John Cullinan

Associate Professor and Chair of Mathematics

Bard College

about                 research                 teaching                 miscellaneous

My main mathematical interests are in arithmetic geometry, representations of finite groups, and the arithmetic of orthogonal polynomials. I also have side interests in the mathematics of voting. My publications, separated roughly by research topic, are as follows (with chronological numbering):

Local-Global Principles for Abelian Varieties and Galois Representations

22. Divisibility of torsion subgroups of abelian surfaces over number fields.
In Preparation.

21. On a probabilistic local-global principal for torsion on elliptic curves (w/ J. Voight).
In Preparation.

19. Fixed-Point Subgroups of GL(3,q).
Submitted, 2018.

6. Symplectic stabilizers with applications to abelian varieties.
International Journal of Number Theory 8 (2012), no. 2, 321-334.

4. Points of small order on three-dimensional abelian varieties; With an appendix by Yuri Zarhin.
Journal of Algebra 324 (2010), no. 3, 565-577.

2. A computational approach to the 2-torsion structure of an abelian threefold.
Mathematics of Computation 78 (2009), no. 267, 1825-1836.

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

Arithmetic of Orthogonal Polynomials

18. On the Jacobians of curves defined by the Generalized Laguerre Polynomials.
To appear in Experimental Mathematics.

17. Algebraic properties of Kaneko-Zagier lifts of supersingular polynomials (w/ F. Hajir).
Proceedings of the American Mathematical Society 145 (2017), no. 6, 2291-2304.

15. On the Newton polygons of Kaneko-Zagier lifts of supersingular polynomials (w/ R. Gajek-Leonard).
Research in Number Theory 2 (2016), Art. 34

11. 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. On the Galois groups of Legendre polynomials (w/ F. Hajir).
Indagationes Mathematicae (N.S.) 25 (2014), no. 3, 534-552.

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

3. 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, 97-108.

Miscellaneous Elliptic Curve

20. A remark on the group structure of elliptic curves in towers of finite fields.
To appear in New York Journal of Mathematics.

16. Real preimages of duplication on elliptic curves.
Missouri Journal of Mathematical Sciences 29 (2017), no. 1, 19-26.

13. 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.

Galois Theory of Towers

12. Arithmetic properties of generalized Rikuna polynomials (w/ Z. Chonoles et al).
Publications Mathematiques de Besancon: Algebre et Theorie des Nombres, 2014/1, 19-33.

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

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

Voting Theory

14. Voting with partially ordered preferences (w/ S. Hsiao).
To appear in Mathematics and Social Justice: Perspectives and Resources for the College Classroom, MAA Edited Volume.

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