# Hasse-Minkowski theorem for quadratic forms on groups

## TL;DR

This paper extends the Hasse-Minkowski theorem to quadratic forms on certain groups related to number fields, proving it holds for ranks 2 and 3 but not higher, thus generalizing classical results.

## Contribution

It establishes the validity of the Hasse-Minkowski theorem for quadratic forms of rank 2 and 3 on specific algebraic groups, and provides counterexamples for higher ranks.

## Key findings

- Hasse-Minkowski theorem holds for rank 2 and 3 forms.
- Counterexamples exist for ranks higher than 3.
- Generalizes classical theorem for binary and ternary forms.

## Abstract

Consider groups such as Mordell-Weil groups of abelian varieties over number fields, odd algebraic $K$-theory groups of number fields, or finitely generated subgroups of the multiplicative groups of number fields. They are all equipped with systems of reduction maps; thus, one can investigate the Hasse-Minkowski theorem for quadratic forms with coefficients in such groups. In this paper, we prove that the theorem holds for the forms whose rank equals $2$ or $3$, and we demonstrate that it does not hold for higher ranks by providing a counterexample. We also show that our results constitute a generalization of the classic Hasse-Minkowski theorem for binary and ternary integral forms.