Eichler transvections of integral lattices #
Let L be an integral lattice with rational form B, and let u ∈ L be isotropic. For a
lattice vector w with B(u, w) = 0, the Eichler transvection of the rational half-norm
x ↦ B(x, x) / 2 is
E_{u,w}(x) = x + B(x, u) • w - B(x, w) • u - (B(w, w) / 2 * B(x, u)) • u.
Every coefficient is an integer on lattice vectors as soon as B(w, w) is even, so these w
form a sublattice L.evenOrthogonal u of L, the intersection of u^⊥ with the even-norm
submodule. Since E_{u,w + k • u} = E_{u,w}, the map w ↦ E_{u,w} descends to a homomorphism
from L.evenOrthogonal u ⧸ ℤ ∙ u into the special isometry group SO(L). When u is not in the
radical of the form, E_{u,w} is trivial exactly when w ∈ ℚ ∙ u; if moreover u is primitive,
the homomorphism is injective.
If L is nondegenerate of rank at least three and its rational space has a nonzero isotropic
vector, then L ∩ u^⊥ contains a vector outside ℚ ∙ u, whose Eichler transvection has infinite
order. Hence SO(L), and with it the isometry group O(L), is infinite. This is the isotropic
half of the infiniteness of O(L) for indefinite lattices of rank at least three; it contrasts
with the finiteness of O(L) for definite lattices. In rank two every Eichler transvection whose
isotropic vector is outside the radical of the form is trivial
(QuadraticMap.transvection_eq_one_of_finrank_eq_two), and the argument gives nothing.
Main definitions #
TauCeti.IntegralLattice.evenOrthogonal: the lattice vectorswwithB(u, w) = 0andB(w, w)even.TauCeti.IntegralLattice.eichlerTransvection: the Eichler transvectionE_{u,w}as a self-isometry ofL.TauCeti.IntegralLattice.eichlerTransvectionHom: the homomorphismw ↦ E_{u,w}fromL.evenOrthogonal u ⧸ ℤ ∙ uintoSO(L).
Main results #
TauCeti.IntegralLattice.eichlerTransvection_apply: the defining formula.TauCeti.IntegralLattice.eichlerTransvection_mem_specialOrthogonalGroup:E_{u,w} ∈ SO(L).TauCeti.IntegralLattice.eichlerTransvection_add:E_{u,w + w'} = E_{u,w} * E_{u,w'}.TauCeti.IntegralLattice.eichlerTransvection_add_zsmul:E_{u,w + k • u} = E_{u,w}.TauCeti.IntegralLattice.eichlerTransvection_eq_one_iff: ifuis not in the radical of the form,E_{u,w} = 1exactly whenw ∈ ℚ ∙ u.TauCeti.IntegralLattice.eichlerTransvectionHom_injective: ifuis primitive and not in the radical of the form,w ↦ E_{u,w}is injective onL.evenOrthogonal u ⧸ ℤ ∙ u.TauCeti.IntegralLattice.not_isOfFinOrder_eichlerTransvection: ifuis not in the radical of the form andw ∉ ℚ ∙ u, thenE_{u,w}has infinite order.TauCeti.IntegralLattice.infinite_specialOrthogonalGroup_of_not_anisotropicandTauCeti.IntegralLattice.infinite_isometry_of_not_anisotropic: a nondegenerate lattice of rank at least three whose rational norm form is isotropic has infiniteSO(L)andO(L).
References #
- M. Eichler, Quadratische Formen und orthogonale Gruppen, Springer (1952).
QuadraticMap.transvection, the Eichler transvection of a quadratic form, which this file restricts to the lattice.
The rational half-norm #
Even vectors orthogonal to a lattice vector #
The even vectors of L orthogonal to u: the w ∈ L with B(u, w) = 0 and B(w, w) even,
that is, the kernel of B(u, -) intersected with the even-norm submodule. For isotropic u these
are the parameters of the integral Eichler transvections E_{u,w}.
Instances For
Every multiple of an isotropic vector u is an even vector orthogonal to u.
Eichler transvections #
The Eichler transvection E_{u,w} of an integral lattice, for an isotropic lattice vector
u and an even lattice vector w orthogonal to it:
E_{u,w}(x) = x + B(x, u) • w - B(x, w) • u - (B(w, w) / 2 * B(x, u)) • u.
Instances For
The Eichler transvection acts by
E_{u,w}(x) = x + B(x, u) • w - B(x, w) • u - (B(w, w) / 2 * B(x, u)) • u.
On the carrier, the Eichler transvection is given by the same formula with integral coefficients.
An Eichler transvection fixes its isotropic vector.
An Eichler transvection fixes every vector orthogonal to both u and w.
An Eichler transvection has determinant one.
An Eichler transvection is a proper isometry.
The Eichler transvection with parameter 0 is the identity.
The Eichler transvections with a fixed isotropic vector u compose additively in w.
The inverse of an Eichler transvection is the Eichler transvection of -w.
Integer multiples of the parameter give powers of an Eichler transvection.
An Eichler transvection whose parameter is a rational multiple of u is the identity.
An Eichler transvection depends on its parameter only modulo ℤ ∙ u:
E_{u,w + k • u} = E_{u,w}.
The Eichler transvections with isotropic vector u, as a homomorphism from the parameter
group L.evenOrthogonal u ⧸ ℤ ∙ u into the special isometry group, sending the class of w to
E_{u,w}. It is injective when u is primitive and not in the radical of the form
(eichlerTransvectionHom_injective).
Instances For
The homomorphism sends the class of w to the Eichler transvection E_{u,w}, as an element
of specialOrthogonalGroup L.
The homomorphism sends the class of w to the Eichler transvection E_{u,w}, as an isometry
of L.
If u is not in the radical of the form (for instance, u ≠ 0 and L is nondegenerate),
then the Eichler transvection E_{u,w} is the identity exactly when w is a rational multiple
of u.
If u is primitive and not in the radical of the form, the Eichler transvections with
isotropic vector u form a copy of the parameter group L.evenOrthogonal u ⧸ ℤ ∙ u inside
SO(L).
If u is not in the radical of the form, an Eichler transvection whose parameter is not a
rational multiple of u has infinite order.
Infinite isometry groups #
If L has rank at least three and u is not in the radical of the form, then some even
lattice vector orthogonal to u is not a rational multiple of u.
The isotropic case of the infiniteness of SO(L). A nondegenerate integral lattice of
rank at least three whose rational norm form represents zero nontrivially has infinitely many
proper isometries, already among its Eichler transvections.
The isotropic case of the infiniteness of O(L). A nondegenerate integral lattice of
rank at least three whose rational norm form represents zero nontrivially has infinitely many
isometries.