$E_8$ and the average size of the 3-Selmer group of the Jacobian of a pointed genus-2 curve

TL;DR

This paper proves that the average size of the 3-Selmer group for genus-2 curves with a marked Weierstrass point is 4, providing new insights into the arithmetic of these curves.

Contribution

It establishes the average size of the 3-Selmer group for a specific class of genus-2 curves, linking it to the structure of the $E_8$ Lie group.

Findings

Average size of 3-Selmer group is 4

Results connect $E_8$ structure to genus-2 curves

Advances understanding of Selmer groups in algebraic geometry

Abstract

We prove that the average size of the 3-Selmer group of a genus-2 curve with a marked Weierstrass point is 4.