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Fix and unify docs about maximal lattices #1307

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Fix and unify docs about maximal lattices #1307

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ddfa6c6
fix and unify docs about maximal lattices
StevellM Dec 2, 2023
077bfdd
Update Lattices.jl
StevellM Dec 2, 2023
ca442e6
Apply suggestions from code review
simonbrandhorst Dec 4, 2023
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10 changes: 7 additions & 3 deletions src/QuadForm/Herm/Lattices.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -680,12 +680,13 @@ function _maximal_integral_lattice(L::HermLat, p, minimal = true)
end

function is_maximal_integral(L::HermLat, p)
@req order(p) == fixed_ring(L) "The ideal does not belong to the fixed ring of the lattice"
valuation(norm(L), p) < 0 && return false, L
return _maximal_integral_lattice(L, p, true)
end

function is_maximal_integral(L::HermLat)
!is_integral(norm(L)) && error("The lattice is not integral")
!is_integral(norm(L)) && return false, L
S = base_ring(L)
f = factor(discriminant(S))
ff = factor(norm(volume(L)))
Expand All @@ -703,6 +704,8 @@ function is_maximal_integral(L::HermLat)
end

function is_maximal(L::HermLat, p)
@req order(p) == fixed_ring(L) "The ideal does not belong to the fixed ring of the lattice"
@req valuation(norm(L), p) >= 0 "The norm of the lattice is not locally integral"
#iszero(L) && error("The lattice must be non-zero")
v = valuation(norm(L), p)
x = elem_in_nf(p_uniformizer(p))^(-v)
Expand All @@ -715,7 +718,7 @@ function is_maximal(L::HermLat, p)
end

function maximal_integral_lattice(L::HermLat)
!is_integral(norm(L)) && error("The lattice is not integral")
@req is_integral(norm(L)) "The norm of the lattice is not integral"
S = base_ring(L)
f = factor(discriminant(S))
ff = factor(norm(volume(L)))
Expand All @@ -730,7 +733,8 @@ function maximal_integral_lattice(L::HermLat)
end

function maximal_integral_lattice(L::HermLat, p)
valuation(norm(L), p) < 0 && error("Lattice is not locally integral")
@req order(p) == fixed_ring(L) "The ideals does not belong to the fixed ring of the lattice"
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@req valuation(norm(L), p) >= 0 "The norm of the lattice is not locally integral"
_, L = _maximal_integral_lattice(L, p, false)
return L
end
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37 changes: 24 additions & 13 deletions src/QuadForm/Lattices.jl
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Expand Up @@ -1989,45 +1989,56 @@ end
is_maximal_integral(L::AbstractLat, p::NfOrdIdl) -> Bool, AbstractLat

Given a lattice `L` and a prime ideal `p` of the fixed ring $\mathcal O_K$ of
`L`, return whether the completion of `L` at `p` is maximal integral. If it is
not the case, the second returned value is a lattice in the ambient space of `L`
whose completion at `p` is a minimal overlattice of $L_p$.
`L`, return whether the completion of `L` at `p` has integral norm and has no
proper overlattice satisfying this property.
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If the norm of `L` is not integral at `p`, the second output is `L` by default.
Otherwise, either `L` is maximal at `p` and the second output is `L`, or the
second output is a lattice `M` in the ambient space of `L` whose completion
at `p` is a minimal overlattice of $L_p$ with integral norm.
"""
is_maximal_integral(::AbstractLat, p)

@doc raw"""
is_maximal_integral(L::AbstractLat) -> Bool, AbstractLat

Given a lattice `L`, return whether `L` is maximal integral. If it is not,
the second returned value is a minimal overlattice of `L` with integral norm.
Given a lattice `L`, return whether `L` has integral norm and has no proper
overlattice satisfying this property.

If the norm of `L` is not integral, the second output is `L` by default.
Otherwise, either `L` is maximal and the second output is `L`, or the second
output is a minimal overlattice `M` of `L` with integral norm.
"""
is_maximal_integral(::AbstractLat)

@doc raw"""
is_maximal(L::AbstractLat, p::NfOrdIdl) -> Bool, AbstractLat

Given a lattice `L` and a prime ideal `p` in the fixed ring $\mathcal O_K$ of
`L`, check whether the norm of $L_p$ is integral and return whether `L` is maximal
at `p`. If it is locally integral but not locally maximal, the second returned value
is a lattice in the same ambient space of `L` whose completion at `p` has integral norm
and is a proper overlattice of $L_p$.
`L` such that the norm of $L_p$ is integral, return whether `L` is maximal
integral at `p`.

If `L` is locally maximal at `p`, the second output is `L`, otherwise it is
a lattice `M` in the same ambient space of `L` whose completion at `p` has
integral norm and is a proper overlattice of $L_p$.
"""
is_maximal(::AbstractLat, p)

@doc raw"""
maximal_integral_lattice(L::AbstractLat, p::NfOrdIdl) -> AbstractLat

Given a lattice `L` and a prime ideal `p` of the fixed ring $\mathcal O_K$ of
`L`, return a lattice `M` in the ambient space of `L` which is maximal integral
at `p` and which agrees with `L` locally at all the places different from `p`.
`L` such that the norm of $L_p$ is integral, return a lattice `M` in the
ambient space of `L` which is maximal integral at `p` and which agrees
with `L` locally at all the places different from `p`.
"""
maximal_integral_lattice(::AbstractLat, p)

@doc raw"""
maximal_integral_lattice(L::AbstractLat) -> AbstractLat

Given a lattice `L`, return a lattice `M` in the ambient space of `L` which
is maximal integral and which contains `L`.
Given a lattice `L` with integral norm, return a maximal integral overlattice
`M` of `L`.
"""
maximal_integral_lattice(::AbstractLat)

Expand Down
11 changes: 6 additions & 5 deletions src/QuadForm/Quad/Lattices.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -440,7 +440,7 @@ function guess_max_det(L::QuadLat, p)
end

function is_maximal_integral(L::QuadLat, p)
@req order(p) == base_ring(L) "Rings do not match"
@req order(p) == base_ring(L) "The ideal does not belong to the base ring of the lattice"
#if iszero(L)
# return true, L
#end
Expand Down Expand Up @@ -532,8 +532,8 @@ function is_maximal_integral(L::QuadLat)
end

function maximal_integral_lattice(L::QuadLat, p)
@req base_ring(L) == order(p) "Second argument must be an ideal of the base ring of L"
@req valuation(norm(L), p) >= 0 "The normal of the lattice must be locally integral"
@req base_ring(L) == order(p) "The ideal does not belong to the base ring of the lattice"
@req valuation(norm(L), p) >= 0 "The norm of the lattice is not locally integral"

ok, LL = is_maximal_integral(L, p)
while !ok
Expand All @@ -544,7 +544,8 @@ function maximal_integral_lattice(L::QuadLat, p)
end

function is_maximal(L::QuadLat, p)
@req order(p) == base_ring(L) "Asdsads"
@req order(p) == base_ring(L) "The ideal does not belong to the base ring of the lattice"
@req valuation(norm(L), p) >= 0 "The norm of the lattice is not locally integral"
#if iszero(L)
# return true, L
#end
Expand Down Expand Up @@ -576,7 +577,7 @@ function maximal_integral_lattice(V::QuadSpace)
end

function maximal_integral_lattice(L::QuadLat)
@req is_integral(norm(L)) "Lattice must be integral"
@req is_integral(norm(L)) "The norm of the lattice is not integral"
for p in bad_primes(L; even = true)
L = maximal_integral_lattice(L, p)
end
Expand Down
36 changes: 17 additions & 19 deletions src/QuadForm/Quad/ZLattices.jl
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Expand Up @@ -1142,14 +1142,13 @@ function _maximal_integral_lattice(L::ZZLat)
end

@doc raw"""
maximal_even_lattice(L::ZZLat, p) -> ZZLat
maximal_even_lattice(L::ZZLat, p::IntegerUnion) -> ZZLat

Given an even lattice `L` and a prime number `p` return an overlattice of `M`
which is maximal at `p` and agrees locally with `L` at all other places.

Recall that $L$ is called even if $\Phi(x,x) \in 2 \mathbb Z$ for all $x in L$.
Given an integer lattice `L` with integral scale and a prime number `p` such that
$L_p$ is even, return an overlattice `M` of `L` which is maximal even at `p` and
which agrees locally with `L` at all other places.
"""
function maximal_even_lattice(L::ZZLat, p)
function maximal_even_lattice(L::ZZLat, p::IntegerUnion)
while true
ok, L = is_maximal_even(L, p)
if ok
Expand All @@ -1161,10 +1160,7 @@ end
@doc raw"""
maximal_even_lattice(L::ZZLat) -> ZZLat

Return a maximal even overlattice `M` of the even lattice `L`.

Recall that $L$ is called even if $\Phi(x,x) \in 2 \mathbb Z$ for all $x in L$.
Note that the genus of `M` is uniquely determined by the genus of `L`.
Given an even integer lattice `L`, return a maximal even overlattice `M` of `L`.
"""
function maximal_even_lattice(L::ZZLat)
@req iseven(L) "The lattice must be even"
Expand All @@ -1175,29 +1171,31 @@ function maximal_even_lattice(L::ZZLat)
end

function maximal_integral_lattice(L::ZZLat)
@req denominator(norm(L)) == 1 "The quadratic form is not integral"
@req denominator(norm(L)) == 1 "The norm of the lattice is not integral"
L2 = rescale(L, 2)
LL2 = maximal_even_lattice(L2)
return rescale(LL2, QQ(1//2))
end


@doc raw"""
is_maximal_even(L::ZZLat, p) -> Bool, ZZLat
is_maximal_even(L::ZZLat, p::IntegerUnion) -> Bool, ZZLat

Return if the (`p`-locally) even lattice `L` is maximal at `p` and an even overlattice `M`
of `L` with $[M:L]=p$ if `L` is not maximal and $1$ else.
Given an integer lattice `L` with integral scale and a prime number `p`,
return whether $L_p$ is even and has no proper overlattice satisfying this
property.

Recall that $L$ is called even if $\Phi(x,x) \in 2 \mathbb{Z}$ for all $x in L$.
If $L_p$ is not even, the second output is `L` by default. Otherwise, either
`L` is maximal at `p` and the second output is `L`, or the second output is
an overlattice `M` of `L` such that $M_p$ is even and $[M:L] = p$.
"""

function is_maximal_even(L::ZZLat, p)
function is_maximal_even(L::ZZLat, p::IntegerUnion)
@req denominator(scale(L)) == 1 "The bilinear form is not integral"
@req p != 2 || mod(ZZ(norm(L)),2) == 0 "The bilinear form is not even"
p != 2 || mod(ZZ(norm(L)), 2) == 0 || return false, L

# o-maximal lattices are classified
# see Kirschmer Lemma 3.5.3
if valuation(det(L), p)<= 1
if valuation(det(L), p) <= 1
return true, L
end
G = change_base_ring(ZZ, gram_matrix(L))
Expand Down
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