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Asymptotics of $n$-universal lattices over number fields

Dayoon Park, Robin Visser, Pavlo Yatsyna, Jongheun Yoon

TL;DR

The paper establishes a precise asymptotic for the logarithm of the minimal rank of $n$-universal lattices over the ring of integers of totally real fields, showing $\log U_{\mathcal{O}_{K}}({n}) = \frac{d}{4}n^{2}\log n + n^{2}(\frac{2\log\Delta_{K}-2d\log(2\pi)-3d}{8}) + O(n\log n)$ for all but finitely many real quadratic fields, and derives limsup results that imply rapid growth of $U_{\mathcal{O}_{K}}({n})$. The authors connect universal lattices to indecomposable lattices, Siegel mass formulas, and automorphism-group bounds to obtain both lower and upper bounds, then prove effective density results showing finiteness of fields with small minimal ranks or small universal criterion sets. They also study gaps between successive minimal ranks, criterion-set sizes, and uniqueness properties of minimal criterion sets, establishing broad finiteness and nonuniqueness phenomena. The work combines analytic, algebraic, and combinatorial methods (Wright’s theorem, mass formulas, discriminant bounds) to illuminate growth and distribution of $n$-universal lattices across totally real fields with explicit discriminant-related constants. Collectively, the results significantly advance understanding of how global field invariants govern universal lattice representations and provide effectively computable finiteness criteria.

Abstract

We prove an explicit asymptotic formula for the logarithm of the minimal ranks of $n$-universal lattices over the ring of integers of totally real number fields. We also show that, for any constant $C > 0$ and $n \geq 3$, there are only finitely many totally real fields with an $n$-universal lattice of rank at most $C$, with all such fields being effectively computable. Similarly, for any $n \geq 3$, we show that there are only finitely many totally real fields admitting an $n$-universal criterion set of size at most $C$, with all such fields likewise being effectively computable.

Asymptotics of $n$-universal lattices over number fields

TL;DR

The paper establishes a precise asymptotic for the logarithm of the minimal rank of -universal lattices over the ring of integers of totally real fields, showing for all but finitely many real quadratic fields, and derives limsup results that imply rapid growth of . The authors connect universal lattices to indecomposable lattices, Siegel mass formulas, and automorphism-group bounds to obtain both lower and upper bounds, then prove effective density results showing finiteness of fields with small minimal ranks or small universal criterion sets. They also study gaps between successive minimal ranks, criterion-set sizes, and uniqueness properties of minimal criterion sets, establishing broad finiteness and nonuniqueness phenomena. The work combines analytic, algebraic, and combinatorial methods (Wright’s theorem, mass formulas, discriminant bounds) to illuminate growth and distribution of -universal lattices across totally real fields with explicit discriminant-related constants. Collectively, the results significantly advance understanding of how global field invariants govern universal lattice representations and provide effectively computable finiteness criteria.

Abstract

We prove an explicit asymptotic formula for the logarithm of the minimal ranks of -universal lattices over the ring of integers of totally real number fields. We also show that, for any constant and , there are only finitely many totally real fields with an -universal lattice of rank at most , with all such fields being effectively computable. Similarly, for any , we show that there are only finitely many totally real fields admitting an -universal criterion set of size at most , with all such fields likewise being effectively computable.

Paper Structure

This paper contains 28 sections, 41 theorems, 147 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally $n$-universal $\mathcal{O}_{K}$-lattices
  4. Indecomposable $\mathcal{O}_{K}$-lattices
  5. Bounds for the Siegel mass constant
  6. The standard lattice $I_n$
  7. Asymptotics for large $n$
  8. Bounds for small $n$
  9. Bounds for $n = 2$
  10. Arbitrary unimodular lattices $L$
  11. Bounds on the number of rank $n$ unimodular $\mathcal{O}_{K}$-lattices
  12. Bounding the orders of automorphism groups
  13. Schur's bounds
  14. Lower bounds for $U_{\mathcal{O}_{K}}({n})$
  15. Case $d \geq 3$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $K$ be a totally real degree $d$ number field of discriminant $\Delta_K$. Then, with the exception of a finite set of at most $559$ real quadratic fields $K$, we have as $n \to \infty$.

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Example 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 61 more