[Paper Review] An arithmetic invariant theory of curves from $E_8$

Avinash Kulkarni|arXiv (Cornell University)|Nov 24, 2017

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper establishes a correspondence between elements of the 2-torsion Selmer group of the Jacobian of a uniquely trigonal genus 4 curve with a rational ramified point and orbits of the split adjoint group of type $E_8$ acting on a specific variety. Using techniques from arithmetic invariant theory, it generalizes earlier results for $E_7$ and hyperelliptic curves, providing a canonical, characteristic-0 invariant-theoretic framework for studying rational points on such curves via $E_8$-representations.

ABSTRACT

Let $k$ be a field of characteristic 0, let $C/k$ be a uniquely trigonal genus 4 curve, and let $P \in C(k)$ be a simply ramified point of the uniquely trigonal morphism. We construct an assignment of an orbit of an algebraic group of type $E_8$ acting on a specific variety to each element of $J_C(k)/2$. The algebraic group and variety are independent of the choice of $(C,P)$. We also construct a similar identification for uniquely trigonal genus 4 curves $C$ with $P \in C(k)$ a totally ramified point of the trigonal morphism. Our assignments are analogous to the assignment of a genus 3 curve with a rational point $(C,P)$ to an orbit of an algebraic group of type $E_7$ exhibited by Jack Thorne. Our assignment is also analogous to one constructed by Bhargava and Gross, who use it determine average ranks of hyperelliptic Jacobians.

Motivation & Objective

• To extend arithmetic invariant theory to trigonal genus 4 curves with rational ramified points using $E_8$-representations.
• To generalize Jack Thorne's $E_7$-based correspondence for plane quartics to the $E_8$ case.
• To construct a canonical assignment of $J_C(k)/2$ to $G(k)$-orbits in a variety associated with $E_8$, independent of curve choice.
• To provide a framework analogous to Bhargava and Gross’s work on hyperelliptic curves, but for trigonal curves.
• To lay groundwork for future study of average Selmer group sizes in this family.

Proposed method

• Adapts Thorne’s method of constructing $G(k)$-orbits from Galois cohomology via theta groups and isomorphisms of standard tuples.
• Uses the Weil pairing and a lattice $\Lambda$ of type $E_8$ to identify $J_C[2] \cong \Lambda/2\Lambda$ as a $k$-group.
• Constructs a canonical isomorphism between theta groups associated to a point $A \in J_C(k)$ and its 2-torsion lift $B$ with $[2]B = A$, defined over $k^{\mathrm{sep}}$.
• Applies a cocycle construction via the Galois action on the isomorphism $F$, identifying the image with $[B^\sigma - B]$, linking to cohomology.
• Uses the canonical identification $\operatorname{Hom}(\Lambda, \mathbb{G}_m)^{\mathrm{rss}}(k) \to T_0(k)$ to define a $G(k)$-orbit in $X_x(k)$ via $\varphi_0^{-1}(\kappa_C^0)$.
• Establishes a bijection between $G(k)\backslash X_x(k)$ and $H^1(k, Z_G(\varphi_0^{-1}(\kappa_C^0)))$, which maps to $H^1(k, J_C[2])$ via $J_C(k)/2$.

Experimental results

Research questions

• RQ1 Can the arithmetic of uniquely trigonal genus 4 curves with a rational ramified point be encoded via $E_8$-group orbits in a way analogous to $E_7$ for plane quartics?
• RQ2 How does the $2$-Selmer group of the Jacobian of such curves relate to the orbit structure of an $E_8$-representation?
• RQ3 Is there a canonical, $k$-independent assignment of elements in $J_C(k)/2$ to $G(k)$-orbits in a variety associated with $E_8$?
• RQ4 What is the role of the Weil pairing and theta group isomorphisms in constructing this correspondence over $k^{\mathrm{sep}}$?
• RQ5 How does this construction generalize Bhargava and Gross’s work on hyperelliptic curves to non-hyperelliptic families?

Key findings

• The paper constructs a canonical, $k$-independent assignment of each element of $J_C(k)/2$ to a $G(k)$-orbit in a variety $X_x$, where $G$ is the split adjoint group of type $E_8$.
• The assignment is injective and factors through the coboundary map $\delta: J_C(k)/2 \to H^1(k, J_C[2])$, establishing a cohomological link.
• The orbit construction is independent of the choice of rational point $P$ or lift $B$ with $[2]B = A$, due to canonical isomorphisms over $k^{\mathrm{sep}}$.
• The map from $J_C(k)/2$ to $G(k)\backslash X_x(k)$ is realized via a Galois cocycle $\sigma \mapsto F^{-1}F^\sigma$, where $F$ is induced by the isomorphism of theta groups.
• The construction generalizes Thorne’s $E_7$-based correspondence for genus 3 curves to the $E_8$-case for genus 4 trigonal curves.
• The method provides a framework that could be used to compute or bound the average size of the $2$-Selmer group of such Jacobians in future work.