Hasse-Minkowski theorem for quadratic forms on groups

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.

Theorems & Definitions (15)

• Lemma 1
• Theorem 2
• Theorem 3
• Proposition 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• proof : Proof of Lemma \ref{['2and3']}
• proof : Proof of Theorem \ref{['thm for 2']}