Extensions of Birch–Merriman and Related Finiteness Theorems

A classical theorem of Birch and Merriman states that, for fixed 𝑛the set of integral binary 𝑛-ic forms with fixed nonzero discriminant breaks into finitely many GL2(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 GL3(ℤ)-orbits with fixed nonzero discriminant. We also prove a similar result for a 27-dimensional representation associated with a family of 𝐾3surfaces.

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.