Rational Points on the Fermat Cubic Surface

TL;DR

This paper establishes a lower bound for the number of rational points on the Fermat cubic surface that matches Manin's prediction and presents a counterexample to Manin's conjecture over the rationals.

Contribution

It provides the first lower bound aligning with Manin's prediction and offers a counterexample to the conjecture over the rationals.

Findings

Lower bound matches Manin's prediction

Counterexample to Manin's conjecture over rationals

Advances understanding of rational points on cubic surfaces

Abstract

We prove a lower bound that agrees with Manin's prediction for the number of rational points of bounded height on the Fermat cubic surface. As an application we provide a simple counterexample to Manin's conjecture over the rationals.