John Voight

Assistant Professor
Department of Mathematics and Statistics
College of Engineering and Mathematical Sciences
University of Vermont

Address:
Department of Mathematics and Statistics
16 Colchester Ave
University of Vermont
Burlington, VT 05401
USA

Office: 16 Colchester Ave, 207C
Email: jvoight@gmail.com
Phone: (802) 656-2271
Homepage: http://www.cems.uvm.edu/~voight/
Office hours: Mondays, 10:00 - 11:30 a.m. and 2:00 - 3:30 p.m.; or please make an appointment!

Research Interests:

• Arithmetic algebraic geometry: Modular curves and moduli spaces, Shimura curves, elliptic curves, computational and algorithmic aspects
• Number theory: Algebraic number theory, quaternion algebras, Arakelov theory, zeta functions of varieties over finite fields, cryptography and coding theory

Teaching:

• Math 241: Analysis in Several Real Variables I (Fall 2009)
• Hons 195N: A Social and Mathematical History of Cryptography (Fall 2009)

Past Courses (UVM):

• Math 255: Elementary Number Theory (Spring 2009)
• Math 20C: Fundamentals of Calculus II (Fall 2008)
• Math 295A/395A: Cryptography (Fall 2008)
• Math 252: Abstract Algebra II (Spring 2008)
• Math 251: Abstract Algebra I (Fall 2007)

Past Courses (Berkeley):

• Math 110: Linear Algebra (Spring 2005)
• Math 115: Introduction to Number Theory (Summer 2004)
• Math 1A: Calculus (Spring 2004)
• Math 250B: Multilinear Algebra (Spring 2003)
• Math 195: Cryptography (Spring 2002)
• Math 1B: Calculus (Fall 2001)

Vitae:

• Curriculum Vitae (updated 16 July 2009)
  [DVI] [PDF] [PS]

Publications:

21. Nondegenerate curves of low genus over small finite fields (with Wouter Castryck), submitted.
    [math.NT/0907.2060] [DVI] [PS] [PDF]
    
20. Nonsolvable number fields ramified only at 3 and 5 (with Lassina Dembélé and Matthew Greenberg), submitted.
    [math.NT/0906.4374] [DVI] [PS] [PDF]
    
19. Rings of low rank with a standard involution and quaternion rings, submitted.
    [math.NT/0904.4310] [DVI] [PS] [PDF]
    
18. Computing systems of Hecke eigenvalues associated to Hilbert modular forms (with Matthew Greenberg), submitted.
    [math.NT/0904.3908] [DVI] [PS] [PDF]
    
17. Algebraic curves uniformized by congruence subgroups of triangle groups (with Pete Clark), in preparation.
16. Computing zeta functions for sparse hypersurfaces using Dwork cohomology (with Steven Sperber), in preparation.
15. Algorithmic identification of quaternion algebras and the matrix ring, in preparation.
14. Algorithmic enumeration of ideal classes for quaternion orders (with Markus Kirschmer), accepted to SIAM J. Comput. (SICOMP)
    [math.NT/0808.3833] [DVI] [PS] [PDF]
    See also the resulting tables.
13. The Gauss higher relative class number problem, Ann. Sci. Math. Québec 32 (2008), no. 2, 221–232.
    [math.NT/0806.0306] [PDF]
    
12. Shimura curves of genus at most two, Math. Comp. 78 (2009), 1155-1172.
    [Linked PDF] [math.NT/0802.0911] [DVI] [PS] [PDF]
    See also the resulting tables and the important erratum.
11. Computing fundamental domains for cofinite Fuchsian groups, J. Théorie Nombres Bordeaux 21 (2009), no. 2, 467-489.
    [math.NT/0802.0196] [PDF]
    See also the supporting code and demo.
10. On nondegeneracy of curves (with Wouter Castryck), Algebra & Number Theory 3 (2009), no. 3, 255-281.
    [Link] [math.AG/0802.0420] [DVI] [PS] [PDF]
9. Enumeration of totally real number fields of bounded root discriminant, Algorithmic number theory, eds. Alfred van der Poorten and Andreas Stein, Lecture notes in computer science, vol. 5011, Springer, Berlin, 2008, 268-281.
   [Link] [math.NT/0802.0194] [DVI] [PS] [PDF]
   See also the resulting tables.
8. Shimura curve computations, to appear in proceedings from the 2006 Clay Mathematics Institute, Arithmetic Geometry summer school.
   [DVI] [PS] [PDF]
7. Heegner points and Sylvester's conjecture (with Samit Dasgupta), to appear in proceedings from the 2006 Clay Mathematics Institute, Arithmetic Geometry summer school.
   [DVI] [PDF]
6. Quadratic forms that represent almost the same primes, Math. Comp. 76 (2007), 1589-1617.
   [math.NT/0410266] [Linked PDF] [DVI] [PS] [PDF]
5. Computing CM points on Shimura curves arising from cocompact arithmetic triangle groups, Algorithmic number theory, eds. Florian Hess, Sebastian Pauli, Michael Pohst, Lecture notes in computer science, vol. 4076, Springer, Berlin, 2006, 406-420.
   [Link] [PS] [PDF]
4. Arithmetic Fuchsian groups and Shimura curves, Quaternion algebras (with David Kohel), Associative orders (with Nicole Sutherland), Handbook of Magma functions, eds. John Cannon and Wieb Bosma, Sydney, ed. 2.14, 2007.
3. Curves over finite fields with many points: an introduction, Computational aspects of algebraic curves, ed. Tanush Shaska, Lecture notes series on computing, vol. 13, World Scientific, Hackensack, NJ, 2005, 124-144.
   [DVI] [PDF] [PS]
2. Quadratic forms and quaternion algebras: Algorithms and arithmetic, Ph.D. thesis, University of California, Berkeley, 2005.
   [DVI] [PDF] [PS]
1. On the nonexistence of odd perfect numbers, MASS Selecta: Teaching and learning advanced undergraduate mathematics, Svetlana Katok, Alexei Sossinsky, and Serge Tabachnikov, eds., American Mathematical Society, Providence, RI, 2003, 293-300.
   [PDF]

Tables:

• Totally real number fields of fixed degree and bounded root discriminant
• Shimura curves of genus at most two
• Definite Eichler orders with class number at most two

Notes and Expository Articles:

• Perfect numbers: An elementary introduction
  [DVI] [PDF] [PS]
• Introduction to stacks
  [DVI] [PDF] [PS]
• Toric surfaces and continued fractions
  [DVI] [PDF] [PS]
• Oberwolfach seminar on Explicit Algebraic Number Theory
  [Link to Notes]
• Aspects of complex multiplication (notes from Zagier)
  [DVI] [PDF] [PS]
• Introduction to group schemes (notes from Schoof)
  [DVI] [PDF] [PS]
• Rational and integral points on higher dimensional varieties (notes from AIM)
  [Link to Notes]
• Algebraic geometry (notes from Hartshorne)
  [DVI] [PDF] [PS]

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