Bounds for Completely Decomposable Jacobians

TL;DR

This paper establishes sharp bounds on the genus and rational points of curves over finite fields with completely decomposable Jacobians, using explicit formula methods, significantly improving previous bounds.

Contribution

It provides the first sharp bounds for genus and rational points of such curves, and relates Frobenius angles to the number of rational points over finite fields.

Findings

Curves over two elements have genus ≤ 26 and ≤ 6 rational points.

Previous genus bounds were reduced from 145 to 26.

A relation between Frobenius angles and rational points is established.

Abstract

A curve over the field of two elements with completely decomposable Jacobian is shown to have at most six rational points and genus at most 26. The bounds are sharp. The previous upper bound for the genus was 145. We also show that a curve over the field of $q$ elements with more than $q^{m/2}+1$ rational points has at least one Frobenius angle in the open interval $(\pi/m,3\pi/m)$. The proofs make use of the explicit formula method.