arXiv subjects

Mark McConnell

Publications and source records attributed to Mark McConnell.

At least 19 recordsLinked to original sources

arXiv2024

Cohomology with Sym^g coefficients for congruence subgroups of SL_4(Z) and Galois representations

We extend the computations in our prior work to find the cohomology in degree five of a congruence subgroup Gamma of SL_4(Z) with coefficients in Sym^g(K^4), twisted by a nebentype character eta, along with the action of the Hecke algebra. This is the top cuspidal degree. In this paper we take K to be a finite field of large characteristic, as a proxy for the complex numbers. For each Hecke eigenclass found, we produce the unique Galois representation that appears to be attached to it. The computations require modifications to our previous algorithms to accommodate the fact that the coefficients are not one-dimensional.

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arXiv2024

Explicit sharbly cycles at the virtual cohomological dimension for SL_n(Z)

Denote the virtual cohomological dimension of SL_n(Z) by t=n(n-1)/2. Let St denote the Steinberg module of SL_n(Q) tensored with Q. Let Sh_* denote the sharbly resolution of the Steinberg module St. By Borel-Serre duality, the one-dimensional Q-vector space H^0(SL_n(Z), Q) is isomorphic to H_t(SL_n(Z),St). We find an explicit generator of H_t(SL_n(Z),St) in terms of sharbly cycles and cosharbly cocycles. These methods may extend to other degrees of cohomology of SL_n(Z).

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arXiv2023

Cohomology at Infinity and the Well-Tempered Complex

We prove the existence of a sequence of commutative diagrams generalizing existing results on the cohomology of the Borel-Serre boundary and well-rounded retract to the context of the well-tempered complex. Our main theorem provides a method for computing in finite terms the action of Hecke operators on the equivariant cohomology of an arithmetic subgroup $\Gamma$ of the special linear group $SL_n$.

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arXiv2023

Tempered Perfect Lattices in the Binary Case

A new algorithm for computing Hecke operators for SL(n,Z) was introduced by MacPherson, McConnell in 2020. The algorithm uses tempered perfect lattices, which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi. The present paper is the first step in characterizing tempered perfect lattices. We obtain a complete classification in the plane, where the Hecke operators are for SL(2,Z) and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields.

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arXiv2023

ASCENT - A balloon-borne hard X-ray imaging spectroscopy telescope using transition edge sensor microcalorimeter detectors

Core collapse supernovae are thought to be one of the main sources in the galaxy of elements heavier than iron. Understanding the origin of the elements is thus tightly linked to our understanding of the explosion mechanism of supernovae and supernova nucleosynthesis. X-ray and gamma-ray observations of young supernova remnants, combined with improved theoretical modeling, have resulted in enormous improvements in our knowledge of these events. The isotope ${}^{44}$Ti is one of the most sensitive probes of the innermost regions of the core collapse engine, and its spatial and velocity distribution are key observables. Hard X-ray imaging spectroscopy with the Nuclear Spectroscopic Telescope Array (NuSTAR) has provided new insights into the structure of the supernova remnant Cassiopeia A (Cas A), establishing the convective nature of the supernova engine. However, many questions about the details of this engine remain. We present here the concept for a balloon-borne follow-up mission called ASCENT (A SuperConducting ENergetic x-ray Telescope). ASCENT uses transition edge sensor gamma-ray microcalorimeter detectors with a demonstrated 55 eV Full Width Half Maximum (FWHM) energy resolution at 97 keV. This 8--16-fold improvement in energy resolution over NuSTAR will allow high resolution imaging and spectroscopy of the ${}^{44}$Ti emission. This will allow a detailed reconstruction of gamma-ray line redshifts, widths, and shapes, allowing us to address questions such as: What is the source of the neutron star "kicks"? What is the dominant production pathway for ${}^{44}$Ti? Is the engine of Cas A unique?

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arXiv2022

COMPTEL data analysis using GammaLib and ctools

More than 20 years after the end of NASA's Compton Gamma-Ray Observatory mission, the data collected by its Imaging Compton Telescope (COMPTEL) still provide the most comprehensive and deepest view of our Universe in MeV gamma rays. While most of the COMPTEL data are archived at NASA's High Energy Astrophysics Science Archive Research Center (HEASARC), the absence of any publicly available software for their analysis means the data cannot benefit from the scientific advances made in the field of gamma-ray astronomy at higher energies. To make this unique treasure again accessible for science, we developed open source software that enables a comprehensive and modern analysis of the archived COMPTEL telescope data. Our software is based on a dedicated plug-in to the GammaLib library, a community-developed toolbox for the analysis of astronomical gamma-ray data. We implemented high-level scripts for building science analysis workflows in ctools, a community-developed gamma-ray astronomy science analysis software framework. We describe the implementation of our software and provide the underlying algorithms. Using data from the HEASARC archive, we demonstrate that our software reproduces derived data products that were obtained in the past using the proprietary COMPTEL software. We furthermore demonstrate that our software reproduces COMPTEL science results published in the literature. This brings the COMPTEL telescope data back into life, allowing them to benefit from recent advances in gamma-ray astronomy, and gives the community a means to unveil its still hidden treasures.

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arXiv2022

The Future of Gamma-Ray Experiments in the MeV-EeV Range

Gamma-rays, the most energetic photons, carry information from the far reaches of extragalactic space with minimal interaction or loss of information. They bring messages about particle acceleration in environments so extreme they cannot be reproduced on earth for a closer look. Gamma-ray astrophysics is so complementary with collider work that particle physicists and astroparticle physicists are often one in the same. Gamma-ray instruments, especially the Fermi Gamma-ray Space Telescope, have been pivotal in major multi-messenger discoveries over the past decade. There is presently a great deal of interest and scientific expertise available to push forward new technologies, to plan and build space- and ground-based gamma-ray facilities, and to build multi-messenger networks with gamma rays at their core. It is therefore concerning that before the community comes together for planning exercises again, much of that infrastructure could be lost to a lack of long-term planning for support of gamma-ray astrophysics. Gamma-rays with energies from the MeV to the EeV band are therefore central to multiwavelength and multi-messenger studies to everything from astroparticle physics with compact objects, to dark matter studies with diffuse large scale structure. These goals and new discoveries have generated a wave of new gamma-ray facility proposals and programs. This paper highlights new and proposed gamma-ray technologies and facilities that have each been designed to address specific needs in the measurement of extreme astrophysical sources that probe some of the most pressing questions in fundamental physics for the next decade. The proposed instrumentation would also address the priorities laid out in the recent Astro2020 Decadal Survey, a complementary study by the astrophysics community that provides opportunities also relevant to Snowmass.

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arXiv2021

Computing Hecke operators for arithmetic subgroups of $\text{Sp}_4$

We outline an algorithm for computing Hecke operators on equivariant cohomology $H^\ast_{Γ_{\text{Sp}}}(X_{\text{Sp}};ρ)$ for the symplectic group $\text{Sp}_4(\mathbb{R})$. To do this, we define a new acyclic cell complex for $\text{Sp}_4(\mathbb{R})$ and make use of the well-tempered complex.

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arXiv2020

Computing Hecke Operators for Arithmetic Subgroups of General Linear Groups

We present an algorithm to compute the Hecke operators on the equivariant cohomology of an arithmetic subgroup $Γ$ of the general linear group $\mathrm{GL}_n$. This includes $\mathrm{GL}_n$ over a number field or a finite-dimensional division algebra. As coefficients, we may use any finite-dimensional local coefficient system. Unlike earlier methods, the algorithm works for the cohomology $H^i$ in all degrees $i$. It starts from the well-rounded retract $\tilde{W}$, a $Γ$-invariant cell complex which computes the cohomology. It extends $\tilde{W}$ to a new well-tempered complex $\tilde{W}^+$ of one higher real dimension, using a real parameter called the temperament. The algorithm has been coded up for $\mathrm{SL}_n(\mathbb{Z})$ for $n=2,3,4$; we present some results for congruence subgroups of $\mathrm{SL}_3(\mathbb{Z})$.

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arXiv2019

The X-ray Polarization Probe mission concept

The X-ray Polarization Probe (XPP) is a second generation X-ray polarimeter following up on the Imaging X-ray Polarimetry Explorer (IXPE). The XPP will offer true broadband polarimetery over the wide 0.2-60 keV bandpass in addition to imaging polarimetry from 2-8 keV. The extended energy bandpass and improvements in sensitivity will enable the simultaneous measurement of the polarization of several emission components. These measurements will give qualitatively new information about how compact objects work, and will probe fundamental physics, i.e. strong-field quantum electrodynamics and strong gravity.

astro-ph.IM↗

arXiv2019

Astro2020 Science White Paper: Using X-Ray Polarimetry to Probe the Physics of Black Holes and Neutron Stars

This white paper highlights compact object and fundamental physics science opportunities afforded by high-throughput broadband (0.1-60 keV) X-ray polarization observations. X-ray polarimetry gives new observables with geometric information about stellar remnants which are many orders of magnitude too small for direct imaging. The X-ray polarimetric data also reveal details about the emission mechanisms and the structure of the magnetic fields in and around the most extreme objects in the Universe. Whereas the Imaging X-ray Polarimetry Explorer (IXPE) to be launched in 2021 will obtain first results for bright objects, a follow-up mission could be one order of magnitude more sensitive and would be able to use a broader bandpass to perform physics type experiments for representative samples of sources.

astro-ph.HE↗

arXiv2018

Cohomology with twisted one-dimensional coefficients for congruence subgroups of SL(4,Z) and Galois representations

We extend the computations in [AGM1, AGM2, AGM3] to find the cohomology in degree five of a congruence subgroup Gamma of SL(4,Z) with coefficients in a field K, twisted by a nebentype character eta, along with the action of the Hecke algebra. This is the top cuspidal degree. In practice we take K to be a finite field of large characteristic, as a proxy for the complex numbers. For each Hecke eigenclass found, we produce a Galois representation that appears to be attached to it. Our computations show that in every case this Galois representation is the only one that could be attached to it. The existence of the attached Galois representations agrees with a theorem of Scholze and sheds light on the Borel-Serre boundary for Gamma. The computations require serious modifications to our previous algorithms to accommodate the twisted coefficients. Nontrivial coefficients add a layer of complication to our data structures, and new possibilites arise that must be taken into account in the Galois Finder, the code that finds the Galois representations. We have improved the Galois Finder so that it reports when the attached Galois representation is uniquely determined by our data.

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arXiv2018

On the cohomology of congruence subgroups of GL3 over the Eisenstein integers

Let F be the imaginary quadratic field of discriminant -3 and OF its ring of integers. Let Gamma be the arithmetic group GL_3 (OF), and for any ideal n subset OF let Gamma_0 (n) be the congruence subgroup of level n consisting of matrices with bottom row (0,0,*) bmod n. In this paper we compute the cohomology spaces H^{nu - 1} (Gamma_0 (n); C) as a Hecke module for various levels n, where nu is the virtual cohomological dimension of Gamma. This represents the first attempt at such computations for GL_3 over an imaginary quadratic field, and complements work of Grunewald--Helling--Mennicke and Cremona, who computed the cohomology of GL_2 over imaginary quadratic fields. In our results we observe a variety of phenomena, including cohomology classes that apparently correspond to nonselfdual cuspforms on GL_3/F.

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arXiv2016

On the growth of torsion in the cohomology of arithmetic groups

Let G be a semisimple Lie group with associated symmetric space D, and let Gamma subset G be a cocompact arithmetic group. Let L be a lattice inside a Z Gamma-module arising from a rational finite-dimensional complex representation of G. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup H_i (Gamma_k; L )_tors as Gamma_k ranges over a tower of congruence subgroups of Gamma. In particular they conjectured that the ratio (log |H_i (Gamma_k ; L)_tors|)/[Gamma : Gamma_k] should tend to a nonzero limit if and only if i= (dim(D)-1)/2 and G is a group of deficiency 1. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including GL_n (Z) for n=3,4,5 and GL_2 (O) for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron--Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron--Venkatesh conjecture.

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arXiv2013

Mod 2 homology for GL(4) and Galois representations

We extend the computations in [AGM4] to find the mod 2 homology in degree 1 of a congruence subgroup Gamma of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is closely related to the cohomology of Gamma with F_2 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 [AGM4]. Our method of computation was justified in [AGM5].

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arXiv2011

Resolutions of the Steinberg module for GL(n)

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 that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses.

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arXiv2010

Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

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 up to 31.

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arXiv2009

A Vigorous Explorer Program

Explorers have made breakthroughs in many fields of astrophysics. The science from both these missions contributed to three Nobel Prizes - Giacconi (2002), Mather, and Smoot (2006). Explorers have: marked the definitive beginning of precision cosmology, discovered that short gamma-ray bursts are caused by compact star mergers and have measured metalicity to redshifts z>6. NASA Explorers do cutting-edge science that cannot be done by facility-class instruments. The Explorer program provides a rapid response to changing science and technology, to enable cutting-edge science at moderate cost. Explorers also enable innovation, and engage & train scientists, managers and engineers, adding human capital to NASA and the nation. The astrophysics Explorer launch rate now being achieved is 1 per 3 years, and budget projections are in the $150M/year range for the next five years. A newly Vigorous Explorer Program should be created to: 1. Reach the long-stated goal of annual astrophysics launches; 2. Find additional launch options for Explorers and actively encourage cost savings in launchers and spacecraft, such as new commercial vehicles and innovative partnerships. 3. Mitigate risk via stronger technical development and sub-orbital programs, and through longer, more thorough, Phase A programs, potentially reducing the need for a 30% contingency; 4. Strive to protect the funding for missions that have reached Phase B, to prevent significant launch slips and cancellations, with a goal of 4 to 5 years from Phase B to launch; 5. Review the project management procedures and requirements to seek cost reductions, including the risk management strategy and the review and reporting process; 6. Review and possibly modify the cost caps for all Explorer classes to optimize scientific returns per dollar. [ABRIDGED]

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