Shells and representation numbers of a lattice #
For a ℤ-submodule L of a real normed space E and a real number t, the shell of squared
norm t is the set of lattice vectors
shell L t = {v ∈ L | ‖v‖² = t},
and the representation number r_L(t) is its cardinality. More generally, for γ : E the
shell of the translate γ + L is cosetShell L γ t = {v ∈ L | ‖γ + v‖² = t}, with cardinality
r_{γ+L}(t). In a real inner product space these are the counts that the theta series
Θ_L(τ) = ∑_{v ∈ L} e^{π i ‖v‖² τ} and the coset theta series θ_{γ+L} expand: grouping the terms
by squared norm, r_L(t) is the coefficient of e^{π i t τ}. That identification is made in
TauCeti.NumberTheory.ModularForms.ThetaSeries.QExpansion.
The shells are sets with a finiteness theorem rather than Finsets, since membership is not
decidable and the finiteness is the content: for a discrete L in a proper space, such as a
finite-dimensional real normed space, every shell lies in a closed ball, which meets L in
finitely many points.
This file proves that every shell is finite and that only finitely many squared norms up to any
bound are represented, that r_L(0) = 1 and that negative numbers are not represented; that
the representation numbers of a translate depend only on its coset (and so descend to
DiscriminantGroup L), are unchanged by γ ↦ -γ,
and are invariant under linear isometries; that an even lattice represents only even natural
numbers and an integral lattice only natural numbers, and that for an even lattice L and
γ ∈ L^∨ the translate γ + L represents only numbers congruent to ‖γ‖² modulo 2; and that
for a full lattice L in a real normed space of dimension n the representation numbers grow at
most like t^{n/2}, from the count of lattice points in a ball, which is O(R^n)
(ZLattice.ncard_closedBall_inter_isBigO).
Main definitions #
ZLattice.shell L t: the lattice vectors of squared normt.ZLattice.representationNumber L t: the numberr_L(t)of lattice vectors of squared normt.ZLattice.cosetShell L γ t: the lattice vectorsvwith‖γ + v‖² = t.ZLattice.cosetRepresentationNumber L γ t: their numberr_{γ+L}(t).ZLattice.cosetRepresentationNumberClass L: the coset representation numberr_γ(t)descended toγ : DiscriminantGroup L.ZLattice.representedNormsLE L t: the squared normss ≤ trepresented byL.
Main statements #
ZLattice.finite_shell,ZLattice.finite_cosetShell: shells of a discrete subgroup of a proper space are finite.ZLattice.finite_representedNormsLE: such a subgroup represents only finitely many squared norms up to any bound.ZLattice.representationNumber_zero:r_L(0) = 1.ZLattice.representationNumber_span_one: the integer lattice represents each positive square twice and no nonsquare natural number.ZLattice.cosetRepresentationNumber_add_mem,ZLattice.cosetRepresentationNumberClass_mk,ZLattice.cosetRepresentationNumber_neg:r_{γ+L}depends only on the cosetγ + L, descends toDiscriminantGroup L(vanishing at0on nonzero classes), andr_{-γ+L} = r_{γ+L}.ZLattice.representationNumber_comap,ZLattice.cosetRepresentationNumber_comap: invariance under linear isometry equivalences.ZLattice.IsEven.shell_eq_empty,ZLattice.IsIntegral.shell_eq_empty: an even lattice has no vector of squared norm outside2ℕ, an integral lattice none outsideℕ.ZLattice.IsEven.cosetShell_eq_empty: for evenLandγ ∈ L^∨, the translateγ + Lhas no vector of squared norm outside‖γ‖² + 2ℤ;ZLattice.IsEven.cosetRepresentationNumber_div_eq_zero_of_not_modEq: if moreoverN ‖γ‖² = 2 n, thenr_{γ+L}(2m / N) = 0unlessm ≡ n (mod N).ZLattice.representationNumber_isBigO:r_L(t) = O(t^{n/2})ast → ∞.
References #
- J.-P. Serre, A Course in Arithmetic, Ch. VII §6.
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chapter 2, §2.3.
Shells and representation numbers #
The shell of squared norm t of L: the vectors v ∈ L with ‖v‖² = t. It is finite when
L is discrete in a proper space (ZLattice.finite_shell).
Instances For
The representation number r_L(t): the number of vectors of L of squared norm t. For a
discrete L in a proper space every shell is finite, so this is a genuine count.
Instances For
The shell of squared norm t of the translate γ + L: the vectors v ∈ L with
‖γ + v‖² = t. For γ in the dual lattice these are the vectors of the coset γ + L of squared
norm t, translated back into L.
Instances For
The representation number r_{γ+L}(t) of the translate γ + L: the number of vectors
v ∈ L with ‖γ + v‖² = t.
Instances For
The shells of the translate 0 + L are the shells of L.
The only lattice vector of squared norm 0 is 0.
A lattice represents 0 exactly once: r_L(0) = 1.
No translate of a lattice has a vector of negative squared norm.
The shell of squared norm t of γ + L lies in the closed ball of radius √t about -γ.
The shell of squared norm t of L lies in the closed ball of radius √t about 0.
Dependence on the coset #
Translating γ by a lattice vector w translates the shells of γ + L by -w.
The representation numbers of γ + L depend only on the coset of γ modulo L.
Translates by vectors congruent modulo L have the same representation numbers.
Negating γ negates the shells of γ + L.
The translates γ + L and -γ + L have the same representation numbers.
Invariance under isometries #
A linear isometry equivalence e carries the shells of the pull-back of L along e to the
shells of L.
Representation numbers of translates are invariant under linear isometry equivalences.
A linear isometry equivalence e carries the shells of the pull-back of L along e to the
shells of L.
Representation numbers are invariant under linear isometry equivalences.
Finiteness #
The shells of a translate of a discrete subgroup of a proper space are finite.
The shells of a discrete subgroup of a proper space are finite.
r_{γ+L}(t) vanishes exactly when the shell is empty.
r_L(t) vanishes exactly when no lattice vector has squared norm t.
The represented squared norms up to a bound #
The squared norms s ≤ t represented by L: the real numbers s ≤ t whose shell in L is
nonempty. It is finite when L is discrete in a proper space
(ZLattice.finite_representedNormsLE).
Instances For
L represents s ≤ t exactly when some vector of L has squared norm s.
A discrete subgroup of a proper space represents only finitely many squared norms up to any bound.
The represented numbers of even and integral lattices #
An integral lattice only represents natural numbers.
For an even lattice L and γ ∈ L^∨, the translate γ + L only represents numbers congruent
to ‖γ‖² modulo 2: the half-norm ‖γ‖² / 2 is well defined modulo ℤ on the coset.
For an even lattice L and γ ∈ L^∨, the representation number r_{γ+L}(t) vanishes unless
t ∈ ‖γ‖² + 2ℤ.
For an even lattice L, γ ∈ L^∨ and a positive integer N with N ‖γ‖² = 2 n, the
representation number r_{γ+L}(2m / N) vanishes unless m ≡ n (mod N).
Coset representation numbers on the discriminant group #
The representation number r_γ(t) = r_{x+L}(t) of a class γ of the discriminant group
L^∨ ⧸ (L ∩ L^∨): the number of vectors v ∈ L with ‖x + v‖² = t, for any representative
x ∈ L^∨ of γ (cosetRepresentationNumberClass_mk).
Instances For
On the class of x ∈ L^∨, cosetRepresentationNumberClass is cosetRepresentationNumber.
At the zero class of DiscriminantGroup L, the coset representation number is r_L(t).
A nonzero class in DiscriminantGroup L does not represent squared norm 0.
Opposite classes in DiscriminantGroup L have the same representation numbers.
No class of DiscriminantGroup L represents a negative squared norm.
Growth of the representation numbers #
The representation numbers of a full lattice in a real normed space of dimension n grow at
most like t^{n/2}: r_L(t) = O(t^{n/2}) as t → ∞.