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 |
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Pages (from-to) | 298-315 |
Number of pages | 18 |
Journal | Journal of Number Theory |
Volume | 202 |
DOIs | |
Publication status | Published - 2019 Sep 1 |
Keywords
- Fourier coefficient
- Galois representation
- Modular form
ASJC Scopus subject areas
- Algebra and Number Theory