Research

My research lies mostly in the areas of arithmetic geometry and Galois theory. I like to study curves and abelian varieties and the geometry of their moduli spaces. I often focus on the p-torsion of Jacobians of curves over finite fields, including results about L-polynomials, maximal curves, and existence of curves with given p-rank, Newton polygon, or Ekedahl-Oort type. Using Galois theory, I study group actions on curves, with an emphasis on wildly ramified actions. In an earlier series of papers, I studied the semi-stable reduction, deformation, and formal patching of curves. More recently, I have branched in several new directions including abelian varieties over function fields and topological obstructions to rational points over number fields.

I am a professor in the math department at Colorado State University. My research is currently supported by NSF grant DMS-15-02227.
Research during summer 2015 was supported by NSA grant 131011. Research during the periods 7/2011-6/2015, 7/2007-6/2011, 7/2004-6/2007 was supported in part by NSF grants DMS-11-01712, DMS-07-01303, DMS 04-00461. I was supported as a VIGRE postdoc at Columbia University from 2000-2003.

Here are links to my publications and preprints.

My arXiv papers

Submitted Fully maximal and fully minimal abelian varieties (with V. Karemaker), 27 pages.

Submitted The de Rham cohomology of the Suzuki curves, 11 pages.

Submitted Current results on Newton polygons of curves, 20 pages, submitted to Questions in Arithmetic Algebraic Geometry, Advanced Lectures in Mathematics Book Series.

Submitted The Galois action and cohomology of a relative homology group of Fermat curves, (with R. Davis, V. Stojanoska, K. Wickelgren), 29 pages.

Submitted On the existence of ordinary and almost ordinary Prym varieties, (with E. Ozman), 28 pages.

In press Reduction of dynatomic curves (with J. Doyle, H. Krieger, A. Obus, S. Rubinstein-Salzedo, L. West), 47 pages.

In press Non-ordinary curves with a Prym variety of low p-rank, (with T. Celik, Y. Elias, B. Gunes, R. Newton, E. Ozman, L. Thomas) 29 pages.

In press Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields, (with L. Berger, C. Hall, R. Pannekoek, J. Park, R. Pries, S. Sharif, A. Silverberg, and D. Ulmer), to appear in Memoirs of the American Mathematical Society, 127 pages.

In press Abhynakar's conjectures in Galois theory: current status and future directions (with D. Harbater, A. Obus, K. Stevenson), to appear in Bulletin of the American Mathematical Society, 44 pages.

2016 Arithmetic of Abelian Varieties in Artin-Schreier Extensions (with D. Ulmer). Transactions of the AMS, 368 , 12:8553-8595.

2016 2016 Galois action on the homology of Fermat curves, (with R. Davis, V. Stojanoska, K. Wickelgren), Directions in Number Theory: proceedings of the 2014 WIN3 workshop, Springer XV, pages 57-86.

2016, Hurwitz theory in positive characteristic, Appendix A, pages 143-150, in Riemann surfaces and algebraic curves: a first course in Hurwitz theory, by R. Cavalieri and E. Miles, London Mathematical Society Student texts #87, Cambridge University Press.

2015 Superspecial rank of supersingular abelian varieties and Jacobians (with Jeff Achter), Journal de Theorie des Nombres de Bordeaux, 27, 3:605-624.

2015 Ekekdahl-Oort type of Jacobians of Hermitian curves (with Colin Weir). Asian Journal of Math. 19-5 (2015), 845-870.

2014 Generic Newton polygons for curves of given p-rank (with Jeff Achter). Algebraic Curves and Finite Fields, Cryptography and Other Applications Editors: Niederreiter, Ostafe, Panario, Winterhof Radon Series on Computational and Applied Mathematics, 16, 1-21.

2013 Newton polygons for a variant of the Kloosterman family (with Bellowin, Garthwaite, Ozman, Williams, Zhu), Contempory Mathematics, 606, (2013), 47-63.

2013 Ekedahl-Oort strata of hyperelliptic curves in characteristic 2 (with Arsen Elkin), Algebra and Number Theory 7-3, (2013), 507-532.

2013 The a-numbers of Jacobians of Suzuki Curves (with Friedlander, Garton, Malmskog, Weir), Proceedings of the AMS, 141, (2013), 3019-3028.

2013 Families of Artin-Schreier curves with Cartier-Manin matrix of constant rank (with Shawn Farnell), Linear Algebra and its Applications, 439 (2013), 7:2158-2166.

2012 The automorphism groups of a family of maximal curves (with Robert Guralnick and Beth Malmskog), Journal of Algebra, 361, (2012), 92-106.

2012 Alternating group covers of the affine line (with Jeremy Muskat), Israel Journal of Mathematics, 187 , (2012), 117-139.

2012 The p-rank stratification of Artin-Schreier curves (with Hui June Zhu), Annales de l'Institut Fourier, 62, (2012) 2: 707-726.

2011 The p-rank strata of the moduli space of hyperelliptic curves (with Jeff Achter), Advances in Mathematics 227 , (2011), 1846-1872.

2011 A survey of Galois theory of curves in characteristic p (with Kate Stevenson), Fields Institute Communications 60 , American Mathematical Society, Providence RI, (2011), 169-191.

2011 Semi-direct Galois covers of the affine line, (with Linda Gruendken, Laura Hall-Seelig, Bo-Hae Im, Ekin Ozman, and Katherine Stevenson) Fields Institute Communications, 60, AMS, Providence RI, (2011), 201-210.

2010 Wild cyclic-by-tame extensions (with Andrew Obus), Journal of Pure and Applied Algebra, 214, (2010), 5:565-573

2009 The p-torsion of curves with large p-rank, International Journal of Number Theory, 5 , (2009), 6:1103-1115.

2008 Curves of given p-rank with trivial automorphism group (with Jeff Achter and Darren Glass), Michigan J. Math., 56 , (2008), 3:583-592.

2008 Note A short guide to p-torsion of abelian varieties in characteristic p, Computational Arithmetic Geometry, CONM, 463, (2008) 121-129.

2008 Monodromy of the p-rank strata of the moduli space of curves (with Jeff Achter), International Mathematics Research Notices, 15, (2008) Art. ID rnn053, 25 pages.

2007 Hyperelliptic curves with a-number 1 in small characteristic (with Arsen Elkin), Albanian Journal of Math, 1, (2007), 4:245-252.

2007 The integral monodromy of hyperelliptic and trielliptic curves (with Jeff Achter), Math. Annalen, 338 (2007), (1):187-206.

2006 Wildly ramified covers with large genus, J. Number Theory, 119 (2006) 2:194-209.

2005 Corrigendum Hyperelliptic curves with prescribed p-torsion (with Darren Glass) Manuscripta, 117 (2005) 3:299-317.

2005 Questions on p-torsion of hyperelliptic curves (with Darren Glass)
in Problems from the 2004 Leiden Workshop, Rendiconti del Seminario Matematico, 113 (2005) 145-149.

2005 On the moduli space of Klein four covers of the projective line (with Darren Glass)
in Computational Aspects of Algebraic Curves, ed. T. Shaska, Lecture Notes Series in Computing, 13 (2005).

2005 Jacobians of quotients of Artin-Schreier curves,
in Recent progress in arithmetic and algebraic geometry, CONM. 386 (2005) 145-156.

2003 Rigidity, reduction, and ramification (with Irene Bouw), Math. Annalen. 326 (2003) 4: 802-824.

2003 Conductors of wildly ramified covers III, Pacific J.Math. 221 (2003) 1:163-182.

2002 Conductors of wildly ramified covers II, C.R. Acad Sci. Paris, Ser. I 335 (2002) 484-487.

2002 Conductors of wildly ramified covers I, C.R. Acad Sci. Paris, Ser. I 335 (2002) 481-484.

2002 Families of wildly ramified covers of curves, Amer. J. Math. 124 (2002), 4:737-768.

2000 Construction of covers with formal and rigid geometry,
in Courbes semi-stables et groupe fondamental en geometrie algebrique, Progr. Math. 187 (2000) 157-167.

My dissertation, "Formal Patching and Deformation of Wildly Ramified Covers of Curves" was completed in May 2000 at the University of Pennsylvania under the supervision of Professor D. Harbater.

Papers in preparation:
pdf Equiramified deformations of covers in positive characteristic, in preparation.

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