In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series [GKZ]. We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series [G2]. Each of them is a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1, 1, 1, 1). Based on such computations, we state two conjectures about MDZV.

We develop a general method for computing the homological Euler characteristic of finite index subgroups Γ of GL m (O K ) where O K is the ring of integers in a number field K. With this method we find, that for large, explicitly computed dimensions m, the homological Euler characteristic of finite index subgroups of GL m (O K ) vanishes. For other cases, some of them very important for spaces of multiple polylogarithms, we compute non-zero homological Euler characteristic. With the same method we find all the torsion elements in GL 3 Z up to conjugation. Finally, our method allows us to obtain a formula for the Dedekind zeta function at −1 in terms of the ideal class set and the multiplicative group of quadratic extensions of the base ring. IntroductionThe homological Euler characteristic of a group Γ with coefficients in a representation V is defined byThe main theoretical result of our thesis is a general method that allows us to calculate χ h (Γ, V ). Toward the end of the introduction we describe briefly this method. Before that we list the most important results of the thesis which demonstrate the scope of the method. Our first result is about vanishing of homological Euler characteristics.Theorem 0.1 Let Γ be a finite index subgroup of GL m (O K ), where O K is the ring of integers in a number field K. Let V be a finite dimensional representation of Γ. Then We also compute the homological Euler characteristic of the arithmetic subgroups Γ 1 (3, N ) and Γ 1 (4, N ) of GL 3 (Z) and GL 4 (Z), respectively, where Γ 1 (m, N ) is the subgroup of GL m (Z) that fixes the vector [0, . . . , 0, 1] mod N .Theorem 0.3 The homological Euler characteristic of Γ 1 (3, N ) and of Γ 1 (4, N ) for N not divisible by 2 and 3 is given bywhere ϕ(N ) is the Euler ϕ-function, and ϕ 2 (N ) is the multiplicative arithmetic function generated byWith the same technique we find all the torsion elements in GL 3 Z up to conjugation (see proposition 3.2). Using our method we, also, compute χ h (GL m (Z), S n V m ) for m = 3 and m = 4. The computation of χ h (GL 3 (Z), S n V 3 ) (see theorem 6.4) agrees with the computation of the H i (GL 3 (Z), S n V 3 ) in [G1], which was used for computation of dimensions of spaces of certain multiple polylogatithms. Also the Euler characteristic of Γ 1 (3, N ) with trivial coefficients when N is an odd prime, greater that 3, agrees with the computation of H i inf (Γ 1 (3, N ), Q) in the corrected version of [G1]. We compute the homological Euler characteristic of GL 2 (Z[i]) and GL 2 (Z[ξ 3 ]) with coefficients in the symmetric powers of the standard representations. Also we compute the homological Euler characteristic of Γ 1 (2, a) for an ideal 2 a in Z[i] and Z[ξ 3 ], respectively. These computations can be used for finding dimensions of spaces of certain elliptic polylogarithms, as it was explained to me by professor A. Goncharov.Theorem 0.4 (a) If 1 + i does not divide a the homological Euler characteristic ofis the multiplicative function defined on the ideals of Z[i], generated by).whe...

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.

In this paper we compute the cohomology groups of GL 4 (Z) with coefficients in symmetric powers of the standard representation twisted by the determinant. This problem arises in Goncharov's approach to the study of motivic multiple zeta values of depth 4. The techniques that we use include Kostant's formula for cohomology groups of nilpotent Lie subalgebras of a reductive Lie algebra, Borel-Serre compactification, a result of Harder on Eisenstein cohomology. Finally, we need to show that the ghost class, which is present in the cohomology of the boundary of the Borel-Serre compactification, disappears in the Eisenstein cohomology of GL 4 (Z). For this we use a computationally effective version for the homological Euler characteristic of GL 4 (Z) with non-trivial coefficients. * 1 n k 1 1 . . . n km m , where k 1 + . . . + k m is called weight and m is called depth. Goncharov has described the cases of depth=2 [G2] and of depth=3 [G3]. He relates the space of motivic multiple zeta values of depth=2 and weight=n to the cohomology groups of GL 2 (Z) with coefficients in the (n − 2)-symmetric power of the standard representation V 2 , namely, toHe calls this a misterious relation between the multiple zeta values of depth=m and the "modular variety"In the paper [G3], he relates the spaces of motivic multiple zeta values of depth=3 and weight=n to the cohomology of GL 3 (Z) with coefficients in the (n−3)-symmetric power of the standard representation V 3 , namely,Goncharov has also related the case of multiple zeta values of depth=4 and weight=n to the computation of the cohomology of GL 4 (Z) with coefficients in the

In this article, several cohomology spaces associated to the arithmetic groups SL 3 (Z) and GL 3 (Z) with coefficients in any highest weight representation M λ have been computed, where λ denotes their highest weight. Consequently, we obtain detailed information of their Eisenstein cohomology with coefficients in M λ . When M λ is not self dual, the Eisenstein cohomology coincides with the cohomology of the underlying arithmetic group with coefficients in M λ . In particular, for such a large class of representations we can explicitly describe the cohomology of these two arithmetic groups. We accomplish this by studying the cohomology of the boundary of the Borel-Serre compactification and their Euler characteristic with coefficients in M λ . At the end, we employ our study to discuss the existence of ghost classes. Communicated by