Skip to content

Latest commit

 

History

History
103 lines (83 loc) · 4.78 KB

primembed.md

File metadata and controls

103 lines (83 loc) · 4.78 KB
Raw
CurrentModule = Oscar

We introduce here the necessary definitions and results which lie behind the methods about primitive embeddings. Most of the content is taken from Nik79.

Nikulin's theory on primitive embeddings

Primitive embeddings

Given an embedding $i\colon S\hookrightarrow T$ of non-degenerate integral integer lattices, we call $i$ primitive if its cokernel $T/i(S)$ is torsion free. Two primitive embeddings $i_1\colon S\hookrightarrow M_1$ and $i_2\colon S \hookrightarrow M_2$ of $S$ into two lattices $M_1$ and $M_2$ are called isomorphic if there exists an isometry $M_1 \to M_2$ which restricts to the identity of $S$. Moreover, if there exists an isometry between $M_1$ and $M_2$ which maps $S$ to itself (not necessarily identically), we say that $i_1$ and $i_2$ defines isomorphic primitive sublattices Nik79.

In his paper, V. V. Nikulin gives necessary and sufficient condition for an even integral lattice $M$ to embed primitively into an even unimodular lattice with given invariants (see Theorem 1.12.2 in Nik79). More generally, the author also provides methods to compute primitive embeddings of any even lattice into an even lattice of a given genus (see Proposition 1.15.1 in Nik79). In the latter proposition, it is explained how to classify such embeddings as isomorphic embeddings or as isomorphic sublattices. Moreover, with enough care, one can generalize the previous results for embeddings in odd lattices.

A general method to compute primitive embeddings between integral lattices can be algorithmically implemented, however it tends to be slow and inefficient in general for large rank or determinant. But, in the case where the discriminant groups are (elementary) $p$-groups, the method can be made more efficient.

We provide 4 kinds of output:

  • A boolean, which only returns whether there exists a primitive embedding;
  • A single primitive embedding as soon as the algorithm computes one;
  • A list of representatives of isomorphism classes of primitive embeddings;
  • A list of representatives of isomorphism classes of primitive sublattices.
primitive_embeddings(::ZZLat, ::ZZLat)

Note that the previous two functions require the first lattice of the input to be unique in its genus. Otherwise, one can specify a genus, or its invariants, as a first input:

primitive_embeddings(::ZZGenus, ::ZZLat)
primitive_embeddings(::TorQuadModule, ::Tuple{Int, Int}, ::ZZLat)

In order to compute such primitive embeddings of a lattice $M$ into a lattice $L$, we follow the proof of Proposition 1.15.1 of Nik79.

Note: for the implementation of the algorithm, we construct a lattice $T$ which is unique in its genus, such that $D_T$ and $D_M(-1)$ are isometric and $O(T)\to O(D_T)$ is surjective. We then classify all primitive extensions of $M\oplus T$ modulo $O(D_T)$ (and modulo $O(M)$ for a classification of primitive sublattices). To classify such primitive extensions, we use Proposition 1.5.1 of Nik79:

primitive_extensions(::ZZLat, ::ZZLat)

We recall that a primitive extension of the orthogonal direct sum of two integral integer lattices $M$ and $N$ is an overlattice $L$ of $M\oplus N$ such that both $M$ and $N$ embed primitively in $L$ (via the natural embeddings $M,N \to M\oplus N\subseteq L$). Such primitive extensions are obtained, and classified, by classifying gluings between anti-isometric subgroups of the respective discriminant groups of $M$ and $N$. The construction of an overlattice is determined by the graph of such gluing.

Equivariant primitive extensions

An equivariant primitive extension of a pair of integer lattices with isometries $(M, f_M)$ and $(N, f_N)$ is a primitive extension of $M$ and $N$ obtained by gluing two subgroups which are respectively $D_{f_M}$ and $D_{f_N}$ stable along a glue map which commutes with these two actions. If such a gluing exists, then the overlattice $L$ of $M\oplus N$ is equipped with an isometry $f_L$ which preserves both $M$ and $N$, and restricts to $f_M$ and $f_N$ respectively.

equivariant_primitive_extensions(::Union{ZZLatWithIsom, ZZLat}, ::Union{ZZLatWithIsom, ZZLat})

Admissible equivariant primitive extensions

The following function is a major tool provided by BH23. Given a triple of even integer lattices with isometry $((A, a), (B, b), (C, c))$ and two prime numbers $p$ and $q$ (possibly equal), if $(A, B, C)$ is $p$-admissible, this function returns representatives of isomorphism classes of equivariant primitive extensions $(A, a)\oplus (B, b)\to (D, d)$ such that the type of $(D, d^q)$ is equal to the type of $(C, c)$ (see type(::ZZLatWithIsom)).

admissible_equivariant_primitive_extensions(::ZZLatWithIsom, ::ZZLatWithIsom, ::ZZLatWithIsom, ::IntegerUnion, ::IntegerUnion)