My research interests are centered around effective methods in arithmetic geometry, arithmetic statistics, and number theory.
Ongoing Projects
- L-functions and Modular Forms Database (LMFDB)
- A census of zeta functions of quartic K3 surfaces over $\mathbf{F}_3$ (with David Harvey)
- Effective computations of Bernoulli numbers (with David Harvey)
- Counting points on singular plane curves (with David Harvey)
- Computation of zeta functions of nondegenerate projective/toric hypersurfaces via controlled reduction in $p$-adic cohomology (with David Harvey and Kiran Kedlaya) see also the following associated software packages:
Publications
-
Computing Zeta Functions of Cyclic Covers in Large Characteristic
(with
Vishal Arul,
Alex Best,
Richard Magner,
and
Nicholas Triantafillou
)
presented at ANTS (Algorithmic Number Theory Symposium)
arXiv:1806.02262, SageMath implementation -
Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in $p$-adic cohomology
(with David Harvey and Kiran Kedlaya)
presented at ANTS (Algorithmic Number Theory Symposium)
arXiv:1806.00368 -
Rigorous computation of the endomorphism ring of a Jacobian
(with
Nicolas Mascot,
Jeroen Sijsling, and
John Voight
)
to appear in Mathematics of Computation.
arXiv:1705.09248, SageMath/Magma implementation -
On the arithmetic of a family of degree-two K3 surfaces
(with
Florian Bouyer,
Dino Festi,
Chris Nicholls, and
Mckenzie West
)
Mathematical Proceedings (2018).
DOI, arXiv:1703.02127 -
On the distribution of the Picard ranks of the reductions of a
K3 surface
(with
Andreas-Stephan Elsenhans and
Jörg Jahnel
)
submitted
arXiv:1610.07823. -
Traces, High powers and One level density for families of curves over finite fields
(with
Alina Bucur,
Chantal David,
João Guerreiro,
David Lowry-Duda
)
Mathematical Proceedings 162(2) (2018).
DOI, arXiv:1610.00164. -
Effective computations of Hasse-Weil zeta functions
Ph.D. thesis pdf -
Variation of Néron-Severi ranks of reductions of K3 surfaces (with Yuri Tschinkel)
Experimental Mathematics 23 (2014), 475-481.
DOI, arXiv:1405.2265, raw data. -
Asymptotic expansions, $L$-values and a new Quantum Modular Form (with Korneel Debaene and João Guerreiro)
The Ramanujan Journal 35 (2014), 141-148.
DOI, arXiv:1311.3469. -
A search for Wilson primes (with Robert Gerbicz
and David Harvey)
Mathematics of Computation 82 (2014), 3071-3091.
DOI, arXiv:1209.3436. -
Faster deterministic integer factorization (with David Harvey)
Mathematics of Computation 83 (2014), 339-345.
DOI, arXiv:1201.2116.
Software
- L-functions and Modular Forms Database (LMFDB)
- Controlled reduction: Computation of zeta functions of projective hypersurfaces (C++) and its SageMath wrapper
- Endomorphisms: Rigorous computation of the endomorphism ring of a Jacobian (SageMath/Magma)
Talks
-
Frobenius Distributions
(slides)
Department of Mathematics, University of Washington, April 2019. -
Upper bounds for the endomorphism algebra of an abelian variety
(slides)
JMM, AMS Special Session on Number Theory, Arithmetic Geometry, and Computation, January 2019 -
Randomness in number theory
(slides)
Department of Mathematics, Colorado Sate University, November 2018 -
Frobenius Distributions
(slides)
Colóquio de Geometria e Aritmética Rio de Janeiro, October 2018. -
Frobenius Distributions
(slides)
Department of Mathematics, Massachusetts Institute of Technology, September 2018. -
Computing zeta functions of nondegenerate hypersurfaces in toric varieties
(slides)
Algorithmic Number Theory Symposium XIII, University of Wisconsin, Madison, July 2018. -
Distributions in arithmetic geometry
(slides)
Communicating Mathematics Effectively, University of Washington, June 2018. -
Computing zeta functions of nondegenerate hypersurfaces in toric varieties
(slides)
Birational Geometry and Arithmetic, ICERM, Brown University, May 2018. -
Computing zeta functions of nondegenerate toric hypersurfaces
(slides)
Department of Mathematics & Statistics, Boston University, December 2017. -
Computing zeta functions of nondegenerate toric hypersurfaces
(slides)
Maine-Québec Number Theory Conference, October 2017 -
Computing zeta functions of nondegenerate toric hypersurfaces
(slides, video)
Banff International Research Station, $p$-adic Cohomology and Arithmetic Applications, October 2017 -
Toric point counting and applications (slides, video)
Banff International Research Station, New Trends in Arithmetic and Geometry of Algebraic Surfaces, March 2017 -
Toric point counting and applications
Department of Mathematics & Statistics, University of Vermont, February 2017 -
Counting points on curves (slides)
Université Laval, Québec, Québec-Maine Number Theory Conference, October 2016 -
Counting points on curves (slides)
University of New South Wales, School of Mathematics and Statistics, July 2016 -
Zeta functions of quartic K3 surfaces over $\mathbf{F}_3$
(slides)
Sage Days 71: Explicit p-adic methods in number theory, Oxford University, March 2016 -
Equidistributions in arithmetic geometry
(slides)
Department of Mathematics, Dartmouth College, Jan 2016 -
Equidistributions in arithmetic geometry
(slides)
Departamento de Matemática, Instituto Superior Técnico, Dec 2015 -
Effective computations of Hasse-Weil zeta functions
(slides)
Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives, ICERM, Brown University, Oct 2015 -
Effective computations of Hasse-Weil zeta functions
Courant Institute of Mathematical Sciences, NYU, May 2015 -
Variation of Néron-Severi ranks of reductions of algebraic surfaces
Department of Mathematics, Rice University, Feb 2015 -
Variation of Néron-Severi ranks of reductions of algebraic surfaces
School of Mathematics and Statistics, UNSW, Oct 2014 -
Variation of Néron-Severi ranks of reductions of algebraic surfaces
Courant Institute of Mathematical Sciences, NYU, Oct 2014 -
The point counting problem
Courant Institute of Mathematical Sciences, NYU, Dec 2012 -
Faster deterministic integer factorization
Courant Institute of Mathematical Sciences, NYU, Jan 2012 -
Étale Cohomology
Courant Institute of Mathematical Sciences, NYU, Mar 2011 -
Étale Cohomology
Departamento de Matemática, Instituto Superior Técnico, July 2010