Number Theory and Arithmetic Geometry Group

Permanent faculty and their fields of interests.

Pete L. Clark, Assistant Professor, Ph.D. Harvard 2003. Arithmetic of abelian varieties; torsion points, endomorphism algebras, Weil-Chatelet groups. Modular curves and Shimura curves. Period-index problems. Pointless varieties and the (anti-) Hasse principle. Geometric approaches to the inverse Galois problem.

Jonathan Hanke, Assistant Professor, Ph. D. Princeton, 1999. Arithmetic of Quadratic forms and their connections to Automorphic Forms, with an emphasis on computation. Topics include Local-Global principles, Local Densities, Mass Formulas, Class Numbers, Explicit Finiteness Theorems, and Computing all integers represented by a given Quadratic Form.

Dino Lorenzini, 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.

Neil Lyall, Assistant Professor, Ph.D., University of Wisconsin, 2004. Application of Fourier analytic techniques to problems in additive combinatorics. Discrete problems in harmonic analysis.

Akos Magyar, Associate Professor, Ph.D., Princeton University, 1996. Discrete problems in harmonic analysis and applications to ergodic and number theory. Polynomial type ergodic theorems, distribution of solutions of diophantine equations.

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.

Post Doctoral Associates and their fields of interest

Patrick Corn, Postdoctoral Associate, Ph.D. U. C. Berkeley, 2005. Rational points on varieties. Rational surfaces. Counterexamples to the Hasse principle.

Matthew L. Smith, Postdoctoral Associate, Ph.D. Michigan, 2007. Arithmetic combinatorics, solution-free sets for non-linear Diophantine systems. The Hardy-Littlewood circle method.

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 Schedule

Number Theory/Arithmetic Geometry: Dino Lorenzini
Wednesday, 3:30p.m., Room 304

VIGRE

In 2001-2002, Andrew Granville organized a VIGRE research group on Prime number races. Members of the group were Jim Solazzo, Zubeyir Cinkir, Brian Lawler, Michael Guy, Mike Beck, and Eric Pine.

In 2002-2003, Dino Lorenzini organized a VIGRE research group on the rank of an elliptic curve. Members of the group are: Bill Rulla, Sungkon Chang, Zubeyir Cinkir, Steven Donnelly, Zach Michaels, Charles Pooh, Joe Rusinko, Rene Shumbusho.

During the summer of 2003, Matt Baker and Robert Rumely ran an REU on Analysis on Metrized Graphs.

In 2004-2005, Dino Lorenzini organized a VIGRE research group on ranks of curves. Members of the group are: Robert Brice, Sungkon Chang, Jerry Hower, Jacob Keenum, Nausheen Lotia, Daeshik Park, Clay Petsche, Charles Pooh, Dong Hoon Shin, and Juhyung Yi.

During the summer of 2005, Akos Magyar ran an REU on Ramsey theory.

In 2005-2006, Robert Rumely organized a VIGRE research group on the Riemann-Siegel Formula.

In the Fall of 2006, Dino Lorenzini organized a VIGRE research group on the Laplacian of a graph.

In the Fall of 2007, Dino Lorenzini organizes a VIGRE research group on Graphs and Geometry.

More information on VIGRE at UGA

Recent graduates and their dissertation.

2007

Zubeyir Cinkir (Robert Rumely), The tau constant of a metrized graph.

2006

Daeshik Park (Robert Rumely), The Fekete-Szego Theorem with Splitting Conditions on the Projective Line of Positive Characteristic p.

2005

Paulo Almeida (A. Granville), Sign Changes of Error Terms Related to Certain Arithmetic Functions.
Sungkon Chang (D. Lorenzini), The arithmetic of twists of the jacobians of superelliptic curves.
Charles Pooh (R. Rumely), Capacity theory and Algebraic integers.

2004

Michael Beck (A. Granville), Square Dependence in Random Integers.
Jim Blair (A. Magyar), On the Embedding of Simplicies into Integer Lattices.
Milton Nash (R. Rumely), Special Values of Dirichlet L- functions.
Eric Pine (A. Granville), Sums of Integer Cubes.
Rene-Michel Shumbusho (D. Lorenzini), Elliptic Curves With Prime Conductor and a Conjecture of Cremona.

2003

Steve Donnelly (R. Rumely), Elements of given order in Tate-Shafarevich groups of elliptic curves.

2000

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

1998

Shuguang Li (C. Pomerance), On Artin's conjecture for composite moduli.
David Penniston (D. Lorenzini), The unipotent part of the generalized jacobian of a curve.

1997

Glenn Fox (A. Granville), A p-adic L-function of Two Variables.
Jon Grantham (C. Pomerance), Frobenius Pseudoprimes.
Kevin James (A. Granville), On Congruences for the Coefficients of Modular Forms and Applications.

1995

Ronnie Burthe (C. Pomerance), The Average Witness is 2.
Fred Cheng (C. Pomerance), An Explicit Upper Bound for the Zeta Function in the Critical Strip.
Anitha Srinivasan (A. Granville), Computations of Class Numbers of Quadratic Fields.

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.