John Cullinan

Associate Professor and Chair of Mathematics

Bard College

My main mathematical interests are in arithmetic geometry, representations of finite groups, and the arithmetic of orthogonal polynomials. My current research focuses on the following topics:

The torsion structure of abelian varieties: The focus of my dissertation was on certain representation-theoretic aspects of the torsion structure of three-dimensional abelian varieties. Since then, I have been interested in classifying the images of mod l and l-adic representations attached to low-dimensional abelian varieties.

Galois properties of orthogonal polynomials: My collaborators and I are interested in the algebraic properties of certain families of orthogonal polynomials. We study the varieties defined by these polynomials, as well as the arithmetic information they encode. Of particular interest to us are the Jacobi polynomials (and their specializations) because of their deep connections to arithmetic.

Publications

18. [abs] [pdf] Subgroups of GL(3,q) with applications to Galois representations Submitted, 2016.

17. [abs] [pdf] On the Newton polygons of Kaneko-Zagier lifts of supersingular polynomials (w/ R. Gajek-Leonard). Research in Number Theory 2(1), 1-16 (2016)

16. [abs] [pdf] Real preimages of duplication on elliptic curves. Missouri Journal of Mathematical Sciences, 29, (1), 19-26 (2017)

15. [abs] [pdf] Voting with partially ordered preferences (w/ S. Hsiao). To appear in the Mathematics and Social Justice edited volume, MAA.

14. [abs] [pdf] Algebraic properties of Kaneko-Zagier lifts of supersingular polynomials (w/ F. Hajir). Proceedings of the American Mathematical Society. 145, (6), 2291-2304 (2017)

13. [abs] [pdf] On the solution to a generalized Fermat equation: arithmetic and combinatorics (w/ A. Barghi). Journal of Combinatorics and Number Theory. 7 (2015) no. 3 157-169

12. [abs] [pdf] Arithmetic properties of generalized Rikuna polynomials (w/ Z. Chonoles, H. Hausman, A.M. Pacelli, S. Pegado, F. Wei). Publications Mathematiques de Besancon: Algebre et Theorie des Nombres. 1 (2014) 19 - 33

11. [abs] [pdf] On the ranks of Jacobians of curves defined by Jacobi polynomials (w/ A. Etropolski and E. Sell). Rocky Mountain Journal of Mathematics. 44 (2014) no. 1 23-33

10. [abs] [pdf] On the Galois groups of Legendre polynomials (w/ F. Hajir). Indagationes Mathematicae. 25 (2014) 534--552

9. [abs] [pdf] A Borda Count procedure for partially ordered ballots (w/ S. Hsiao and D. Polett). Social Choice and Welfare. 42 (2014) no. 4 913--926

8. [abs] [pdf] Specializations of generalized Rikuna polynomials (w/ C. Cass, A. Rasmussen, and D. Trifunovski). International Journal of Number Theory 10 (2014), no. 3 585--600

7. [abs] [pdf] Primes of prescribed congruence class in short intervals (w/ F. Hajir). Integers 12 (2012) #A56.

6. [abs] [pdf] Symplectic stabilizers with applications to abelian varieties. International Journal of Number Theory 8 (2012), no. 1, 321-334.

5. [abs] [pdf] Ramification in iterated towers for rational functions (w/ F. Hajir). Manuscripta Mathematica 137 (2012), no. 3-4, 273-286.

4. [abs] [pdf] Points of small order on three-dimensional abelian varieties; With an appendix by Yuri Zarhin. Journal of Algebra, 324 (2010), 565-577.

3. [abs] [pdf] Algebraic properties of a family of Jacobi polynomials (w/ F. Hajir and E. Sell). Journal de Theorie des Nombres de Bordeaux 21 (2009), no. 1, 71-82.

2. [abs] [pdf] A computational approach to the 2-torsion structure of an abelian threefold. Mathematics of Computation 78 (2009), 1825-1836.

1. [abs] [pdf] Local-global properties of torsion points on three-dimensional abelian varieties. Journal of Algebra 311 (2007), no. 2, 736-774.

Here are some links to math talks I like very much: Serre's Minerva lectures 1, 2, and 3 and Serre on Writing Mathematics Badly; Katz's IHES colloquium; the FLT Conference.

In spring 2013 I collaborated with Marina Noronha from the CCS Bard Hessel Museum on a curatorial project. Part of the exhibition used the main theorem of this paper to select artwork for the show. The site for the show is here.

In 2010, Allison Pacelli and I organized an MAA PREP workshop on Algebraic Number Theory. We used to have a wiki devoted to the workshop, but technical difficulties prevent us from continuing to host it. However, the files are available here (zipped).

I enjoy writing reviews for MathSciNet. You can read the reviews I've written here, as well as those of my papers. You need a site license to access these reviews.