) X>, the sequence has at least sign changes and takes at least 'large values'. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan-Gross-Prasad conjecture, we prove the bound < a(F,S) F, (S) k2 - {1}{2} (S) 18 - > for fundamental matrices S.

AB - Let F be a Siegel cusp form of degree <2>, even weight k ≥ 2>, and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients of F at fundamental matrices S (i.e., with <-4 (S)> equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with <(S) X>, the sequence has at least sign changes and takes at least 'large values'. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan-Gross-Prasad conjecture, we prove the bound < a(F,S) F, (S) k2 - {1}{2} (S) 18 - > for fundamental matrices S.

KW - Fourier coefficients

KW - L-function

KW - Siegel modular form

KW - sign changes

UR - http://www.scopus.com/inward/record.url?scp=85119268374&partnerID=8YFLogxK

U2 - 10.1017/S1474748021000542

DO - 10.1017/S1474748021000542

M3 - Article

AN - SCOPUS:85119268374

SN - 1474-7480

JO - Journal Of The Institute Of Mathematics Of Jussieu

JF - Journal Of The Institute Of Mathematics Of Jussieu

ER -

ON FUNDAMENTAL FOURIER COEFFICIENTS of SIEGEL CUSP FORMS of DEGREE 2

Abstract

Keywords

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