Independence between coefficients of two modular forms.

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 ∈ 2 Z 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 prime p } , where a f (n) (resp. a g (n)) denotes the n th 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 = c g 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. [ABSTRACT FROM AUTHOR]