UCalgary Algebra and Number Theory Seminar: Fatemehzahra Janbazi

A classical theorem of Birch and Merriman states that, for fixed $n$the set of integral binary $n$-ic forms with fixed nonzero discriminant breaks into finitely many ${\mathrm{G}\mathrm{L}}_2(\mathbb{Z})$-orbits. In this talk, I’ll present several extensions of this finiteness result. 

In joint work with Arul Shankar, we study a representation-theoretic generalization to ternary $n$-ic forms and prove analogous finiteness theorems for ${\mathrm{G}\mathrm{L}}_3(\mathbb{Z})$-orbits with fixed nonzero discriminant. We also prove a similar result for a 27-dimensional representation associated with a family of $K3$surfaces. 

In joint work with Sajadi, we take a geometric perspective and prove a finiteness theorem for Galois-invariant point configurations on arbitrary smooth curves with controlled reduction. This result unifies classical finiteness theorems of Birch–Merriman, Siegel, and Faltings.