A new approach to classification of integral quadratic forms over dyadic local fields

Abstract

Abstract. In 1963, O'Meara solved the classification problem for lattices over dyadic local fields in terms of Jordan decompositions. In this paper we translate his result in terms of good BONGs. BONGs (bases of norm generators) were introduced in 2003 as a new way of describing lattices over dyadic local fields. This result and the notions we introduce here are a first step towards a solution of the more difficult problem of representations of lattices over dyadic fields.

Representations of quadratic lattices over dyadic local fields

“…that i+1 = n k + 1 < n k+1 . By [B3,Lemma 3.7(ii)] α i + α i+1 > 2e implies f k ⊂ 4a k+1 w −1 k+1 so by O'Meara's 93:28(iii) we get [B3,Lemma 3.7(iii)], α i + α i+1 > 2e implies that f k ⊂ 4a k+1 w −1 k+1 or f k+1 ⊂ 4a k+1 w −1 k+1 . The first case was treated above.…”

“…For the following cases, b. and c., we have both [B3,Lemma 2.7(iii) and Corollary 2.8(i)] it is enough to prove that…”

“…Corollary 2.10 Suppose that M, N satisfy the condition (i) of the main theorem. If [B3,Lemma 2.7(ii)]…”

“…is a norm generator for N p r k while a ′ is a norm generator for both M p r k and Ox. By [B3,Lemma 2.11] we have wM p r k = wN p r k + w(Ox)…”

Positive definite almost regular ternary quadratic forms over totally real number fields

“…One of the main problems in the arithmetic theory of quadratic forms is the representation problem that asks for a complete determination of the set Q(L). The first step is to determine the set of elements in F that are represented by all localizations of L. This local version of the representation problem is completely solved by O'Meara in the non-dyadic and 2-adic cases [12], and a solution for the general dyadic case is recently announced by Beli [1]. However, it is well known that the existence of local representations at all primes do not always guarantee a global representation.…”