Number/Representation Theory

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 will present several extensions of this finiteness result from both the representation theoretic and the geometric perspectives, along with an application of these generalizations.

On the representation theoretic side, we study ternary $n$-ic forms and prove analogous finiteness theorems for $\mathrm{GL}{_3}(\mathbb{Z})$ orbits with fixed nonzero discriminant. We also establish a corresponding result for a $27$ dimensional representation associated with a family of K3 surfaces (joint with A. Shankar).

On the geometric side, we prove a finiteness theorem for Galois invariant point configurations on arbitrary smooth curves with controlled reduction, unifying the classical finiteness theorems of Birch and Merriman, Siegel, and Faltings (joint with F. Sajadi).

As an application, we consider families of pencils of curves with fixed discriminant, where the representation theoretic and geometric extensions, together with the Birch and Merriman theorem, play a central role in establishing finiteness.