A positive proportion of plane cubics fail the Hasse principle

TL;DR

This paper demonstrates that a positive proportion of plane cubic forms over integers fail the Hasse principle, while also showing that many satisfy it, revealing significant distribution patterns among genus one models.

Contribution

It provides the first positive proportion results for failures and successes of the Hasse principle among various genus one models over integers.

Findings

A positive proportion of plane cubics fail the Hasse principle.

A positive proportion of genus one models satisfy the Hasse principle.

Results extend to binary quartics and intersections of quadrics.

Abstract

When all ternary cubic forms over $\mathbb Z$ are ordered by the heights of their coefficients, we show that a positive proportion of them fail the Hasse principle, i.e., they have a zero over every completion of $\mathbb Q$ but no zero over $\mathbb Q$. We also show that a positive proportion of all ternary cubic forms over $\mathbb Z$ nontrivially satisfy the Hasse principle, i.e., they possess a zero over every completion of $\mathbb Q$ and also possess a zero over $\mathbb Q$. Analogous results are proven for other genus one models, namely, for equations of the form $z^2=f(x,y)$ where $f$ is a binary quartic form over $\mathbb Z$, and for intersections of pairs of quadrics in $\mathbb P^3$.