Cubic polynomials represented by norm forms

TL;DR

This paper proves that certain cubic equations represented by norm forms satisfy the Hasse principle, using sieve methods, under specific conditions on the polynomial and the number field.

Contribution

It establishes the Hasse principle for a class of cubic equations represented by norm forms, extending previous results with a novel sieve-based proof approach.

Findings

Hasse principle holds for specified cubic norm form equations

Sieve methods effectively prove local-global principles in this context

Results apply to irreducible cubics under certain hypotheses

Abstract

We show that for an irreducible cubic $f\in\mathbb Z[x]$ and a full norm form $\mathbf N(x_1,\ldots,x_k)$ for a number field $K/\mathbb Q$ satisfying certain hypotheses the variety $f(t)=\mathbf N(x_1,\ldots,x_k)\ne 0$ satisfies the Hasse principle. Our proof uses sieve methods.