**Project leader:** Professor Jayce Getz**Project manager:** Chung-Ru Lee**Team members: **Trung Can, Ben Nativi, and Gary Zhou

**Project summary: **Consider a finite set of polynomials in several variables with rational coefficients. The set of tuples of complex numbers where the polynomials all vanish is called its zero locus. If the zero locus is nonempty we can ask if the zero locus contains a point with entries that are all rational numbers. If this is so one says the zero locus has a rational point. In this project the three undergraduate students and their graduate student mentor proved the existence of rational points for certain special zero loci that are equipped with actions of general linear, orthogonal, or symplectic groups. This has applications to both relative trace formulae and arithmetic invariant theory, and generalizes work of Jacquet and Friedberg and J. Thorne.