Independence between coefficients of two modular forms

Abstract

Let k be an even integer and Sk be the space of cusp forms of weight k on SL2(Z). Let S = circle plus S-k is an element of 2z(k). For f, g is an element of 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 is an element of P-1 (C) vertical bar x = [a(f)(p) : a(g) (p)] for some prime p}, 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 SL2(Z). We apply it to study the number of representations of a positive integer by a quadratic form. (C) 2019 Elsevier Inc. All rights reserved.