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Full-Text Articles in Number Theory
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 …
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.
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 …
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.
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.