The q-expansions of theta series #
Let L be a discrete subgroup of a finite-dimensional real normed space E and γ : E. Grouping
the terms of the theta series θ_{γ+L}(τ) = ∑_{v ∈ L} exp (π i ‖γ + v‖² τ) by the squared norm
‖γ + v‖² writes it as a power series in a q-parameter, with the representation numbers
r_{γ+L}(t) = #{v ∈ L | ‖γ + v‖² = t} (ZLattice.cosetRepresentationNumber) as coefficients.
This needs the squared norms to lie in a discrete progression: if h ‖γ + v‖² ∈ 2ℤ for every
v ∈ L, for some h > 0, then with q_h = exp (2 π i τ / h)
θ_{γ+L}(τ) = ∑_{m ≥ 0} r_{γ+L}(2m / h) q_h^m,
since the term of v is q_h^m for m = h ‖γ + v‖² / 2, and the vectors with a given m form
the finite shell of squared norm 2m / h. By uniqueness of q-expansion coefficients, the
coefficients of qExpansion h (θ_{γ+L}) are these representation numbers.
The three cases of interest are the following.
- For an even lattice
L(h = 1), inq = exp (2 π i τ),Θ_L(τ) = ∑_{m ≥ 0} r_L(2m) q^m: them-th coefficient counts the vectors of half-normm. This is the expansion compared with theq-expansions ofE₄,Δand the other level-one modular forms. - For an integral lattice
L(h = 2), inq^{1/2} = exp (π i τ),Θ_L(τ) = ∑_{m ≥ 0} r_L(m) q^{m/2}, the shape of the expansion ofjacobiTheta. - For an even lattice
L,γ ∈ L^∨and a positive multipleNoflevel L(in particularN = level LwhenLis a full lattice), inq_N = exp (2 π i τ / N),θ_{γ+L}(τ) = ∑_{m ≥ 0} r_{γ+L}(2m / N) q_N^m, and on the discriminant groupA_L,θ_δ(τ) = ∑_{m ≥ 0} r_δ(2m / N) q_N^m. For an even full lattice, the coefficient ofq_N^mvanishes unlessm mod Nmaps toq_L(δ)underZMod.toRatAddCircle N(ZLattice.IsEven.qExpansion_level_thetaCoset_coeff_eq_zero,ZLattice.IsEven.qExpansion_level_thetaCosetClass_coeff_eq_zero). This support condition describes the expansion at the cusp∞only; the analysis at the other cusps also needs the theta transformation formulas.
Main statements #
ZLattice.hasSum_thetaCoset_qParam,ZLattice.qExpansion_thetaCoset_coeff: the expansion ofθ_{γ+L}inq_hwhenh ‖γ + v‖² ∈ 2ℤfor allv ∈ L, and its coefficients.ZLattice.IsEven.hasSum_thetaSeries_qParam,ZLattice.IsEven.qExpansion_thetaSeries_coeff:Θ_L = ∑ r_L(2m) q^mfor evenL.ZLattice.IsIntegral.hasSum_thetaSeries_qParam_two,ZLattice.IsIntegral.qExpansion_two_thetaSeries_coeff:Θ_L = ∑ r_L(m) q^{m/2}for integralL.ZLattice.IsEven.hasSum_thetaCoset_qParam,ZLattice.IsEven.qExpansion_thetaCoset_coeff,ZLattice.IsEven.hasSum_thetaCoset_level_qParam,ZLattice.IsEven.qExpansion_level_thetaCoset_coeff:θ_{γ+L} = ∑ r_{γ+L}(2m / N) q_N^mfor evenLandγ ∈ L^∨, including atN = level Land onDiscriminantGroup L.ZLattice.IsEven.qExpansion_thetaCoset_coeff_eq_zero,ZLattice.IsEven.qExpansion_level_thetaCoset_coeff_eq_zero,ZLattice.IsEven.qExpansion_level_thetaCosetClass_coeff_eq_zero: that coefficient vanishes outside the residue class ofq_L(γ)moduloN.
References #
- J.-P. Serre, A Course in Arithmetic, Ch. VII §6.
- W. Ebeling, Lattices and Codes, 3rd ed., Springer, 2013, Chapter 3.
The q-expansion of a coset theta series. If h ‖γ + v‖² ∈ 2ℤ for every v ∈ L, with
h > 0, then θ_{γ+L}(τ) = ∑_{m ≥ 0} r_{γ+L}(2m / h) q_h^m with q_h = exp (2 π i τ / h).
The coefficients of the q-expansion of a coset theta series: if h ‖γ + v‖² ∈ 2ℤ for every
v ∈ L, with h > 0, the m-th coefficient of the expansion of θ_{γ+L} in
q_h = exp (2 π i τ / h) is r_{γ+L}(2m / h).
Even lattices #
The q-expansion of the theta series of an even lattice: in q = exp (2 π i τ),
Θ_L(τ) = ∑_{m ≥ 0} r_L(2m) q^m, where r_L(2m) is the number of vectors of half-norm m.
The coefficients of the q-expansion of the theta series of an even lattice: the m-th
coefficient in q = exp (2 π i τ) is the number r_L(2m) of vectors of half-norm m.
Integral lattices #
The q-expansion of the theta series of an integral lattice, with period 2: in
q^{1/2} = exp (π i τ), Θ_L(τ) = ∑_{m ≥ 0} r_L(m) q^{m/2}.
The coefficients of the q-expansion with period 2 of the theta series of an integral
lattice: the m-th coefficient in q^{1/2} = exp (π i τ) is the number r_L(m) of vectors of
squared norm m.
Coset theta series of even lattices #
The q_N-expansion of a coset theta series of an even lattice: for γ ∈ L^∨ and a positive
integer N with N ‖γ‖² / 2 ∈ ℤ, in q_N = exp (2 π i τ / N),
θ_{γ+L}(τ) = ∑_{m ≥ 0} r_{γ+L}(2m / N) q_N^m.
The coefficients of the q_N-expansion of a coset theta series of an even lattice: for
γ ∈ L^∨ and a positive integer N with N ‖γ‖² / 2 ∈ ℤ, the m-th coefficient in
q_N = exp (2 π i τ / N) is r_{γ+L}(2m / N).
The support of the q_N-expansion of a coset theta series. For an even lattice L,
γ ∈ L^∨ and a positive integer N with N ‖γ‖² = 2 n, the coefficient of q_N^m in the
expansion of θ_{γ+L} vanishes unless m ≡ n (mod N), that is, unless m / N is congruent to
‖γ‖² / 2 modulo ℤ.
For an even lattice L, γ ∈ L^∨ and a positive multiple N of level L,
θ_{γ+L}(τ) = ∑_{m ≥ 0} r_{γ+L}(2m / N) q_N^m.
For an even lattice L, γ ∈ L^∨ and a positive multiple N of level L, the m-th
coefficient of qExpansion N (θ_{γ+L}) is r_{γ+L}(2m / N).
For an even lattice L, δ : DiscriminantGroup L and a positive multiple N of level L,
θ_δ(τ) = ∑_{m ≥ 0} r_δ(2m / N) q_N^m.
For an even lattice L, δ : DiscriminantGroup L and a positive multiple N of level L,
the m-th coefficient of qExpansion N (θ_δ) is r_δ(2m / N).
For an even lattice L, a nonzero class δ : DiscriminantGroup L, and a positive multiple N
of level L, the constant term of qExpansion N (θ_δ) is 0.
The q_N-expansion of a coset theta series at the level. For an even full lattice L of
level N = level L and γ ∈ L^∨, θ_{γ+L}(τ) = ∑_{m ≥ 0} r_{γ+L}(2m / N) q_N^m.
For an even full lattice L and γ ∈ L^∨, the m-th coefficient of the
q_{level L}-expansion of θ_{γ+L} is r_{γ+L}(2m / level L).
The q_N-expansion of a discriminant-class theta series at the level. For an even full
lattice L of level N = level L and δ : DiscriminantGroup L,
θ_δ(τ) = ∑_{m ≥ 0} r_δ(2m / N) q_N^m.
For an even full lattice L and δ : DiscriminantGroup L, the m-th coefficient of the
q_{level L}-expansion of θ_δ is r_δ(2m / level L).
For an even full lattice L and a nonzero class δ : DiscriminantGroup L, the constant term
of qExpansion (level L) (θ_δ) is 0 (even when q_L(δ) = 0).
For an even full lattice L, x ∈ L^∨, and a positive multiple N of level L, the m-th
coefficient of qExpansion N (θ_{x+L}) vanishes unless m mod N maps to q_L(x + L) under
ZMod.toRatAddCircle N.
Support congruence for q_{level L}-expansions of coset theta series. For an even full
lattice L and x ∈ L^∨, the m-th coefficient of qExpansion (level L) (θ_{x+L}) vanishes
unless m mod level L maps to q_L(x + L) under ZMod.toRatAddCircle (level L).
For an even full lattice L, δ : DiscriminantGroup L, and a positive multiple N of
level L, the m-th coefficient of qExpansion N (θ_δ) vanishes unless m mod N maps to
q_L(δ) under ZMod.toRatAddCircle N.
Support congruence for q_{level L}-expansions on the discriminant group. For an even full
lattice L and δ : DiscriminantGroup L, the m-th coefficient of qExpansion (level L) (θ_δ)
vanishes unless m mod level L maps to q_L(δ) under ZMod.toRatAddCircle (level L).