Theta Series of Even Unimodular Lattices as Modular Forms Encoding Point Counts — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Theta series of even unimodular lattices (E8, Leech) are modular forms whose coefficients encode lattice point counts, with divisibility properties tied to the lattice's structure. | MATH: For an even unimodular lattice L of rank n, the theta series ΘL(τ) = ∑v ∈ L qv · v / 2 (with q = e2π i τ) is a modular form of weight n/2 for SL2(ℤ). For E8 (rank 8), ΘE8 = 1 + 240 ∑k=1∞ σ3(k) qk, where σ3(k) is the sum of cubes of divisors. For the Leech lattice (rank 24), ΘΛ24 = E12(τ) − 65520/691 Δ(τ), where E12 is the Eisenstein series of weight 12 and Δ is the discriminant (cusp form). The coefficient of qm in ΘΛ24 is divisible by 1440 for all m ≥ 1 — this follows from the fact that ΘΛ24 ≡ 1 pmod1440 as a modular form, a consequence of the lattice being e
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026theta,
title = {{Theta Series of Even Unimodular Lattices as Modular Forms Encoding Point Counts — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587142},
url = {https://zenodo.org/records/23587142},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Theta Series of Even Unimodular Lattices as Modular Forms Encoding Point Counts — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587142
UR - https://zenodo.org/records/23587142
AB - FINDING: Theta series of even unimodular lattices (E8, Leech) are modular forms whose coefficients encode lattice point counts, with divisibility properties tied to the lattice's structure. | MATH: For an even unimodular lattice L of rank n, the theta series Θ_L(τ) = ∑_v ∈ L q^v · v / 2 (with q = e^2π i τ) is a modular form of weight n/2 for SL_2(ℤ). For E8 (rank 8), Θ_E8 = 1 + 240 ∑_k=1^∞ σ_3(k) q^k, where σ_3(k) is the sum of cubes of divisors. For the Leech lattice (rank 24), Θ_Λ_24 = E_12(τ) − 65520/691 Δ(τ), where E_12 is the Eisenstein series of weight 12 and Δ is the discriminant (cusp form). The coefficient of q^m in Θ_Λ_24 is divisible by 1440 for all m ≥ 1 — this follows from the fact that Θ_Λ_24 ≡ 1 pmod1440 as a modular form, a consequence of the lattice being e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -