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

TL;DR

This paper investigates modular forms with non-zero central L-values and establishes linear independence of their Fourier coefficients, extending previous theorems to higher weights and providing bounds on related newforms.

Contribution

It generalizes VanderKam's theorem to higher weights and proves linear independence of Hecke operators on modular symbols for large primes.

Findings

Hecke operators act linearly independently on modular symbols for large primes.

Provides bounds on the number of newforms with non-vanishing central L-values.

Establishes linear independence of reductions of modular forms modulo primes.

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 $T_1, T_2, \ldots, T_D$ act linearly independently on the winding elements inside the space of weight $2k$ cuspidal modular symbol $\mathbb{S}_{2k}(\Gamma_0(p))$ with $k\geq 1$ for $D^2\ll p$. 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 $l\not=p$.