Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 298-315 |
| Number of pages | 18 |
| Journal | Journal of Number Theory |
| Volume | 202 |
| DOIs | |
| Publication status | Published - 2019 Sep 1 |
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Keywords
- Fourier coefficient
- Galois representation
- Modular form
ASJC Scopus subject areas
- Algebra and Number Theory
Cite this
Independence between coefficients of two modular forms. / Choi, Dohoon; Lim, Subong.
In: Journal of Number Theory, Vol. 202, 01.09.2019, p. 298-315.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Independence between coefficients of two modular forms
AU - Choi, Dohoon
AU - Lim, Subong
PY - 2019/9/1
Y1 - 2019/9/1
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
UR - http://www.scopus.com/inward/citedby.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 -