Number Theory and Arithmetic Geometry Group

Permanent faculty and their fields of interests.

  • William Alford, Associate Professor, Ph.D. Tulane, 1963. Factoring and other number theory problems by computer.
  • Matthew Baker, Assistant Professor, Ph.D. U.C. Berkeley, 1999. Galois actions on torsion points. Modular curves and their Jacobians. Discreteness problems for arithmetic heights. Linear series and vector bundles in characteristic p. Arithmetic of curves and their Jacobians.

  • Sybilla Beckmann, Associate Professor, Ph.D. U. Penn., 1986. Galois theory. The inverse galois problem, that is, to determine whether every finite group is the galois group of some extension of the rational numbers. Arithmetic information on branched coverings, such as fields of definition. Tilings of the plane. Mathematics education.

  • Andrew Granville, Barrow Professor, Ph.D. Queens,1987. Distribution of primes. Sieving intervals. Distribution of `smooth' numbers. Properties of binomial coefficients. Cyclotomic fields. Carmichael numbers. Exponential sums. Integer solutions to Diophantine equations. Binary quadratic forms and the elementary theory of elliptic curves. Questions related to factoring and primality testing. Symbolic computation and `computing by homomorphisms'. Computational complexity, particularly lower bounds. Power series and the combinatorics of coefficients. Counting lattice points.

  • Dan Lieman, Associate Professor, Ph.D. Brown, 1992. Special values of L-series, Fourier coefficients of metaplectic forms, analytic theory of automorphic and metaplectic forms, exponential sums, sparse polynomials and cryptography, applications of number theory to cryptography.

  • Dino Lorenzini, Associate Professor, Ph.D. U.C. Berkeley, 1988. Rational torsion points on abelian varieties. Groups of components of Néron models of abelian varieties. Modular curves and their jacobians. Degenerations of curves. Monodromy transformations associated to families of curves.

  • Robert Rumely, Professor, Ph.D. Princeton, 1978. Capacity theory, arithmetic intersection theory. Decidability of arithmetic theories. Model-theoretic algebra. Primality testing, primes in arithmetic progressions, zeroes of Dirichlet L-series.

  • Postdoctoral Associates and their fields of interests.

  • Stephen Astels, Ph.D. University of Waterloo, 1999. Cantor sets & continued fractions. Distributions of large primes. Square-free numbers and square-free parts of numbers.

  • Nathan Ng, Ph.D. University of British Columbia, 2000. Riemann zeta function. Zeroes of L-functions. Random matrix theory. Prime numbers.

  • Tom Tucker, Ph.D. U.C. Berkeley, 1998. Diophantine approximation. Algebraic points on varieties.

  • Algebraic Geometry Group.

    Our number theory group is complemented by a large group in algebraic geometry, including Valery Alexeev, William Graham, Elham Izadi, Roy Smith, and Robert Varley. For more information, look up the Geometry Group.

    Weekly Seminars.

    Number Theory: Mondays at 3:30.

    Arithmetic Geometry: Wednesdays at 3:30.

    Algebraic Geometry: Wednesdays at 2:20.

    Recent graduates and their dissertation.

  • Gang Yu (C. Pomerance), Average size of the 2-Selmer group of certain elliptic curves over Q (2000).
  • Mark Watkins (C. Pomerance), Class Numbers of Imaginary Quadratic Fields (2000).
  • Dina Khalil (A. Granville), On the p-divisibility of class numbers of quadratic fields (2000).
  • Pamela Cutter (A. Granville), Finding Prime Pairs with Particular Gaps and Squarefree Parts of Polynomials (2000).
  • Ernest Croot III (A. Granville), Unit fractions (2000).
  • Shuguang Li (C. Pomerance), On Artin's conjecture for composite moduli (1998).

  • David Penniston (D. Lorenzini), The unipotent part of the generalized jacobian of a curve (1998).

  • Glenn Fox (A. Granville), A p-adic L-function of Two Variables (1997).

  • Jon Grantham (C. Pomerance), Frobenius Pseudoprimes (1997).

  • Kevin James (A. Granville), On Congruences for the Coefficients of Modular Forms and Applications (1997).

  • Ronnie Burthe (C. Pomerance), The Average Witness is 2 (1995).

  • Fred Cheng (C. Pomerance), An Explicit Upper Bound for the Zeta Function in the Critical Strip (1995).

  • Anitha Srinivasan (A. Granville), Computations of Class Numbers of Quadratic Fields (1995).
  • Graduate students in the news.

  • Ernest Croot III solves well-known Erdos problem .
  • The research of Ernest Croot III is featured in Fractions to Make Egyptian Scribe Blanch, Science, vol 278, 10/10/1997.
  • The research of Pam Cutter is featured in a May 31st, 1997, article in Science News .
  • The research of Jon Grantham is featured in a June 13th, 1997, article in the Christian Science Monitor .

  • Good news: our graduate program in number theory is ranked 10th by US News and World Report .

    You may access from here the University of Georgia Mathematics department and the web pages of our other research groups. Our Graduate Bulletin with information for prospective graduate students is also available on line. If you are interested in graduate studies in number theory and would like further information on our group, do not hesitate to contact any of us.

    You should check out Keith Matthews' excellent Number Theory Web site, for lots of information on the `Queen of Mathematics'.