Christopher Keyes

Research Interests

I am interested in arithmetic statistics, the study of the distribution of arithmetic objects including primes, number fields, points on varieties, and families of equations.

Lately, my research has centered on questions of the form how often does an equation of a given shape have (certain kinds of) solutions? Some "shapes" of equations I find myself think about include hyperelliptic and superelliptic curves, hypersurfaces in projective space, and stacky curves arising from generalized Fermat equations. The solutions of interest are typically rational or integral points, but higher degree points and local (i.e. p-adic) solutions are also interesting.

Poster advertising our paper

Publications and Preprints

  1. Local-global principles for stacky curves (joint with Juanita Duque-Rosero, Andrew Kobin, Manami Roy, Soumya Sankar, and Yidi Wang). In preparation.
  2. How often does a cubic hypersurface have a rational point? (joint with Lea Beneish) Submitted. (arxiv, code)
  3. Fields generated by points on superelliptic curves (joint with Lea Beneish). Submitted. (arxiv)
  4. On the density of locally soluble superelliptic curves (joint with Lea Beneish). Finite Fields and Their Applications, Volume 85, 102128, 2023. (arxiv, journal, code, slides)
  5. Mertens' theorem for Chebotarev sets (joint with Santiago Arango-Piñeros and Daniel Keliher). International Journal of Number Theory, Volume 18, Issue 8, 2022. (journal, arxiv)
  6. Growth of points on hyperelliptic curves. Journal de Théorie des Nombres de Bordeaux, Volume 34, Issue 1, pp. 271 - 194, 2022. (journal, arxiv)
  7. Bounding the number of arithmetical structures on graphs (joint with Tomer Reiter). Discrete Mathematics, Volume 344, Issue 9, 2021. (journal, arxiv, slides, video)

Invited Talks

  1. TBD. Spring of Rational Points II, University of Bath, May 13, 2025.
  2. TBD. Great Western Number Theory Seminar, University of Reading, April 4, 2025.
  3. Towards Artin's conjecture on p-adic quintic forms. Manchester Number Theory Seminar, University of Manchester, March 18, 2025.
  4. Towards Artin's conjecture on p-adic quintic forms. London Number Theory Seminar, University College London, March 5, 2025.
  5. How often does a cubic hypersurface have a point? Number Theory Seminar, University of Illinois Chicago, October 15, 2024.
  6. How often does a cubic hypersurface have a point? Arithmetic Geometry Seminar, University of Bath, May 21, 2024.
  7. How often does a cubic hypersurface have a point? Utrecht Geometry Centre Seminar, Utrecht University, April 9, 2024.
  8. How often does a cubic hypersurface have a point? Number Theory Seminar, University of Cambridge, February 20, 2024.
  9. Explicit local solubility in families of varieties. Number Theory Seminar, King's College London, November 30, 2023.
  10. Explicit local solubility in families of varieties. Linfoot Seminar, University of Bristol, November 29, 2023.
  11. Local solubility in families of superelliptic curves. Algebra Seminar, University of North Texas, September 15, 2023.
  12. Local solubility in families of superelliptic curves. Number Theory Seminar, UC Irvine, January 19, 2023.
  13. Local solubility in families of superelliptic curves. AMS Special Session on Arithmetic Statistics, Joint Math Meetings, January 4 - 7, 2023.
  14. Local solubility in families of superelliptic curves. Number Theory Seminar, UC San Diego, December 1, 2022.
  15. Local solubility in families of superelliptic curves. Number Theory Seminar, Ohio State University, November 21, 2022.
  16. Local solubility in families of superelliptic curves. Number Theory Seminar, University of Georgia, November 2, 2022.
  17. Local solubility in families of superelliptic curves. Arithmetic Geometry and Number Theory Seminar, UC Berkeley, October 24, 2022.
  18. Local solubility in families of superelliptic curves. Algebra, Geometry, and Number Theory Seminar, University of South Carolina, April 8, 2022.
  19. On the proportion of everywhere locally soluble superelliptic curves. (Secret) Algebra, Geometry, and Number Theory Seminar, Tufts University, November 18, 2021. (slides)
  20. Fields generated by points on superelliptic curves (joint talk with Lea Beneish). UW Number Theory Seminar, University of Washington (held virtually), June 8, 2021.

Contributed Talks

  1. Local solubility in families of superelliptic curves. Connecticut Number Theory Conference (CTNT), University of Connecticut, June 10, 2022.
  2. On the proportion of everywhere locally soluble superelliptic curves. Upstate Number Theory Conference, Union College, October 23, 2021.
  3. Fields generated by points on superelliptic curves. Young Researchers in Number Theory (Y-RANT), University of Bristol (held virtually), August 20, 2021.
  4. Mertens' product theorem for primes in Chebotarev sets. Front Range Number Theory Day (held virtually), April 24, 2021.
  5. An upper bound for the number of arithmetical structures on a graph. Mid-Atlantic Seminar on Numbers (MASON) V (held virtually), March 27, 2021.
  6. An upper bound for the number of arithmetical structures on a graph. PAlmetto Joint Arithmetic, Modularity, and Analysis Series (PAJAMAS), University of South Carolina (held virtually), December 6, 2020. (slides)
  7. Growth of points on hyperelliptic curves. Tufts Undergraduate Research Symposium, Tufts University, May 3, 2018.
  8. Growth of points on hyperelliptic curves. PAlmetto Number Theory Series (PANTS) XXVIII, University of Tennessee Knoxville, September 17, 2017.

Travel, past and future

Oct/Nov 2025 Focused Research Group, BIRS, Banff, Canada
September 2025 Geometric and Analytic Number Theory, University of Bath, UK
June/July 2025 Research in Groups, ICMS, Edinburgh, UK
June 2025 Research in Residence, CIRM, Luminy, France
January 2025 Joint Math Meetings, Seattle, WA, USA
December 2024 Research in Teams, BIRS, Banff, Canada
June 2024 Research in Groups, ICMS, Edinburgh, UK
April 2024 Utrecht Geometry Seminar, Utrecht, Netherlands
June 2023 MRC on Stacks, Java Center, NY, USA
May 2023 INTEGERS Conference, Athens, GA, USA
March 2023 Arizona Winter School, Tucson, AZ, USA
February 2023 MSRI Introductory Workshop on Diophantine Geometry, Berkeley, CA, USA
January 2023 Joint Math Meetings, Boston, MA, USA