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

CONSTRUCTING WEYL GROUP MULTIPLE DIRICHLET SERIES
(2010-01) Chinta, G; Gunnells, PE
Let be a reduced root system of rank . A Weyl group multiple Dirichlet series for is a Dirichlet series in complex variables , initially converging for sufficiently large, that has meromorphic continuation to and satisfies functional equations under the transformations of corresponding to the Weyl group of . A heuristic definition of such a series was given by Brubaker, Bump, Chinta, Friedberg, and Hoffstein, and they have been investigated in certain special cases by others. In this paper we generalize results by Chinta and Gunnells to construct Weyl group multiple Dirichlet series by a uniform method and show in all cases that they have the expected properties.
Cohomology of congruence subgroups of SL4(Z)
(2002-01) 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) 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.
Some elementary Ramanujan graphs
(2005-01) 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.
A smooth space of tetrahedra
(2002-01) Babson, E; Gunnells, PE; Scott, R
We construct a smooth symmetric compactification of the space of all labeled tetrahedra in 3.
Hecke operators and Q-groups associated to self-adjoint homogeneous cones
(2003-01) Gunnells, PE; McConnell, M
Let G be a reductive algebraic group associated to a self-adjoint homogeneous cone defined over , and let ΓG be an appropriate neat arithmetic subgroup. We present two algorithms to compute the action of the Hecke operators on for all i. This simultaneously generalizes the modular symbol algorithm of Ash-Rudolph (Invent. Math. 55 (1979) 241) to a larger class of groups, and proposes techniques to compute the Hecke-module structure of previously inaccessible cohomology groups.
Robert MacPherson and arithmetic groups
(2006-01) 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.
TORSION IN THE COHOMOLOGY OF CONGRUENCE SUBGROUPS OF SL(4, Z) AND GALOIS REPRESENTATIONS
(2010-01) 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.
On the p-parts of quadratic Weyl group multiple Dirichlet series
(2008-01) Chinta, G; Friedberg, S; Gunnells, PE
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 (si) sufficiently large, which has meromorphic continuation to r and satisfies functional equations under the transformations of r corresponding to the Weyl group of Φ. Two constructions of such series are available, one [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series I, in: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, S. Friedberg, D. Bump, D. Goldfeld, and J. Hoffstein, eds., Proc. Symp. Pure Math. 75 (2006), 91–114.] [B. Brubaker, D. Bump, and S. Friedberg, Twisted Weyl group multiple Dirichlet series: the stable case, in: Eisenstein Series and Applications, Gan, Kudla, and Tschinkel, eds., Progr. Math. 258 (2008), 1–26.] [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293–316.] [B. Brubaker, D. Bump, and S. Friedberg, Weyl group multiple Dirichlet series II, The stable case, Invent. Math. 165 (2006), no. 2, 325–355.] based on summing products of n-th order Gauss sums, the second [G. Chinta and P. E. Gunnells, Weyl group multiple Dirichlet series constructed from quadratic characters, Invent. Math. 167 (2007), no. 2, 327–353.] based on averaging a certain group action over the Weyl group. In each case, the essential work occurs at a generic prime p; the local factors, satisfying local functional equations, are then pieced into a global object. In this paper we study these constructions and the relationship between them. First we extend the averaging construction to obtain twisted Weyl group multiple Dirichlet series, whose p-parts are given by evaluating certain rational functions in r variables. Then we develop properties of such a rational function, giving its precise denominator, showing that the nonzero coefficients of its numerator are indexed by points that are contained in a certain convex polytope, determining the coefficients corresponding to the vertices, and showing that in the untwisted case the rational function is uniquely determined from its polar behavior and the local functional equations. We also give evidence that in the case Φ = Ar, the p-part obtained here exactly matches the p-part of the twisted multiple Dirichlet series introduced in [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293–316.] when n = 2.
Weyl group multiple Dirichlet series constructed from quadratic characters
(2007-01) Chinta, G; Gunnells, PE
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].