Theta series near the cusps 0 and a / c #
Let L be a full lattice in a real inner product space E of even dimension n = 2k. This file
computes the leading behaviour of the theta series of L and of its translates as the variable
approaches a rational point of the boundary of ℍ.
The cusp 0. Let ε → 0 in ℍ in such a way that im (-1/ε) = im ε / |ε|² → ∞; this is the
filter comap (S • ·) atImInfty, the preimage of atImInfty under the involution
S : τ ↦ -1/τ. The real part of ε may vary along the way. Then for every v : E,
ε^k θ_{v+L}(ε) → i^k (covol L)⁻¹.
This is the inversion law ZLattice.thetaCoset_S_smul read at τ = -1/ε: since ε τ = -1,
ε^k θ_{v+L}(ε) = i^k (covol L)⁻¹ ∑_{m ∈ L^∨} e(⟪v, m⟫) exp (π i ‖m‖² τ), and as im τ → ∞ every
term of the sum tends to 0, except the term 1 of m = 0.
The cusps a / c. Let L be even, a ∈ ℤ and c a positive integer. Splitting L into its
classes y + cL, the factor exp (π i ‖x‖² a / c) is the same for all x ∈ y + cL, namely the
Gauss summand e(a ‖y‖² / (2c)) of ZLattice.gaussTerm, so
Θ_L(a/c + τ) = ∑_{y ∈ L/cL} e(a ‖y‖² / (2c)) θ_{y+cL}(τ).
More generally, for γ ∈ L^∨ the factor is e(a ‖γ + y‖² / (2c)) on the class γ + y + cL of the
coset γ + L, and θ_{γ+L}(a/c + τ) splits in the same way into the series θ_{γ+y+cL}(τ).
Since covol (cL) = c^{2k} covol L, the asymptotic at the cusp 0 applied to each lattice coset
y + cL gives ε^k Θ_L(a/c + ε) → i^k G_L(a, c) / (c^{2k} covol L), where G_L(a, c) is the
Gauss sum ZLattice.gaussSum. Along the imaginary axis ε = i t this reads
t^k Θ_L(a/c + i t) → G_L(a, c) / (c^{2k} covol L) as t → 0⁺.
Both statements hold, with the same proof, for any lattice M with cL ≤ M ≤ L^∨ in place of L:
since L is even and ⟪y, ℓ⟫ ∈ ℤ for y ∈ M and ℓ ∈ L, the factor exp (π i ‖x‖² a / c) is
constant on y + cL, equal to e(a ‖y‖² / (2c)). For M = L^∨ at a cusp b / a this gives
Θ_{L^∨}(b/a + τ) = ∑_{y ∈ L^∨/aL} e(b ‖y‖² / (2a)) θ_{y+aL}(τ),
and ε^k Θ_{L^∨}(b/a + ε) → i^k (∑_{y ∈ L^∨/aL} e(b ‖y‖² / (2a))) / (a^{2k} covol L).
Comparing the limit at a/c with the one obtained from the inversion law at a/c + i t, which
moves the point near the cusp -c/a of the dual theta series, is the analytic proof of the
reciprocity law for the Gauss sums of an even lattice
(ZLattice.IsEven.gaussSum_eq_mul_dualGaussSum). The statement is additive, not multiplicative:
G_L(a, c) may vanish while Θ_L(a/c + i t) does not.
Main statements #
ZLattice.tendsto_pow_mul_thetaCoset_comap_S,ZLattice.tendsto_pow_mul_thetaSeries_comap_S:ε^k θ_{v+L}(ε) → i^k (covol L)⁻¹asεtends to the cusp0.ZLattice.IsEven.thetaSeries_vadd_eq_sum_of_le,ZLattice.IsEven.tendsto_pow_mul_thetaSeries_vadd_comap_S_of_le: the splitting ofΘ_M(a/c + τ)into theta series of translates ofcL, and the asymptotic ofΘ_Mat the cuspa / c, for any latticeMwithcL ≤ M ≤ L^∨.ZLattice.IsEven.thetaCoset_vadd_eq_sum,ZLattice.IsEven.thetaSeries_vadd_eq_sum: the splitting ofθ_{γ+L}(a/c + τ), forγ ∈ L^∨, and ofΘ_L(a/c + τ)into theta series of translates ofcL, weighted by Gauss summands.ZLattice.IsEven.tendsto_pow_mul_thetaSeries_vadd_comap_S,ZLattice.IsEven.tendsto_pow_mul_thetaSeries_nhdsGT: the asymptotic ofΘ_Lat the cuspa / c.ZLattice.IsEven.thetaSeries_dual_vadd_eq_sum,ZLattice.IsEven.tendsto_pow_mul_thetaSeries_dual_vadd_comap_S: the splitting ofΘ_{L^∨}(b/a + τ)into theta series of translates ofaL, and the asymptotic ofΘ_{L^∨}at the cuspb / a.
References #
- B. Schoeneberg, Elliptic Modular Functions, Springer, 1974, Chapter IX.
- J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer, 1973, Appendix 4.
Theta series of a translate near the cusp 0: if dim E = 2k, then for v : E,
ε^k θ_{v+L}(ε) → i^k (covol L)⁻¹ as ε → 0 with im (-1/ε) → ∞.
Theta series near the cusp 0: if dim E = 2k, then ε^k Θ_L(ε) → i^k (covol L)⁻¹ as
ε → 0 with im (-1/ε) → ∞.
Splitting the theta series of a dual coset at the cusp a / c. For an even lattice L,
γ ∈ L^∨, a ∈ ℤ and a positive integer c,
θ_{γ+L}(a/c + τ) = ∑_{y ∈ L/cL} e(a ‖γ‖² / (2c)) e((a ‖y‖² / 2 + ⟪y, a γ⟫) / c) θ_{γ+y+cL}(τ):
on each class γ + y + cL the factor exp (π i ‖x‖² a / c) is e(a ‖γ + y‖² / (2c)), which is
e(a ‖γ‖² / (2c)) times the twisted Gauss summand of y at a γ ∈ L^∨.
Splitting a theta series at the cusp a / c over an intermediate lattice. Let L be even,
a ∈ ℤ, c a natural number and M a lattice with cL ≤ M ≤ L^∨. If g is a function on
M / cL taking the value e(a ‖v‖² / (2c)) at the class of each v ∈ M, then
Θ_M(a/c + τ) = ∑_{y ∈ M/cL} g(y) θ_{y+cL}(τ): since ⟪M, L⟫ ⊆ ℤ and L is even, the factor
exp (π i ‖x‖² a / c) is constant on each class y + cL.
Splitting a theta series at the cusp a / c. For an even lattice L, a ∈ ℤ and a
positive integer c, Θ_L(a/c + τ) = ∑_{y ∈ L/cL} e(a ‖y‖² / (2c)) θ_{y+cL}(τ): on each class
y + cL the factor exp (π i ‖x‖² a / c) is the Gauss summand of y.
Splitting the theta series of the dual lattice at the cusp b / a. For an even lattice
L, b ∈ ℤ and a natural number a,
Θ_{L^∨}(b/a + τ) = ∑_{y ∈ L^∨/aL} e(b ‖y‖² / (2a)) θ_{y+aL}(τ): on each class y + aL the factor
exp (π i ‖x‖² b / a) is the dual Gauss summand of y.
Theta series near the cusp a / c over an intermediate lattice: if dim E = 2k, L is
even, a ∈ ℤ, c is a positive integer, cL ≤ M ≤ L^∨, and g is a function on M / cL taking
the value e(a ‖v‖² / (2c)) at the class of each v ∈ M, then
ε^k Θ_M(a/c + ε) → i^k (∑_{y ∈ M/cL} g(y)) / (c^{2k} covol L) as ε → 0 with
im (-1/ε) → ∞.
Theta series of an even lattice near the cusp a / c: if dim E = 2k, then for a ∈ ℤ
and a positive integer c, ε^k Θ_L(a/c + ε) → i^k G_L(a, c) / (c^{2k} covol L) as ε → 0 with
im (-1/ε) → ∞, where G_L(a, c) is the Gauss sum of L.
The dual theta series of an even lattice near the cusp b / a: if dim E = 2k, then for
b ∈ ℤ and a positive integer a,
ε^k Θ_{L^∨}(b/a + ε) → i^k (∑_{y ∈ L^∨/aL} e(b ‖y‖² / (2a))) / (a^{2k} covol L) as ε → 0 with
im (-1/ε) → ∞.
Theta series of an even lattice on a vertical line: if dim E = 2k, then for a ∈ ℤ and a
positive integer c, t^k Θ_L(a/c + i t) → G_L(a, c) / (c^{2k} covol L) as t → 0⁺, where
G_L(a, c) is the Gauss sum of L.