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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Search Results

Now showing 1 - 10 of 26

Open Access
Modular forms and elliptic curves over the field of fifth roots of unity
(2010-01) Gunnells, PE; Hajir, F; Yasaki, Dan
Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F
Open Access
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].
Open Access
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.
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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.
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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.
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Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions
(2003-01) Gunnells, PE; Sczech, R
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.
Open Access
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.
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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.
Open Access
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.
Open Access
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.