Modular forms with non-vanishing central values and linear independence of Fourier coefficients

Abstract

In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to modular symbols of higher weights. We prove that for sufficiently large primes p, Hecke operators T-1,T-2,& mldr;,T-D act linearly independently on the winding elements inside the space of weight 2k cuspidal modular symbol S-2k(Gamma(0)(p)) with k >= 1 for D-2 << p. This gives a bound on the number of newforms with non-vanishing arithmetic L-functions at their central critical points and linear independence on the reductions of these modular forms for prime modulo l not equal p.