### Abstract

Let k be an even integer and S_{k} be the space of cusp forms of weight k on SL_{2}(Z). Let S=⊕_{k∈2Z}S_{k}. 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∈P^{1}(C)|x=[a_{f}(p):a_{g}(p)]for some primep}, where a_{f}(n)(resp. a_{g}(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 SL_{2}(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 |

### Fingerprint

### Keywords

- Fourier coefficient
- Galois representation
- Modular form

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*202*, 298-315. https://doi.org/10.1016/j.jnt.2019.01.005

**Independence between coefficients of two modular forms.** / Choi, Dohoon; Lim, Subong.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 202, pp. 298-315. https://doi.org/10.1016/j.jnt.2019.01.005

}

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 -