# Number Theory and Arithmetic Geometry

**Permanent faculty and their fields of interests**

Benjamin Bakker, *Assistant Professor, Ph.D. Princeton University, 2010*. Geometric and arithmetic aspects of hyperbolicity in moduli spaces. Torsion structures on abelian varieties. Moduli of sheaves on curves and K3 surfaces. Birational geometry of hyperkahler varieties. Derived categories.

Pete L. Clark, *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.

Daniel Krashen, *Associate Professor, Ph.D. University of Texas 2001*. Finite dimensional division algebras, quadratic forms, and their interplay with algebraic groups and homogeneous varieties. Algebraic cycles and motives. Moduli and configuration spaces.

Dino Lorenzini, *Distinguished Research **Professor, Ph.D. U.C. Berkeley, 1988*. Rational points on algebraic varieties. Torsion points on abelian varieties. Néron models of abelian varieties. Modular curves and their jacobians. Models of curves and wild ramification. Wild quotient singularities of surfaces.

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

Akos Magyar, *Professor, Ph.D. Princeton, 1996*. Discrete harmonic analysis. Arithmetic combinatorics. Analytic methods for diophantine problems.

Giorgis Petridis, *Assistant Professor, Ph.D. University of Cambridge, 2011*. Arithmetic combinatorics and their application to exponential sums. Random graphs.

Paul Pollack,* Associate Professor, Ph.D. Dartmouth, 2008*. Classical problems in number theory, with an emphasis on elementary and analytic methods. Arithmetic functions and their iterates; perfect numbers and their relatives. Multiplicative number theory. The number-theoretic work of Paul Erdos.

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.

**Emiriti professors**

Carl Pomerance, *Distinguished Research Professor, PhD. Harvard, 1972*, retired from UGA in 1999.

**Post Doctoral Associates and their fields of interest**

Abbey Bourdon, *Postdoctoral Associate, Ph.D. Wesleyan University, 2014. *Torsion point fields of abelian varieties, rational torsion, and Galois representations associated to abelian varieties. Fundamental groups in arithmetic geometry.

Andrew Niles, *Limited-Term Assistant Professor, Ph.D. UC Berkeley, 2014*. Moduli of curves and other objects in arithmetic geometry. Algebraic stacks. Compactifications of moduli in positive characteristic. Compactified Jacobians.

**Recent graduates and their dissertations**

2016

**Jacob Hicks** (Pete L. Clark) , *Quadratic Forms Over Hasse Domains: Finiteness of the Hermite Constant.***Ken Jacobs **(Robert Rumely), *Asymptotic Behavior of Arithmetic Equivariants in non-Archimedean Dynamics.***Lee Troupe** (Paul Pollack), *Three applications of sieve methods in analytic number theory.*

2015

**Allan Lacy** (Pete L. Clark), *On the Index of Genus One Curves.*

2014

**John Doyle** (Robert Rumely), *Dynamics of Quadratic Polynomials over Quadratic Fields.*

2013

**David Krumm **(Dino Lorenzini), *Quadratic Points on Modular Curves.*

2012

**Alex Rice** (Neil Lyall),* Improvements and Extensions of Two Theorems of Sarkozy*.**James Stankewicz** (Pete L. Clark/Dino Lorenzini),* Twists of Shimura Curves.***Nathan Walters **(Robert Rumely), *Some Capacity-Theoretic Results Extended to Algebraic Curves.*

2009

**Jeremiah Hower** (Dino Lorenzini), On elliptic curves and arithmetical graphs.

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*.