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- Cohomology of arithmetic groups (4)
- Algebraic eraser (2)
- Automorphic forms (2)
- Braid group cryptography (2)
- Colored Burau key agreement protocol (2)
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- Cryptography for RFID systems (2)
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- Gindikin-Karpelevich formula (1)
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- Kazhdan-Lusztig cells (1)
- Kazhdan–Lusztig cells (1)
- Littelmann patterns (1)
- Module category (1)
- Space of tetrahedra (1)
- Space of triangles (1)
- Table of marks (1)
- Torsion cohomology classes (1)
- Whittaker functions (1)
Articles 1 - 24 of 24
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Crystal Graphs, Tokuyama's Theorem, And The Gindikin-Karpelevic Formula For G2, Holley Friedlander, Louis Gaudet, Paul E. Gunnells
Crystal Graphs, Tokuyama's Theorem, And The Gindikin-Karpelevic Formula For G2, Holley Friedlander, Louis Gaudet, Paul E. Gunnells
Paul Gunnells
We conjecture a deformation of the Weyl character formula for type G2 in the spirit of Tokuyama’s formula for type A . Using our conjecture, we prove a combinatorial version of the Gindikin–Karpelevič formula for G2 , in the spirit of Bump–Nakasuji’s formula for type A .
Kazhdan-Lusztig Cells In Planar Hyperbolic Coxeter Groups And Automata, Mikhail V. Belolipetsky, Paul E. Gunnells, Richard A. Scott
Kazhdan-Lusztig Cells In Planar Hyperbolic Coxeter Groups And Automata, Mikhail V. Belolipetsky, Paul E. Gunnells, Richard A. Scott
Paul Gunnells
Let C be a one- or two-sided Kazhdan–Lusztig cell in a Coxeter group (W, S), and let Red(C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Red(C) is regular. In this paper, we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.
Mod 2 Homology For Gl(4) And Galois Representations, Avner Ash, Paul E. Gunnells, Mark Mcconnell
Mod 2 Homology For Gl(4) And Galois Representations, Avner Ash, Paul E. Gunnells, Mark Mcconnell
Paul Gunnells
We extend the computations in [AGM11] to find the mod 2 homology in degree 1 of a congruence subgroup Γ of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is related to the cohomology of Γ with F2 coefficients in the top cuspidal degree. These computations require a modification of the algorithm to compute the action of the Hecke operators, whose previous versions required division by 2. We verify experimentally that every mod 2 Hecke eigenclass found appears to have an attached Galois representation, giving evidence for a conjecture in …
A Table Of Elliptic Curves Over The Cubic Field Of Discriminant −23, Steve Donnelly, Paul Gunnells, Ariah Klages-Mundt, Dan Yasaki
A Table Of Elliptic Curves Over The Cubic Field Of Discriminant −23, Steve Donnelly, Paul Gunnells, Ariah Klages-Mundt, Dan Yasaki
Paul Gunnells
Abstract. Let F be the cubic field of discriminant −23 and OF its ring of integers. Let be the arithmetic group GL2(OF ), and for any ideal n ⊂ OF let 0(n) be the congruence subgroup of level n. In [16], two of us (PG and DY) computed the cohomology of various 0(n), along with the action of the Hecke operators. The goal of [16] was to test the modularity of elliptic curves over F. In the present paper, we complement and extend the results of [16] in two ways. First, we tabulate more elliptic curves than were found …
Metaplectic Demazure Operators And Whittaker Functions, Gautam Chinta, Paul Gunnells, Anna Pusk´As
Metaplectic Demazure Operators And Whittaker Functions, Gautam Chinta, Paul Gunnells, Anna Pusk´As
Paul Gunnells
Abstract. In [CG10] the first two named authors defined an action of a Weyl group on rational functions and used it to construct multiple Dirichlet series. These series are related to Whittaker functions on an n-fold metaplectic cover of a reductive group. In this paper, we define metaplectic analogues of the Demazure and Demazure-Lusztig operators. We show how these operators can be used to recover the formulas from [CG10], and how, together with results of McNamara [McN], they can be used to compute Whittaker functions on metaplectic groups over p-adic fields.
Torus Orbits On Homogeneous Varieties And Kac Polynomials Of Quivers, Paul Gunnells, Emmanuel Letellier, Fernando Rodriguez Villegas
Torus Orbits On Homogeneous Varieties And Kac Polynomials Of Quivers, Paul Gunnells, Emmanuel Letellier, Fernando Rodriguez Villegas
Paul Gunnells
In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.
On The Cohomology Of Linear Groups Over Imaginary Quadratic Fields, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schurmann, Mathieu Dutour Sikiric, Dan Yasaki
On The Cohomology Of Linear Groups Over Imaginary Quadratic Fields, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schurmann, Mathieu Dutour Sikiric, Dan Yasaki
Paul Gunnells
Let be the group GLN(OD), where OD is the ring of integers in the imaginary quadratic field with discriminant D < 0. In this paper we investigate the cohomology of for N = 3, 4 and for a selection of discriminants: D −24 when N = 3, and D = −3,−4 when N = 4. In particular we compute the integral cohomology of up to p-power torsion for small primes p. Our main tool is the polyhedral reduction theory for developed by Ash [4, Ch. II] and Koecher [18]. Our results extend work of Staffeldt [29], who treated the case n = 3, D = −4. In a sequel [11] to this paper, we will apply some of these results to the computations with the K-groups K4(OD), when D = −3,−4.
On Hilbert Modular Threefolds Of Discriminant 49, Lev A. Borisov, Paul E. Gunnells
On Hilbert Modular Threefolds Of Discriminant 49, Lev A. Borisov, Paul E. Gunnells
Paul Gunnells
Let K be the totally real cubic field of discriminant 49 , let \fancyscriptO be its ring of integers, and let p⊂\fancyscriptO be the prime over 7 . Let Γ(p)⊂Γ=SL2(\fancyscriptO) be the principal congruence subgroup of level p . This paper investigates the geometry of the Hilbert modular threefold attached to Γ(p) and some related varieties. In particular, we discover an octic in P3 with 84 isolated singular points of type A2 .
Modular Forms And Elliptic Curves Over The Cubic Field Of Discriminant - 23, Paul E. Gunnells, Dan Yasaki
Modular Forms And Elliptic Curves Over The Cubic Field Of Discriminant - 23, Paul E. Gunnells, Dan Yasaki
Paul Gunnells
Let F be the cubic field of discriminant –23 and let O Ϲ F be its ring of integers. By explicitly computing cohomology of congruence subgroups of 〖GL〗_2(O) , we computationally investigate modularity of elliptic curves over F.
Generalised Burnside Rings, G-Categories And Module Categories, Paul E. Gunnells, Andrew Rose, Dmitriy Rumynin
Generalised Burnside Rings, G-Categories And Module Categories, Paul E. Gunnells, Andrew Rose, Dmitriy Rumynin
Paul Gunnells
This note describes an application of the theory of generalised Burnside rings to algebraic representation theory. Tables of marks are given explicitly for the groups S4 and S5 which are of particular interest in the context of reductive algebraic groups. As an application, the base sets for the nilpotent element F4(a3) are computed.
Defeating The Kalka–Teicher–Tsaban Linear Algebra Attack On The Algebraic Eraser, Dorian Goldfeld, Paul E. Gunnells
Defeating The Kalka–Teicher–Tsaban Linear Algebra Attack On The Algebraic Eraser, Dorian Goldfeld, Paul E. Gunnells
Paul Gunnells
The Algebraic Eraser (AE) is a public key protocol for shar- ing information over an insecure channel using commutative and non- commutative groups; a concrete realization is given by Colored Burau Key Agreement Protocol (CBKAP). In this paper, we describe how to choose data in CBKAP to thwart an attack by Kalka–Teicher–Tsaban.
Resolutions Of The Steinberg Module For Gl(N), Avner Ash, Paul E. Gunnells, Mark Mcconnell
Resolutions Of The Steinberg Module For Gl(N), Avner Ash, Paul E. Gunnells, Mark Mcconnell
Paul Gunnells
We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are definitive. In particular, in [AGM4] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and …
On The Cryptanalysis Of The Generalized Simultaneous Conjugacy Search Problem And The Security Of The Algebraic Eraser, Paul E. Gunnells
On The Cryptanalysis Of The Generalized Simultaneous Conjugacy Search Problem And The Security Of The Algebraic Eraser, Paul E. Gunnells
Paul Gunnells
The Algebraic Eraser (AE) is a cryptographic primitive that can be used to obscure information in certain algebraic cryptosystems. The Colored Burau Key Agreement Protocol (CBKAP), which is built on the AE, was introduced by I. Anshel, M. Anshel, D. Goldfeld, and S. Lemieux in 2006 as a protocol suitable for use on platforms with constrained computational resources, such as RFID and wireless sensors. In 2009 A. Myasnikov and A. Ushnakov proposed an attack on CBKAP that attempts to defeat the generalized simultaneous conjugacy search problem, which is the public-key computational problem underlying CBKAP. In this paper we investigate the …
Torsion In The Cohomology Of Congruence Subgroups Of Sl (4. Z) And Galois Representations, Avner Ash, Paul E. Gunnells, Mark Mcconnell
Torsion In The Cohomology Of Congruence Subgroups Of Sl (4. Z) And Galois Representations, Avner Ash, Paul E. Gunnells, Mark Mcconnell
Paul Gunnells
We report on the computation of torsion in certain homology theories 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 K4 Of The Gaussian And Eisenstein Integers, Mathieu Dutour Sikiric, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schürmann, Dan Yasaki
On K4 Of The Gaussian And Eisenstein Integers, Mathieu Dutour Sikiric, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schürmann, Dan Yasaki
Paul Gunnells
Abstract. In this paper we investigate the structure of the algebraic K-groups K4(Z[i]) and K4(Z[ρ]), where i := √ −1 and ρ := (1 + √ −3)/2. We exploit the close connection between homology groups of GLn(R) for n 6 5 and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which GLn(R) acts. Our main results are (i) K4(Z[i]) is a finite abelian 3-group, and (ii) K4(Z[ρ]) is trivial.
Toric Modular Forms And Nonvanishing Of L-Functions, Lev A. Borisov, Paul E. Gunnells
Toric Modular Forms And Nonvanishing Of L-Functions, Lev A. Borisov, Paul E. Gunnells
Paul Gunnells
In a previous paper \cite{BorGunn}, we defined the space of toric forms $\TTT(l)$, and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f,1)≠0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.
Evaluation Of Dedekind Sums, Eisenstein Cocycles, And Special Values Of L-Functions, Pe Gunnells, R Sczech
Evaluation Of Dedekind Sums, Eisenstein Cocycles, And Special Values Of L-Functions, Pe Gunnells, R Sczech
Paul Gunnells
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 …
A Smooth Space Of Tetrahedra, E Babson, Pe Gunnells, R Scott
A Smooth Space Of Tetrahedra, E Babson, Pe Gunnells, R Scott
Paul Gunnells
This is the pre-published version harvested from ArXiv. We construct a smooth symmetric compactification of the space of all labeled tetrahedra in 3.
Elliptic Functions And Equations Of Modular Curves, Lev A. Borisov, Paul E. Gunnells, Sorin Popescu
Elliptic Functions And Equations Of Modular Curves, Lev A. Borisov, Paul E. Gunnells, Sorin Popescu
Paul Gunnells
Let P≥5 be a prime. We show that the space of weight one Eisenstein series defines an embedding into P(p−3)/2 of the modular curve X1(p) for the congruence group Γ1(p) that is scheme-theoretically cut out by explicit quadratic equations.
On Toric Varieties And Modular Forms, Paul Gunnells
On Toric Varieties And Modular Forms, Paul Gunnells
Paul Gunnells
No abstract provided.
Modular Symbols And Hecke Operators, Paul E. Gunnells
Modular Symbols And Hecke Operators, Paul E. Gunnells
Paul Gunnells
We survey techniques to compute the action of the Hecke operators on the cohomology of arithmetic groups. These techniques can be seen as generalizations in different directions of the classical modular symbol algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in papers of the author and the author with Mark McConnell. Some results are unpublished work of Mark McConnell and Robert MacPherson.
Computing Special Values Of Partial Zeta Functions, Gautam Chinta, Paul E. Gunnells, Robert Sczech
Computing Special Values Of Partial Zeta Functions, Gautam Chinta, Paul E. Gunnells, Robert Sczech
Paul Gunnells
We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the Eisenstein cocycle Ψ, a group cocycle for GL n (ℤ); the special values are computed as periods of Ψ, and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.
Wonderful Blowups Associated To Group Actions, Lev A. Borisov, Paul Gunnells
Wonderful Blowups Associated To Group Actions, Lev A. Borisov, Paul Gunnells
Paul Gunnells
A group action on a smooth variety provides it with the natural stratification by irreducible components of the fixed point sets of arbitrary sub-groups. We show that the corresponding maximal wonderful blowup in the sense of MacPherson-Procesi has only abelian stabilizers. The result is inspired by the abelianization algorithm of Batyrev.
Eisenstein Series Twisted By Modular Symbols For The Group Sln, Dorian Goldfield, Paul Gunnells
Eisenstein Series Twisted By Modular Symbols For The Group Sln, Dorian Goldfield, Paul Gunnells
Paul Gunnells
We define Eisenstein series twisted by modular symbols for the group SLn, generalizing a construction of the first author \cite{goldfeld1, goldfeld2}. We show that, in the case of series attached to the minimal parabolic subgroup, our series converges for all points in a suitable cone. We conclude with examples for SL2 and SL3.