## Avner Ash

### Professor of Mathematics, Boston College

**Office**: 550 Maloney

**Phone**: please use e-mail instead

**E-mail**: ashav@bc.edu

**Office hours**: As posted on my office door or by appointment via email

## Books

*Fearless Symmetry 2nd ed. ***(with Robert Gross) Princeton University Press (2008)**

*Elliptic Tales ***(with Robert Gross) Princeton University Press (2012)**

*Summing It Up ***(with Robert Gross) Princeton University Press (2016)**

*Smooth Compactifications of Locally Symmetric Varietes 2nd ed. ***(with D. Mumford, M. Rapoport and Y. Tai and with the collaboration of P. Scholze) Cambridge University Press (2010)**

## Selected publications and preprints

**Cohomology with twisted one-dimensional coefficients for congruence subgroups of SL(4,Z) and Galois representations*** (with P. Gunnells and M. McConnell), submitted.*

**Lattice-packing by spheres and eutactic forms*** (with R. Gross), to appear Experimental Mathematics.*

**Sums of Galois representations and arithmetic homology*** (joint with Darrin Doud), to appear TAMS.*

**Reducible Galois representations and arithmetic homology for GL(4)*** (joint with Darrin Doud), to appear Annales Mathématiques Blaise Pascal.*

**Vanishing of the cohomology of the rational points of a reductive Q-group in high dimensions*** (with A. Putman and S. Sam), Compositio Mathematica. 154 (2018) 1111-1130.*

**Comparison of Steinberg modules for a field and a subfield***, J of Algebra 507 (2018), 200-224.*

**Even Galois representations and the cohomology of Gl _{2}(Z)**

**(joint with Darrin Doud) Annales Mathématiques Québec 43 (2019) 1-35.****On the growth of torsion in the cohomology of arithmetic groups***(joint with Paul Gunnells, Mark McConnell and Dan Yasaki), Journal of the Mathematical Institute at the University of Paris Jussieu, published online: 21 March 2018, pp. 1-33.*

**Towards an explicit eigencurve for GL(3)**** **(joint with David Pollack) Int. J. of Number Th. 13 (2017) 1213-1231.

**Relaxation of strict parity for reducible Galois representations attached to the homology of Gl _{3}(Z)**

**(joint with Darrin Doud) Int. J. Number Th. 12 (2016) 361-381****Galois representations attached to tensor products of arithmetic cohomology**** **(joint with Darrin Doud) J. Alg. 465 (2016) 81-99

**Mod 2 homology for GL(4) and Galois representations**** **(joint with Paul Gunnells and Mark McConnell) J Number Theory 146 (2015) 4-22

**Highly reducible Galois representations attached to the homology of GL(n,Z)**** **(joint with Darrin Doud) Proc AMS 143 (2015) 3801-3813

**Reducible Galois representations and the homology of Gl _{3}(Z)**

**(joint with Darrin Doud) IMRN 2014 (2014) 1379-1408****Resolutions of the Steinberg representation for GL(n)**** **(joint with Paul Gunnells and Mark McConnell) J of Alg 349 (2012) 380-390

**Direct sums of characters of mod p characters of Gal(Q-bar/Q) and the homology of Gl _{n}(Z)**

**Comm in Alg 41 (2013) 1751-1775****Hecke operators and the stable homology of GL(n)**** **Comm in Alg 40 (2012) 1369-1387

**A constructive approach to p-adic deformations of arithmetic homology**** **Acta Arithmetica 149 (2011) 305-319

**p-adic deformations of arithmetic cohomology--Draft 2007 ***(joint with Glenn Stevens)*

**Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations ***(joint with Paul Gunnells and Mark McConnell) J. Alg. 325 (2011) 404-415*

**Cohomology of congruence subgroups of GL _{4}(Z). III **

**(**joint with Paul Gunnells and Mark McConnell) Math. Comp. 79 (2010) 1811-1831**Cohomology of congruence subgroups of GL _{4}(Z) II **

**(**joint with Paul Gunnells and Mark McConnell) Journal of Number Theory 128 (2008) 2263-2274**Frequencies of successive pairs of prime residues ****(**joint with Laura Beltis, Robert Gross and Warren Sinnott) Experimental Math. J. 20 (2011) 400-411

**Equality of polynomial and field discriminants ****(**joint with Jos Brakenhoff and Theodore Zarrabi) Experimental Math. J. 16 (2007) 367-374

**Frequencies of successive tuples of Frobenius classes ****(**joint with Brandon Bate and Robert Gross) Experimental Math. J. 18 (2009) 55-63

**Rigidity of p-adic cohomology classes of congruence subgroups of GL(n,Z)**** (**joint with David Pollack and Glenn Stevens) Proc LMS (3) 96 (2008) 367-388

**Everywhere unramified automorphic cohomology for GL _{3}(Z)**

**(**joint with David Pollack) Int. Journal of Number Theory 4 (2008) 663-675**Set-theoretic Hida projectors ***Proc. Amer. Math. Soc. 137 (2008) 1235-1237*

**Hecke L-functions and the distribution of totally positive integers ***joint with Solomon Friedberg Canadian J. Math. 59 (2007) 673-695*

**A _{6}-extensions of Q and the mod p cohomology of GL_{3}(Z)**

**(**joint with David Pollack and Warren Sinnott) Journal of Number Theory, 115 (2005) no. 1, 176-196**Non-vanishing of alternants ***Linear Algebra Applications, 411 (2005) 348-355*

**SL _{3}(F_{2})-extensions of Q and arithmetic cohomology modulo 2 **

**(**joint with David Pollack and Dayna Soares) Experimental Mathematics, 13 (2004) no. 3, 298-307**Smith theory and Hecke operators **Journal of Algebra, 259 (2003) no. 1, 43-58

**Cohomology of congruence subgroups of SL _{4}(Z)**

**(**joint with Paul Gunnells and Mark McConnell) Journal of Number Theory, 94 (2002) no. 1, 181-212**Galois representations with conjectural connections to arithmetic cohomology**** (**joint with David Pollack and Darren Doud) Duke Mathematics Journal, 112 (2002) no. 3, 521-579

**An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod p cohomology of GL(n,Z) ****(**joint with Warren Sinnott) Duke Mathematics Journal, 105 (2000) no. 1, 1-24

**Generalized non-abelian reciprocity laws: a context for Wiles' proof (**joint with Robert Gross) Bull. London Math. Soc., 32 (2000), no. 4, 385-397

**Modular representations of GL(3,F _{p}), symmetric squares, and mod-p cohomology of GL(3,Z) (**joint with Pham Huu Tiep) Journal of Algebra, 222 (1999) no. 2, 376-399

**Monomial Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(p-1,Z) **Math. Ann. 315 (1999), no. 2, 263-28

**Galois representations, Hecke operators, and the mod-p cohomology of GL(3,Z) with twisted coefficients ****(**joint with Gerald Allison and Eric Conrad) Experimental Mathematics, 7 (1998), no. 4, 361-390

**Cohomology at infinity and the well-rounded retract for general linear groups ****(**joint with Mark McConnell) Duke Mathematics Journal 90 (1997) no. 3, 549-576

**p-adic deformations of cohomology classes of subgroups of GL(n,Z) ****(**joint with Glenn Stevens) Journees Arithmetiques, (Barcelona, 1995) Collect. Math. 48 (1997), no. 1-2, 1-30