One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

Groupes et algébres de lie

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

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Reflection Groups And Coxeter Groups

Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s1, . . . , sr, initially converging for Re(si) sufficiently large, that has meromorphic continuation to C and satisfies functional equations under the transformations of C corresponding to the Weyl group of Φ. A heuristic definition of such series was given in [BBC06], and they have been investigated in certain special cases in [BBC06,BBF06,BBFH07, BBF08, BBFa, CG07, CFG08, CG, Chi05]. In this paper we generalize results in [CG] to construct Weyl group multiple Dirichlet series by a uniform method, and show in all cases that they have the expected properties.

/pdf/constructing-weyl-group-multiple-dirichlet-series-4np0ntzfgj.pdf

Constructing Weyl group multiple Dirichlet series

https://people.math.umass.edu/~gunnells/pubs/AGM-part-2/AGM-part-2.pdf

Cohomology of congruence subgroups of SL(4,Z) II

We construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet series in greater than two variables having the properties predicted in [2].

/pdf/weyl-group-multiple-dirichlet-series-constructed-from-5eynxg0k4f.pdf

Weyl group multiple Dirichlet series constructed from quadratic characters

We define higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums as well as Zagier's sums, and we show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications. First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the Eisenstein cocycle introduced by R. Sczech. Hence we obtain a polynomial time algorithm for computing these special values. Second, we show how to use our techniques to compute certain special values of the Witten zeta function, and we compute some explicit examples.

https://works.bepress.com/paul_gunnells/20/download/

Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions

The spherical metaplectic Whittaker function on the double cover of Sp (2r, F), where F is a nonarchimedean local field, is considered from several different points of view. Previously, an expression, similar to the Casselman–Shalika formula, had been given by Bump, Friedberg, and Hoffstein as a sum is over the Weyl group. It is shown that this coincides with the expression for the p-parts of Weyl group multiple Dirichlet series of type B r as defined by the averaging method of Chinta and Gunnells. Two conjectural expressions as sums over crystals of type B are given and another as the partition function of a free-fermionic six-vertex model system.

https://people.math.umass.edu/~gunnells/pubs/spincrystal/spincrystal.pdf

Paul E. Gunnells

Papers

Constructing Weyl group multiple Dirichlet series

Cohomology of congruence subgroups of SL(4,Z) II

Weyl group multiple Dirichlet series constructed from quadratic characters

Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions

Metaplectic Whittaker Functions and Crystals of Type B