Automorphic L-Functions in the Weight Aspect

Abstract

Let S_k(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let S_k(Γ)⁺ be the set of all Hecke eigenforms from this space with the first Fourier coefficient a_f(1) = 1. For f ∈ S_k(Γ)⁺, consider the Hecke L-function L(s, f). Let

$$S\left( {k \leqslant K} \right) = \bigcup\limits_{\mathop {12 \leqslant k \leqslant K}\limits_{k\quad \operatorname{even} } } {Sk\left( \Gamma \right)^ + .}$$

It is proved that for large K,

$$\sum\limits_{f \in S\left( {k \leqslant K} \right)} {L(\frac{1}{2},f)^4 \ll K^{2 + \varepsilon } } ,$$

where ε > 0 is arbitrary. For f ∈ S_k(Γ)⁺, let L(s, sym ² f) denote the symmetric square L-function. It is proved that as k → ∞ the frequence

$$\frac{{\# \left\{ {f|f \in S_k (\Gamma )^ + ,L(1,\operatorname{sym} ^2 f) \leqslant x} \right\}}}{{\# \left\{ {f|f \in S_k (\Gamma )^ + } \right\}}}$$

converges to a distribution function G(x) at every point of continuity of the latter, and for the corresponding characteristic function an explicit expression is obtained. Bibliography: 17 titles.