The theta series of a lattice #

For a ℤ-submodule L of a real normed space E and τ in the upper half-plane ℍ, the theta series of L is

Θ_L(τ) = ∑_{v ∈ L} exp (π i ‖v‖² τ),

and for γ : E the theta series of the translate γ + L is

θ_{γ+L}(τ) = ∑_{v ∈ L} exp (π i ‖γ + v‖² τ).

In the applications E is a real inner product space, so that ‖v‖² is the positive definite quadratic form of the inner product; the definitions and the results here use only the norm. The exponent is π i ‖v‖² τ rather than 2 π i ‖v‖² τ: for an even lattice the q-expansion in q = exp (2 π i τ) is then indexed by the half-norms ‖v‖² / 2. In rank one, for L = ℤ ⊆ ℝ, Θ_L is the Jacobi theta function jacobiTheta. When γ lies in the dual lattice, θ_{γ+L} is the coset theta series of the coset γ + L.

This file defines both series and proves that they converge and are holomorphic, for a discrete subgroup L (of any rank, not necessarily full) of a finite-dimensional real normed space. The term of index v has norm exp (-π ‖γ + v‖² im τ), so on every half-plane {τ | y ≤ im τ}, 0 < y, the terms are dominated by the summable Gaussian exp (-π y ‖γ + v‖²) of ZLattice.summable_exp_neg_mul_norm_add_sq. Hence the series converge uniformly on these half-planes and locally uniformly on ℍ, and are holomorphic, both as functions of z : ℂ on {z | 0 < im z} and in the manifold sense on ℍ, which is what a bundled modular form requires.

The series θ_{γ+L} depends only on the class of γ modulo L, and θ_{-γ+L} = θ_{γ+L}. For a subgroup M ≤ L, splitting L into the classes of L modulo M writes θ_{γ+L} as the sum of the series θ_{γ+y+M} over those classes, and in particular Θ_L as the sum of the θ_{y+M}.

Main definitions #

ZLattice.thetaSeries L: the theta series Θ_L : ℍ → ℂ.
ZLattice.thetaCoset L γ: the theta series θ_{γ+L} : ℍ → ℂ of the translate γ + L.

Main statements #

Complex.norm_exp_pi_mul_I_mul_ofReal_mul: ‖exp (π i t z)‖ = exp (-π t im z) for real t, so the term of index v has norm exp (-π ‖γ + v‖² im τ).
ZLattice.thetaCoset_eq_of_sub_mem, ZLattice.thetaCoset_neg: θ_{γ+L} depends only on γ modulo L, and θ_{-γ+L} = θ_{γ+L}.
ZLattice.summable_norm_thetaCoset, ZLattice.summable_thetaCoset: the series converge absolutely at every τ : ℍ.
ZLattice.hasSum_thetaCoset_add_out, ZLattice.hasSum_thetaCoset_out: for M ≤ L, θ_{γ+L} is the sum of the θ_{γ+y+M}, and Θ_L the sum of the θ_{y+M}, over the classes y of L modulo M.
ZLattice.hasSumUniformlyOn_thetaCoset: uniform convergence on {τ | y ≤ im τ} for 0 < y.
ZLattice.hasSumLocallyUniformly_thetaCoset: locally uniform convergence on ℍ.
ZLattice.differentiableOn_thetaCoset, ZLattice.mdifferentiable_thetaCoset: holomorphy.

Each statement about thetaCoset has a thetaSeries counterpart, obtained from ZLattice.thetaCoset_zero : thetaCoset L 0 = thetaSeries L.

References #

J.-P. Serre, A Course in Arithmetic, Ch. VII §6.
W. Ebeling, Lattices and Codes, Ch. 2.