The Borisov--Nuer conjecture and root decompositions in even unimodular Lorentzian lattices

Abstract

We prove that every vector of $\mathrm{II}_{8n+1,1}$, with $n\ge1$, 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.