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

A hermitian space VV over E/KE/K is a finite-dimensional EE-vector space, together with a sesquilinear (with respect to the involution of E/KE/K) morphism Φ ⁣:V×VE\Phi \colon V \times V \to E. In the trivial case E=KE=K, Φ\Phi is therefore a KK-bilinear morphism and we called (V,Φ)(V, \Phi) a quadratic hermitian space over KK.

We will always work with an implicit canonical basis e1,,ene_1, \ldots, e_n of VV. In view of this, hermitian spaces over E/KE/K are in bijection with hermitian matrices with entries in EE, with respect to the involution x\overline{\phantom{x}}. In particular, there is a bijection between quadratic hermitian spaces over KK and symmetric matrices with entries in KK. For any basis B=(v1,,vn)B = (v_1, \ldots, v_n) of (V,Φ)(V, \Phi), we call the matrix GB=(Φ(vi,vj))1i,jnEn×nG_B = (\Phi(v_i, v_j))_{1 \leq i, j \leq n} \in E^{n \times n} the Gram matrix of (V,Φ)(V, \Phi) associated to BB. If BB is the implicit fixed canonical basis of (V,Φ)(V, \Phi), we simply talk about the Gram matrix of (V,Φ)(V, \Phi).

For a hermitian space VV, we refer to the field EE as the base ring of VV and to x\overline{\phantom{x}} as the involution of VV. Meanwhile, the field KK is referred to as the fixed field of VV.

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 VV be a space over E/KE/K. A finitely generated OE\mathcal O_E-submodule LL of VV is called a hermitian lattice. By extension of vocabulary if VV is quadratic (i.e. E=KE=K), LL is called a quadratic hermitian lattice. We call VV the ambient space of LL and LOEEL\otimes_{\mathcal O_E} E the rational span of LL.

For a hermitian lattice LL, we refer to EE as the base field of LL and to the ring OE\mathcal O_E as the base ring of LL. We also call x ⁣:EE\overline{\phantom{x}} \colon E \to E the involution of LL. Finally, we refer to the field KK fixed by this involution as the fixed field of LL and to OK\mathcal O_K as the fixed ring of LL.

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