Introduction

This chapter deals with quadratic and hermitian spaces, and lattices there of. Note that even though quadratic spaces/lattices are theoretically a special case of hermitian spaces/lattices, a particular distinction is made here. As a note for knowledgeable users, only methods regarding hermitian spaces/lattices over degree 1 and degree 2 extensions of number fields are implemented up to now.

Definitions and vocabulary

We begin by collecting the necessary definitions and vocabulary. The terminology follows mainly [Kir16]

Quadratic and hermitian spaces

Let $K$ be a number field and let $E$ be a finitely generated etale algebra over $K$ of dimension 1 or 2, i.e. $E=K$ or $E$ is a separable extension of $K$ of degree 2. In both cases, $E/K$ is endowed with an $K$-linear involution $\bar{\phantom{x}}:E\to E$ for which $K$ is the fixed field (in the case $E=K$, this is simply the identity of $K$).

A hermitian space $V$ over $E/K$ is a finite-dimensional $E$-vector space, together with a sesquilinear (with respect to the involution of $E/K$) morphism $\mathrm{\Phi }:V\times V\to E$. In the trivial case $E=K$, $\mathrm{\Phi }$ is therefore a $K$-bilinear morphism and we called $(V,\mathrm{\Phi })$ a quadratic hermitian space over $K$.

We will always work with an implicit canonical basis $e_1,\dots ,e_n$ of $V$. In view of this, hermitian spaces over $E/K$ are in bijection with hermitian matrices with entries in $E$, with respect to the involution $\bar{\phantom{x}}$. In particular, there is a bijection between quadratic hermitian spaces over $K$ and symmetric matrices with entries in $K$. For any basis $B=(v_1,\dots ,v_n)$ of $(V,\mathrm{\Phi })$, we call the matrix $G_B=(\mathrm{\Phi }(v_i,v_j){)}_{1\le i,j\le n}\in E^{n\times n}$ the Gram matrix of $(V,\mathrm{\Phi })$ associated to $B$. If $B$ is the implicit fixed canonical basis of $(V,\mathrm{\Phi })$, we simply talk about the Gram matrix of $(V,\mathrm{\Phi })$.

For a hermitian space $V$, we refer to the field $E$ as the base ring of $V$ and to $\bar{\phantom{x}}$ as the involution of $V$. Meanwhile, the field $K$ is referred to as the fixed field of $V$.

By abuse of language, non-quadratic hermitian spaces are sometimes simply called hermitian spaces and, in contrast, quadratic hermitian spaces are called quadratic spaces. In a general context, an arbitrary space (quadratic or hermitian) is referred to as a space throughout this chapter.

Quadratic and hermitian lattices

Let $V$ be a space over $E/K$. A finitely generated ${\mathcal{O}}_E$-submodule $L$ of $V$ is called a hermitian lattice. By extension of vocabulary if $V$ is quadratic (i.e. $E=K$), $L$ is called a quadratic hermitian lattice. We call $V$ the ambient space of $L$ and $L{\otimes }_{{\mathcal{O}}_E}E$ the rational span of $L$.

For a hermitian lattice $L$, we refer to $E$ as the base field of $L$ and to the ring ${\mathcal{O}}_E$ as the base ring of $L$. We also call $\bar{\phantom{x}}:E\to E$ the involution of $L$. Finally, we refer to the field $K$ fixed by this involution as the fixed field of $L$ and to ${\mathcal{O}}_K$ as the fixed ring of $L$.

Once again by abuse of language, non-quadratic hermitian lattices are sometimes simply called hermitian lattices and quadratic lattices refer to quadratic hermitian lattices. Therefore, in a general context, an arbitrary lattice is referred to as a lattice in this chapter.

References

Many of the implemented algorithms for computing with quadratic and hermitian lattices over number fields are based on the Magma implementation of Markus Kirschmer, which can be found here.

Most of the definitions and results are taken from:

[Kir16] : Definite quadratic and hermitian forms with small class number. Habilitationsschrift. RWTH Aachen University, 2016. pdf

[Kir19] : Determinant groups of hermitian lattices over local fields, Archiv der Mathematik, 113 (2019), no. 4, 337–347. pdf