Polymath Jr. 2021

Diophantine m-tuples and elliptic curves

Project lead by Seoyoung Kim, Queen's University (Canada, Ontario)


joint with Professor Steve J. Miller and grad. mentor Trajan Hammonds.


A Diophantine m-tuple, first studied by Diophantus of Alexandria, is a set of integers with the property of the product any two distinct elements is one less than a square, such as {1,3,8,120}. We are going to explore Diophantine m-tuples using the theory of elliptic curves and K3 surfaces, and moreover, aim to find new instances of Bias conjecture. In his thesis, Professor Steve J. Miller found biases in the distribution of coefficients of the L-series in families of elliptic curves. Furthermore, in SMALL, the results have extended to several other families; the goal is to continue to explore this.

        1. Miller, Steven Joel 1- and 2-level densities for families of elliptic curves: Evidence for the underlying group symmetries. Thesis (Ph.D.)–Princeton University. 2002. pdf

        2. Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families (with Blake Mackall, Christina Rapti and Karl Winsor), Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures (David Kohel and Igor Shparlinski, editors), Contemporary Mathematics 663, AMS, Providence, RI 2016. pdf

        3. Lower-Order Biases Second Moments of Dirichlet Coefficients in Families of L-Functions (with Megumi Asada, Ryan Chen, Eva Fourakis, Yujin Kim, Andrew Kwon, Jared Lichtman, Blake Mackall, Eric Winsor, Karl Winsor, Jianing Yang, Kevin Yang; appendices with Roger Weng and Michelle Wu), to appear in Experimental Mathematics, pdf

        4. Rank and Bias in Families of Hyperelliptic Curves via Nagao's Conjecture (with Trajan Hammonds, Seoyoung Kim, Benjamin Logsdon, Alvaro Lozano-Robledo), to appear in the Journal of Number Theory pdf (arxiv) or pdf

        5. Biases in Moments of the Dirichlet Coefficients in One- and Two-Parameter Families of Elliptic Curves (with Y. Yeng and an appendix by J. Wu), submitted to The PUMP Journal of Undergraduate Research. pdf

        6. Diophantine triples and K3 surfaces, by Matija Kazalicki, Bartosz Naskręcki. pdf


Feel free to contact me via sk206@queensu.ca if you want to more about the project!