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 -