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Effective equidistribution of norm one elements in CM-fields

Shabnam Akhtari, Jeffrey D. Vaaler, Martin Widmer

arXiv:2507.10387v1math.NT

TL;DR

The paper studies the structure and height-ordered distribution of S_K, the subgroup whose elements embed into the complex unit circle. For CM-fields it identifies S_K with a norm kernel and proves effective equidistribution with a power-saving error term, while showing that equidistribution fails across many quadratic CM-fields.

  • Problem

    The paper addresses the structure of S_K and the lack of effective height-ordered equidistribution results for its unit-circle embeddings.

    Source: §1 · pp. 1–3

  • Method

    The authors combine the norm-kernel description, Hilbert 90, Weil-height analysis, geometry-of-numbers lattice counting, and a suitable fundamental domain.

    Source: §1 · p. 4

  • Results

    Theorem 1.1 gives asymptotics with a power-saving error term, and the embeddings are simultaneously and independently equidistributed with an explicit discrepancy bound.

    Source: §1 · p. 3

  • Takeaways & Limitations

    For a fixed CM-field, height-ordered elements of S_K admit effective distribution estimates, whereas the corresponding union over many quadratic CM-fields is not equidistributed.

    Source: §1 · p. 3

    The equidistribution conclusion does not extend to the union of rational and imaginary quadratic points across many quadratic CM-fields.

    Source: §1 · p. 3

Abstract

from arXiv · show

For a number field $K$ let $\mathcal{S}_K$ be the maximal subgroup of the multiplicative group $K^\times$ that embeds into the unit circle under each embedding of $K$ into the complex numbers. The group $\mathcal{S}_K$ can be seen as an archimedean counterpart to the group of units $\mathcal{O}_K^\times$ of the ring of integers $\mathcal{O}_K$. If $K=\mathbb{Q}(\mathcal{S}_K)$ is a CM-field then $\mathcal{S}_K/{\mathop{\rm Tor}\nolimits}(K^\times)$ is a free abelian group of infinite rank. If $K=\mathbb{Q}(\mathcal{S}_K)$ is not a CM-field then $\mathcal{S}_K=\{\pm 1\}$. In the former case $\mathcal{S}_K$ is the kernel of the relative norm map from $K^\times$ to the multiplicative subgroup $k^\times$ of the maximal totally real subfield $k$ of $K$.

1. INTRODUCTION

The paper characterizes the structure of S_K for CM-fields and proves effective equidistribution of its elements on unit circles when ordered by Weil height. It also develops counting and quotient methods supporting the result and identifies settings where equidistribution fails.

Group structure

  • S_K is {±1} when Q(S_K) is not a CM-field, while S_K/Tor(K×) has countably infinite rank when Q(S_K) is a CM-field.For CM-fields, S_K is also the kernel of the relative norm map N_K/k: K× → k×.Source: §1 · pp. 1–2

Effective equidistribution

  • The main theorem gives asymptotics with a power-saving error term for elements of S_K whose embeddings lie in prescribed arcs of the unit circle.The theorem applies to CM-fields of degree 2N and H ≥ 2, with L defined as log H for N = 1 and 1 for N ≥ 2.Source: §1 · pp. 2–3

Relation to prior work

  • The authors distinguish their approach from earlier work because Batyrev and Tschinkel’s general methods may yield the main term but do not provide effective results or an explicit power-saving error term.Earlier related approaches include Petersen and Sinclair’s analytic method and possible combinations involving Peyre’s observation.Source: §1 · p. 3

Effective equidistribution

  • Theorem 1.1 implies simultaneous independent equidistribution under the independent embeddings σ1, …, σN, together with an explicit discrepancy bound.The elements are enumerated by the absolute multiplicative Weil height.Source: §1 · pp. 1, 3

Scope and limitation

  • Equidistribution fails for the union of rational and imaginary quadratic points on the unit circle, which is not restricted to a fixed CM-field.For the quadratic-point set S2, the displayed asymptotic implies non-equidistribution when points are ordered by Weil height.Source: §1 · pp. 3–4

Proof strategy

  • The method combines Hilbert 90, Weil-height analysis, geometry-of-numbers lattice counting, and a height-accessible fundamental domain for the action of k×.The fundamental-domain construction is stated to extend, with minor modifications, to norm kernels for arbitrary quadratic extensions K/k.Source: §1 · p. 4

2. BASICS ON CM-FIELDS

This section characterizes CM-fields through complex conjugation and the group S_K, then identifies S_K with the kernel of the relative norm to the maximal totally real subfield.

  • A number field K is CM if and only if a nontrivial automorphism τ satisfies σ∘τ=ρ∘σ for every complex embedding σ.Source: §2 · p. 5
  • The composite of finitely many CM-fields is again a CM-field, enabling the characterization of Q(S_K) through finitely many elements of S_K.Source: §2 · p. 5
  • S_K≠{±1} exactly when Q(S_K) is a CM-field, while otherwise S_K={±1}.Source: §2 · p. 5
  • For a CM-field K with maximal totally real subfield k, S_K is the kernel of the norm map N_K/k:K×→k×.Source: §2 · p. 6
  • The relative norm N_K/k maps every element of K× to a totally positive element of k×.Source: §2 · p. 6
  • Hilbert’s Theorem 90 gives S_K=Im ψ and identifies S_K with K×/k×, where ψ has kernel k×.Source: §2 · p. 6

3. THE GROUP STRUCTURE OF SK

The section proves that, for a CM-field, the torsion-free quotient of S_K is a free abelian group of countably infinite rank by combining height properties with non-finite generation.

  • S_K/Tor(K×) is a free abelian group of countably infinite rank.Source: §3 · pp. 6–7
  • The quotient is not finitely generated because K×/k× is not finitely generated and is isomorphic to the relevant quotient of S_K.Source: §3 · p. 6
  • The logarithmic Weil height defines a discrete norm on S_K and its subgroups, forcing the quotient to be free.Source: §3 · p. 7

4. LATTICE POINT COUNTING

This section develops lattice-point estimates for sets with Lipschitz boundaries, providing quantitative control of lattice points inside bounded regions.

  • A set S is in Lip(D, M, L) if S is a subset of R^D covered by the images of M maps from [0,1]^(D−1), each satisfying |ϕ_i(x) − ϕ_i(y)| ≤ L|x − y|.Source: §4 · p. 7
  • The section’s principal lattice estimate is attributed to [15, Lemma 3.1].Source: §4 · pp. 7–8
  • Sets with Lipschitz boundaries are measurable and admit lattice-point estimates when contained in a Euclidean ball.Source: §4 · p. 7
  • When the origin is excluded, the lattice-point error can be bounded using the shortest nonzero lattice-vector length λ1.Source: §4 · p. 8

5. PRELIMINARIES

The section constructs fundamental domains and expanding regions in Minkowski space, then verifies the geometric regularity needed for lattice-point counting.

  • The Minkowski embedding uses one representative from each of the N pairs of complex-conjugate embeddings of K.Source: §5 · p. 8
  • The logarithmic unit lattice admits a reduced-basis fundamental domain F, which extends to a fundamental domain for the full unit group action.Source: §5 · p. 8
  • The region S_F(J;T) imposes logarithmic-radius coordinates from F(T) and argument coordinates from a product set J.Source: §5 · p. 8
  • These regions are measurable and are used to count lattice points as a step toward Theorem 1.1.Source: §5 · p. 9
  • For interval products I, the set S_F(I*;1) is bounded, excludes the origin, and has a boundary covered by finitely many Lipschitz maps.Source: §5 · p. 9

6. CONSTRUCTING A SUITABLE FUNDAMENTAL DOMAIN

The section constructs a suitable fundamental domain by decomposing ideals and proving that the map ψ gives a bijective parametrization of the relevant norm-one elements.

6. CONSTRUCTING A SUITABLE FUNDAMENTAL DOMAIN

  • Ideals in OK are decomposed using a base ideal from k, a divisor of the ramified-prime product P, and an ideal from IP.The set IP excludes ideals with divisors defined over k and requires coprimality to P.Source: §6 · p. 10
  • The decomposition AOKDB=A′OKD′B′ uniquely determines A, D, and B when the factors satisfy the stated conditions.This uniqueness underlies the disjointness arguments used later in the construction.Source: §6 · p. 10
  • At each complex embedding, ψ transforms arguments according to σn(ψ(β))=σn(β)ρ(σn(β)), so doubled argument intervals control membership in the target domain.The argument relation implies that ψ(β) satisfies the required interval conditions when β lies in the chosen domain.Source: §6 · pp. 11–12
  • The map ψ defined in (6.18) is bijective on the constructed domain and maps it onto SK(I, H).Surjectivity and injectivity are established in Lemmas 6.6 and 6.7, while Lemma 6.8 guarantees disjointness of the resulting decomposition.Source: §6 · pp. 12–13
  • Injectivity follows because equal images differ by an element of k×, while ideal uniqueness reduces this factor to a unit and the archimedean conditions reduce it to 1.The interval condition excludes the remaining possibility β′=−β.Source: §6 · pp. 12–13

7. SIEVING

The section applies sieving and inclusion–exclusion to reduce the count of norm-one elements with prescribed ideal conditions to manageable ideal sums.

7. SIEVING

  • Counting elements in Z∗(COKD)∩SC,D is reduced to an ordinary lattice-point counting problem by simple sieving arguments.The section fixes a finite target set and separates elements according to ideal divisibility conditions.Source: §7 · p. 13
  • The set Z∗(A) is characterized by excluding intersections with AAOK for proper ideals A of Ok and requiring the remaining factor to lie in IP.This identifies the sieved set with elements whose ideal factorization has the required non-base-field component.Source: §7 · p. 13
  • Inclusion–exclusion over prime ideals Q removes elements divisible by ideals defined over k while respecting coprimality with the ramified-prime product P.The resulting identities use the Möbius functions on ideals of Ok and OK.Source: §7 · pp. 14–15
  • Lemma 7.6 combines the preceding sieve identities to provide the counting formula used in the proof.The combination relies on Lemmas 7.1, 7.2, 7.4, and 7.5.Source: §7 · p. 15
  • The ideal sums are simplified using identities involving norms of the ramified-prime factors and ζk(2).The identities distinguish the case P=OK and use that the prime factors of P are exactly those dividing DK/k.Source: §7 · pp. 15–16

8. PROOF OF THEOREM 1.1

The proof of Theorem 1.1 combines the sieve reduction with lattice-point estimates under archimedean constraints, yielding the theorem’s main asymptotic constant.

8. PROOF OF THEOREM 1.1

  • The remaining count is converted into counting elements of an ideal satisfying prescribed archimedean conditions.The sieve formula from Lemma 6.9 and the estimates from Section 8 reduce the theorem to this constrained counting problem.Source: §8 · pp. 16–17
  • Embedding a non-zero ideal F into CN∼=R2N produces a lattice whose determinant and shortest-vector bounds support the required lattice-point estimates.The boundary of the admissible region is controlled by a Lipschitz condition, and the region lies in a zero-centred ball.Source: §8 · pp. 16–17
  • Lemma 8.2 gives an error estimate with L=log H when N=1 and L=1 when N≥2.For N=1, the argument restricts the ideal sum using Nk/Q(A)≤H and introduces an H log H bound.Source: §8 · p. 17
  • The leading constant in the main term is AK|I|, completing the proof of Theorem 1.1.This follows after combining the ideal sums, volume calculation, and lattice determinant estimates.Source: §8 · pp. 17–18

9. MINIMAL HEIGHTS IN COSETS OF k×

This section studies minimal Weil heights in multiplicative cosets of k× and characterizes exactly which cosets intersect the relative norm-one subgroup.

9. MINIMAL HEIGHTS IN COSETS OF k×

  • For an arbitrary quadratic extension K/k, the automorphism τ fixes k, the norm is NK/k(α)=ατ(α), and Hilbert’s Theorem 90 identifies Im ψ with SK/k.The kernel of ψ is k×, yielding an induced isomorphism relevant to the coset analysis.Source: §9 · p. 18
  • The section determines elements of minimal Weil height among cosets of k× that intersect SK/k.The height argument uses invariance under τ and compares H(αγ), H(α−1γ), and H(αγ−1).Source: §9 · pp. 18–19
  • A coset of k× in K× intersects SK/k exactly when it is the image under ˆψ−1 of a square in SK/k.This criterion is Lemma 9.3 and follows from the characterization ψ(α)=α2 for α∈SK/k.Source: §9 · p. 19
  • The identity ˆψ−1(β2)=ˆψ−1(ψ(β))=βk× links squares in SK/k to the corresponding multiplicative cosets.This is the algebraic step used in the forward direction of Lemma 9.3.Source: §9 · p. 19
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