On the growth of modular symbols

∈ Γ let ||γ|| = max(|a|, |b|, |c|, |d|). In this paper we show that for given f the modular symbol as logarithmic growth, i.e., | 〈γ, f〉 | ≤ A log ||γ||+B for some A,B ≥ 0. In [2, 3], D. Goldfeld conjectured that the modular symbol has moderate growth if also f is allowed to vary among the normalized newforms. The result of the present paper reduces Goldfeld’s conjecture to a statement on the growth of the modular symbol on a set of generators of the group Γ0(N). Note that, as f is a cusp form, the modular symbol vanishes on parabolic elements, that is, 〈p, f〉 = 0 for every parabolic element p of Γ.