Number Theory and Arithmetic Geometry

Permanent faculty and their fields of interests.

Pete L. ClarkAssociate 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 KrashenAssociate 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 LorenziniProfessor, 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 LyallAssociate ProfessorPh.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.

Paul Pollack, Assistant 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 RumelyProfessor, 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

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.

Joseph VandeheyPostdoctoral Associate, Ph.D. University of Illinois at Urbana-Champaign, 2013. Elementary and analytic number theory and the ergodic theory of numbers. Normal numbers, continued fractions, asymptotic analysis, exponential sums and integrals.

Recent graduates and their dissertation.


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


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


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.


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


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


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


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.


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.


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


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.


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.


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.


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 BlanchScience, 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.