A positive proportion of plane cubics fail the Hasse principle

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 ${\mathbb{P}}^3$.

15 pages

References in corpus (2)

• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• The average number of elements in the 4-Selmer groups of elliptic curves is 7

Cited by in corpus (8)

• Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces
• Obstructions to the Hasse principle in families
• Many cubic surfaces contain rational points
• A positive proportion of cubic curves over Q admit linear determinantal representations
• Number fields with prescribed norms
• Rational points on del Pezzo surfaces of degree four
• The Hasse norm principle for abelian extensions
• Height of rational points on random Fano hypersurfaces