Gunnells, Paul

Job Title

Professor, Department of Mathematics and Statistics

Last Name

Gunnells

First Name

Paul

Discipline

Algebraic Geometry

Expertise

Algebraic Geometry
Number Theory
Topology of Singular Spaces

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Now showing 1 - 10 of 26

TORSION IN THE COHOMOLOGY OF CONGRUENCE SUBGROUPS OF SL(4, Z) AND GALOIS REPRESENTATIONS
(2010-01-01) Ash, A; Gunnells, PE
We report on the computation of torsion in certain homology the-ories of congruence subgroups of SL(4, Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels 31.
Cohomology of congruence subgroups of SL4(Z)
(2002-01-01) Ash, A; Gunnells, PE; McConnell, M
Let N>1 be an integer, and let Γ=Γ0(N)SL4( ) be the subgroup of matrices with bottom row congruent to (0, 0, 0, *) modN. We compute H5(Γ; ) for a range of N and compute the action of some Hecke operators on many of these groups. We relate the classes we find to classes coming from the boundary of the Borel–Serre compactification, to Eisenstein series, and to classical holomorphic modular forms of weights 2 and 4.
COHOMOLOGY OF CONGRUENCE SUBGROUPS OF SL4(Z). III
(2010-01-01) Ash, A; Gunnells, PE; McConnell, M
In two previous papers we computed cohomology groups for a range of levels , where is the congruence subgroup of consisting of all matrices with bottom row congruent to mod . In this note we update this earlier work by carrying it out for prime levels up to . This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to for prime coming from Eisenstein series and Siegel modular forms.
Toric modular forms of higher weight
(2003-01-01) Borisov, LA; Gunnells, PE
A smooth space of tetrahedra
(2002-01-01) Babson, E; Gunnells, PE; Scott, R
We construct a smooth symmetric compactification of the space of all labeled tetrahedra in 3.
Robert MacPherson and arithmetic groups
(2006-01-01) Gunnells, PE
We survey contributions of Robert MacPherson to the theory of arithmetic groups. There are two main areas we discuss: (i) explicit reduction theory for Siegel modular threefolds, and (ii) constructions of compactifications of locally symmetric spaces. The former is joint work with Mark McConnell, the latter with Lizhen Ji.
Some elementary Ramanujan graphs
(2005-01-01) Gunnells, PE
We give elementary constructions of two infinite families of Ramanujan graphs of unbounded degree. The first uses the geometry of buildings over finite fields, and the second uses triangulations of modular curves.
WEYL GROUP MULTIPLE DIRICHLET SERIES OF TYPE A2
(2007-01-01) Chinta, G; Gunnells, PE
A Weyl group multiple Dirichlet series is a Dirichlet series in several complex variables attached to a root system . The number of variables equals the rank r of the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group W of . In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using nth order Gauss sums attached to the root system of type A2. The basic technique is that of [8, 9]; namely, we construct a rational function in r variables invariant under a certain action of W, and use this to build a “local factor” of the global series.
METAPLECTIC ICE
(2010-01-01) Brubaker, B; Bump, D; Chinta, G; Friedberg, S; Gunnells, PE
We study spherical Whittaker functions on a metaplectic cover of GL(r + 1) over a nonarchimedean local field using lattice models from statistical mechanics. An explicit description of this Whittaker function was given in terms of Gelfand-Tsetlin patterns in [5, 17], and we translate this description into an expression of the values of the Whittaker function as partition functions of a six-vertex model. Properties of theWhittaker function may then be expressed in terms of the commutativity of row transfer matrices potentially amenable to proof using the Yang-Baxter equation. We give two examples of this: first, the equivalence of two different Gelfand-Tsetlin definitions, and second, the effect of the Weyl group action on the Langlands parameters. The second example is closely connected with another construction of the metaplectic Whittaker function by averaging over a Weyl group action [9, 10].
CELLS IN COXETER GROUPS I
(2010-01-01) Belolipetsky, M; Gunnells, PE