Extensions of Birch-Merriman and Related Finiteness Theorems

Abstract: 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 $\mathrm{GL}_2(\mathbb{Z})$-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 $\mathrm{GL}_3(\mathbb{Z})$-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.

commutative algebraalgebraic geometrygroup theorynumber theoryrings and algebrasrepresentation theory

Audience: researchers in the topic

Calgary Algebra and Number Theory Seminar

Series comments: This seminar series is partially supported by the Pacific Institute for the Mathematical Sciences (PIMS).