TY - JOUR

T1 - Independence between coefficients of two modular forms

AU - Choi, Dohoon

AU - Lim, Subong

N1 - Funding Information:
This paper was supported by Samsung Research Fund, Sungkyunkwan University, 2018. The authors are grateful to the referee for helpful comments and corrections. The authors also thank Jeremy Rouse for useful comments on the previous version of this paper.
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2019/9

Y1 - 2019/9

N2 - Let k be an even integer and Sk be the space of cusp forms of weight k on SL2(Z). Let S=⊕k∈2ZSk. For f,g∈S, we let R(f,g)be the set of ratios of the Fourier coefficients of f and g defined by R(f,g):={x∈P1(C)|x=[af(p):ag(p)]for some primep}, where af(n)(resp. ag(n))denotes the nth Fourier coefficient of f (resp. g). In this paper, we prove that if f and g are nonzero and R(f,g)is finite, then f=cg for some constant c. This result is extended to the space of weakly holomorphic modular forms on SL2(Z). We apply it to study the number of representations of a positive integer by a quadratic form.

AB - Let k be an even integer and Sk be the space of cusp forms of weight k on SL2(Z). Let S=⊕k∈2ZSk. For f,g∈S, we let R(f,g)be the set of ratios of the Fourier coefficients of f and g defined by R(f,g):={x∈P1(C)|x=[af(p):ag(p)]for some primep}, where af(n)(resp. ag(n))denotes the nth Fourier coefficient of f (resp. g). In this paper, we prove that if f and g are nonzero and R(f,g)is finite, then f=cg for some constant c. This result is extended to the space of weakly holomorphic modular forms on SL2(Z). We apply it to study the number of representations of a positive integer by a quadratic form.

KW - Fourier coefficient

KW - Galois representation

KW - Modular form

UR - http://www.scopus.com/inward/record.url?scp=85061818821&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2019.01.005

DO - 10.1016/j.jnt.2019.01.005

M3 - Article

AN - SCOPUS:85061818821

VL - 202

SP - 298

EP - 315

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -