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The Borisov--Nuer conjecture and root decompositions in even unimodular Lorentzian lattices
Abstract
We prove that every vector of II8n+1,1, with n≥1, is a sum of two roots if and only if n=1 or $2$. This settles the Borisov--Nuer conjecture and implies the existence of Ulrich line bundles for every very ample polarization on a complex unnodal Enriques surface. For vectors of negative norm, we express the number of root decompositions as a Fourier coefficient of a vector-valued theta series, reducing existence to positivity. The main analytic difficulty is to obtain estimates uniform over lattices of unbounded determinant. We address this by combining a reduction based on primitive sublattices of codimension one and least determinant with explicit bounds for theta norms and Fourier coefficients of cusp forms.
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BibTeX
@misc{wang2026borisovnuer,
title = {{The Borisov--Nuer conjecture and root decompositions in even unimodular Lorentzian lattices}},
author = {Haowu Wang and Shaoyun Yi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38414},
eprint = {2610.38414},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38414},
}
RIS
TY - UNPB
TI - The Borisov--Nuer conjecture and root decompositions in even unimodular Lorentzian lattices
AU - Haowu Wang
AU - Shaoyun Yi
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38414
UR - https://arxiv.org/abs/2610.38414
AB - We prove that every vector of II_8n+1,1, with n≥1, is a sum of two roots if and only if n=1 or $2$. This settles the Borisov--Nuer conjecture and implies the existence of Ulrich line bundles for every very ample polarization on a complex unnodal Enriques surface. For vectors of negative norm, we express the number of root decompositions as a Fourier coefficient of a vector-valued theta series, reducing existence to positivity. The main analytic difficulty is to obtain estimates uniform over lattices of unbounded determinant. We address this by combining a reduction based on primitive sublattices of codimension one and least determinant with explicit bounds for theta norms and Fourier coefficients of cusp forms.
ER -