Noncommutative modular symbols and Eisenstein series

Abstract

We form real-analytic Eisenstein series twisted by Manin's noncommutative modular symbols. After developing their basic properties, these series are shown to have meromorphic continuations to the entire complex plane and satisfy functional equations in some cases. This theory neatly contains and generalizes earlier work in the literature on the properties of Eisenstein series twisted by classical modular symbols.These results depend on properties of Eisenstein series twisted by modular symbols, that is, functions of the formand their generalizations, as in [Gol99]. Here z is in the upper half plane and s is a complex number, initially with real part bigger than 1 to ensure convergence. The function E(z, s; f ) is not automorphic in z, but it does satisfy the more complicated relationand extend this action to the group ring C[Γ] by linearity. The relation (1.3) led Chinta-Diamantis-O'Sullivan [CDO02] to define a second-order modular form on Γ of weight k as a holomorphic function on the upper half plane which satisfies f k (γ 1 − I)(γ 2 − I) = 0, for all γ 1 , γ 2 ∈ Γ. (1.4) An nth-order modular form of weight k on Γ satisfies f k (γ 1 − I)(γ 2 − I)⋯(γ n − I) = 0, for all γ i ∈ Γ. (1.5) Higher-order modular forms were also independently defined by Kleban and Zagier [KZ03]. Their motivation comes from the study of modular properties in crossing probabilities on 2 dimensional lattices. One can similarly define higher-order (real-analytic) Eisenstein series generalizing the second-order series in (1.2) as in [PR04] and [JO08]. Very general series of this form are studied by Diamantis and Sim [DS08]. Manin [Man06] has initiated the development of a theory of noncommutative modular symbols. These arise as iterated integrals of cusp forms and Eisenstein series. The principal motivation for their study comes from multiple zeta values. See, for example, the paper of Choie and Ihara [CI13] which gives explicit formulas for iterated integrals in terms of multiple Hecke L-functions. Generalizing in another direction, Horozov [Hor15] defines noncommutative Hilbert modular symbols.In the present paper we construct Eisenstein series twisted by Manin's noncommutative modular symbols. The result is a generating series whose coefficients contain the higher-order Eisenstein series studied in [PR04, JO08, DS08] as well as further new series not studied before. We find that the automorphic properties and functional equations of the earlier higher-order Eisenstein series are elegantly and succinctly encapsulated in our new formalism.

The distribution of Manin's iterated integrals of modular forms

“…An even more crucial tool in this paper is a higher order Eisenstein series twisted by Manin's iterated integrals, which we denote by E v,w (z, s) (See (8) for the definition). Such a series was first defined by Chinta, Horozov and O'Sullivan in [20] generalizing further an Eisenstein series twisted by modular symbols introduced by Goldfeld [32,31]. Goldfeld's Eisenstein series has been studied intensively in [53,54,55,13,36,56].…”

“…Chinta, Horozov, and O'Sullivan defined in [20] an Eisenstein series twisted by Manins noncommutative modular symbols. This series is a generalization of a certain Eisenstein series twisted by modular symbols introduced by Goldfeld [32,31].…”

“…This series is a generalization of a certain Eisenstein series twisted by modular symbols introduced by Goldfeld [32,31]. In this section we introduce (a slight variant of) the Eisenstein series introduced in [20] and prove several new results about it.…”

“…It follows from [20,Cor 2.6] that E * * a (z, s, k, Ω V , Ω W ) converges absolutely and uniformly for (s) ≥ 1 + ε, for any ε > 0. We emphasize that this means precisely that the coefficients, the higher order twisted Eisensteins series,…”

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

“…Recently in [20] these higher-order Eisenstein series played a key role in the proof of conjectures by Mazur, Rubin and Stein on the statistics of modular symbols. These series have been generalized further in [4].…”