Algebraic Points on Varieties

This project centers on understanding the arithmetic of solutions to systems of polynomial equations, i.e., varieties. A key tool in the project is to use the limiting geometric structure of solutions of large complexity, thereby allowing the PI to study solutions of increasing complexity in a uniform manner. Understanding the arithmetic of varieties has many applications including to cryptography and to coding theory.

This project also funds mentoring and training of early career mathematicians, particularly those from groups who have been historically excluded from mathematics. In addition to training Ph.D. students at their own institution, the PI also co-organizes the Roots of Unity workshop series and the Women in Numbers conference series.

More specifically, the main research focus of the proposal is to organize and, in the case of a rank 0 curve, even describe all algebraic points on a curve. This includes characterizing the local splitting behavior of the residue fields of points that appear in a fixed linear system. In addition, the PI will use the Abel-Jacobi map to package all algebraic points on a curve with rank 0 Jacobian in terms of a finite set of complete linear systems.

This project builds on the PI's prior work on isolated and parameterized points and on degree sets over Henselian fields. The proposal also includes complementary projects that explore the behavior of algebraic points on surfaces. These complementary projects focus on particular classes of surfaces of negative Kodaira dimension and surfaces of Kodaira dimension 0 with a view to understanding the different phenomena that can arise for higher dimensional varieties.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.