### Integers Conference 2025

**Permanent faculty and their fields of interests**

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.

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, *Professor*, *Ph.D. University of Wisconsin, 2004*. Arithmetic combinatorics and harmonic analysis.

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

Giorgis Petridis, *Associate Professor, Ph.D. University of Cambridge, 2011*. Combinatorial number theory, combinatorial geometry, and discrete mathematics.

Paul Pollack,* 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.

Jiuya Wang, *Assistant Professor, Ph.D. University of Wisconsin, 2018*. Algebraic number theory. Arithmetic Statistics. Group Theory. Representation Theory. Distribution of number fields and class groups.

**Emeriti professors**

Carl Pomerance,* Distinguished Research Professor, Ph.D. Harvard, 1972. *Retired from UGA in 1999. Number theory.

Robert Rumely,* Professor, Ph.D. Princeton, 1978. *Retired from UGA in 2017. 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 interest**

Steven Fan, *Postdoctoral Research and Teaching Associate, Ph.D. Dartmouth University, 2023*.

Hyuk Jun Kweon,* Limited Term Assistant Professor, Ph.D. MIT, 2021. *Arithmetic geometry. Bounds on the Néron–Severi group, Bounds on second cohomology groups and abelianization of fundamental groups. Computing Picard schemes and Albanese varieties.

Gabrielle Scullard,* Limited Term Assistant Professor, Ph.D. Penn. State, 2024. *Arithmetic geometry. Computing endomorphism rings of supersingular elliptic curves. Applications to isogeny-based cryptography.

**Current graduate students**

Paco Adajar, Rishika Agrawal, Peter Cassels, Swaroop Hegde, Dustin Kasser, Reddy Ponagandla, Casia Siegel, Akash Singha Roy.

**Number Theory Seminar: Mondays, 3:00-4:30**

Speakers are invited to consider the first half hour of the talk as general background on their field of interest. The seminar takes a short break after one hour so that participants who have later commitments can leave if necessary. The third half-hour can be reserved for technical details aimed at specialists.

The seminar was founded in 1972 by Carl Pomerance, Mary Dauenhaur and John Hunsucker, the year Dauenhaur and Pomerance were hired at UGA. The participants read Tate's unpublished lecture notes "Rational points on elliptic curves'', Philips Lectures, Haverford College, Haverford, Pa., 1961. This led Penney and Pomerance to search and find elliptic curves of rank at least 6 and at least 7 over the rationals, record ranks at the time.

**56 PhD Graduates and their Dissertations**

*2024*

**Patrick Akande** (Paul Pollack), *Three results on the distribution of arithmetic functions.*

**Ilkiz Bildik** (Pete L. Clark), *Torsion subgroups of CM elliptic curves in degree 2qp.*

**Aleksander Shmakov*** *(Daniel Litt)*, Cohomology of Local Systems on Siegel Threefolds with Square-Free Parahoric Level.*

**Haiyang Wang ** (Dino Lorenzini),* Three topics in Number Theory.*

*2023*

**Tyler Genao** (Pete L. Clark), *Typical and Polynomial Bounds on Torsion of Elliptic Curves, and Torsion Upon Base Change.*

**Freddy Saia** (Pete L. Clark),* A Volcanic Approach to CM Points on Shimura Curves.*

**Peter Woolfitt** (Neil Lyall/Akos Magyar),* On k-degenerate Distance Graphs in Finite Fields and a Class of 2-degenerate Distance Graphs in the Integer Lattice .*

*2022*

**Komal Agrawal** (Paul Pollack), *On Some Problems Concerning Integer Recurring Sequences.*

**Arvind Suresh** (Dino Lorenzini), *Realizing Galois Representations in Abelian Varieties by Specialization.*

*2021*

**Matthew Just **(Paul Pollack), *Asymptotic Expansions for some Counting Functions in Group Theory.*

**Mentzelos Melistas **(Dino Lorenzini), *Reduction and Torsion Points of Abelian Varieties.*

**Makoto Suwama **(Dino Lorenzini), *Two Topics in Algebra and Number Theory.*

*2020*

**Kubra Benli **(Paul Pollack), *Three Topics in Analytic Number Theory.*

*2019*

**Noah Lebowitz-Lockard **(Paul Pollack), *The distribution of some special arithmetic functions.*

**Lori Watson** (Pete L. Clark),* Hasse Principle violations in twist families of hyperelliptic and superelliptic curves.*

*2018*

**Marko Milosevic** (Pete L. Clark), *Torsion of elliptic curves. *

*2017*

**Hans D. Parshall** (Neil Lyall/Akos Magyar),* **Point configurations over finite fields**.*

*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 Mora** (Pete L. Clark),* **On the index of genus one curves**.*

*2014*

**John Doyle** (Robert Rumely),* **Dynamics of quadratic polynomials over quadratic fields**.*

**Katherine Thompson** (Jonathan Hanke/Daniel Krashen) *Explicit Representation Results of Quadratic Forms Over Q and Q(sqrt(5)).*

*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 metrized graphs**.*

*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),** ***On the arithmetic of twists 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 triangles into integer lattices**.*

**Milton Nash** (R. Rumely),* **Special values of Hurwitz zeta functions and 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*

**Stephen Donnelly** (R. Rumely), *Finding 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.*

*1993*

**James Haglund **(R. Canfield), *Compositions, rook placements, and permutations of vectors.*

*1992*

**Renet Lovorn **(C. Pomerance), *Rigorous, Subexponential Algorithms for Discrete Logarithms Over Finite Fields*

*1955*

**David Walker** (G. Huff), *On the diophantine equation a^x-b^y=1.*

The first doctoral degrees in mathematics at the University of Georgia were awarded in 1951, and one of them was in number theory.

*1951*

**William D. Peeples** (G. Huff), *Elliptic curves and rational distance sets.*