Properties of Eisenstein series formed with modular symbols

Abstract

In this work a new kind of non-holomorphic Eisenstein series, first introduced by Goldfeld, is studied. For an arbitrary Fuchsian group of the first kind we fix a holomorphic cusp form and consider Eisenstein series constructed with the modular symbol associated with this cusp form. We develop the theory analogously with that of the usual Eisenstein series starting with its meromorphic continuation to the entire complex plane. A functional equation is then obtained relating the values at s to those at 1− s. Introduction Let H = {z ∈ C : Im z > 0} be the upper half plane and let Γ ⊂ SL2(R) be a fixed non co-compact Fuchsian group of the first kind, (for example Γ(N), Γ0(N)), acting on H. For simplicity assume that Γ has a unique cusp at infinity with stability group Γ∞ = { ± ( 1 m 0 1 ) ,m ∈ Z } . For each γ in Γ we shall label its matrix elements ( γa γb γc γd ) . Let f(z) be an element of S2(Γ), the space holomorphic cusp forms of weight 2 for Γ. Following [Go] we define a modified Eisenstein series (0.1) E∗(z, s) = ∑ γ∈Γ∞\Γ 〈 γ, f 〉Im(γz)s, z ∈ H, where for γ ∈ Γ the modular symbol is given by 〈 γ, f 〉 = −2πi ∫ γw w f(τ) dτ, the definition being independent of w ∈ H. Note that since 〈 γ1γ2, f 〉 = 〈 γ1, f 〉+ 〈 γ2, f 〉 the series is not automorphic. The transformation rule is E∗(γz, s) = E∗(z, s)− 〈 γ, f 〉E(z, s), for all γ ∈ Γ where E(z, s) is the usual Eisenstein series for Γ. This new type of non-holomorphic Eisenstein series was introduced by Goldfeld in order to study the distribution properties of modular symbols 〈 γ, f 〉 as γ ranges over the group Γ. The series (0.1) converges for Re(s) > 2 and Goldfeld hypothesised that it should have an analytic continuation and a functional equation. In this paper Selberg’s method, (described in [Iw], [He]), is extended to establish the following results. Typeset by AMS-TEX 1