Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.38414

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

Haowu Wang·Shaoyun Yi
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.NTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Haowu Wang, Shaoyun Yi. 2026-10-28. The Borisov--Nuer conjecture and root decompositions in even unimodular Lorentzian lattices. https://arxiv.org/abs/2610.38414

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Modularity over $\mathbb{C}$ implies modularity over $\mathbb{Q}$

We give an account of Mazur's proof that, for an elliptic curve over $\mathbb{Q}$, if it admits a nonconstant mapping from $X(N)$ defined over the complex numbers $\mathbb{C}$, for some $N$, then it also admits a nonconstant mapping from $X_0(M)$ for some (possibly different) $M$ defined over the rational numbers $\mathbb{Q}$. We also briefly discuss two open questions of Khare concerning uniformisations of elliptic curves by noncongruence modular curves. This expository note is based on the author's expository talk in March 2022 during the 2nd Trimester Program on "Modularity and the Generalised Fermat Equation" held online and organised by Bhaskaracharya Pratishthana in Pune, India.

math.NT↗
arXiv2026

On pairs of $r$-primitive and $k$-normal elements with prescribed traces over finite fields

Given $\mathbb{F}_{q^{n}}$, a field with $q^n$ elements, where $q $ is a prime power and $n$ is positive integer. For $r_1,r_2,m_1,m_2 \in \mathbb{N}$, $k_1,k_2 \in \mathbb{N}\cup \{0\}$, a rational function $F = \frac{F_1}{F_2}$ in $\mathbb{F}_{q}[x]$ with deg($F_i$) $\leq m_i$; $i=1,2,$ satisfying some conditions, and $a,b \in \mathbb{F}_{q}$, we construct a sufficient condition on $(q,n)$ which guarantees the existence of an $r_1$-primitive, $k_1$-normal element $ε\in \mathbb{F}_{q^n}$ such that $F(ε)$ is $r_2$-primitive, $k_2$-normal with $\operatorname{Tr}_{\mathbb{F}_{q^n}/\mathbb{F}_q}(ε) = a$ and $\operatorname{Tr}_{\mathbb{F}_{q^n}/\mathbb{F}_q}(ε^{-1}) = b$. For $m_1=10, \; m_2=11,\; r_1 = 3, \; r_2 = 2, \; k_1=2,\;k_2 = 1$, we establish bounds on $q$, for various $n$, to determine the existence of such elements in $\mathbb{F}_{q^{n}}$. Furthermore, we identify all such pairs $(q,n)$ excluding 10 possible values of $(q,n)$, in fields of characteristics 13.

math.NT↗
arXiv2026

Uniform existential definitions of valuations in function fields in one variable

We study function fields of curves over a base field $K$ which is either a global field or a large field having a separable field extension of degree divisible by $4$. We show that, for any such function field, Hilbert's 10th Problem has a negative answer, the valuation rings containing $K$ are uniformly existentially definable, and the finitary holomorphy rings are definable by a universal-existential formula. In order to obtain these results, we develop further the usage of local-global principles for quadratic forms in function fields to definability of certain subrings. We include a first systematic presentation of this general method, without restriction on the characteristic.

math.NT↗