Generalised quadratic forms over totally real number fields

TL;DR

This paper introduces a new class of generalized quadratic forms over totally real number fields, linking their arithmetic to systems of quadrics over rationals using a number field version of the Hardy-Littlewood circle method.

Contribution

It develops a novel framework for generalized quadratic forms over totally real fields and connects it to classical quadratic systems via advanced analytic methods.

Findings

New class of quadratic forms over totally real fields

Connection established between these forms and rational quadratic systems

Application of Hardy-Littlewood circle method over number fields

Abstract

We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy-Littlewood circle method over number fields.