Arithmetic Groups and the Lehmer Conjecture

Let $\mathbf{G}$ be a connected semisimple $\mathbb{R}$-group with $\operatorname{rank}_{\mathbb{R}}(\mathbf{G})\geq 2$. Then, $G=\mathbf{G}(\mathbb{R})$ is a semisimple Lie group and Margulis’ celebrated arithmeticity theorem [Mar75] states that every irreducible lattice $\Gamma\subset G$ is arithmetic (see §3 for precise definitions and statements).

The starting point of this paper is the following consequence of arithmeticity for irreducible non-cocompact lattices [Mar91, IX, (4.21-A)]:

Margulis [Mar91, IX, (4.21-B)] then conjectured that an analogous statement would hold for cocompact lattices. In fact, Margulis indicates that this conjecture would follow from a weaker form of Lehmer’s conjecture, which we recall below (see namely Conjecture (C)).

For the purpose of this paper, it will be useful to work with the following statement.

Let $P\in\mathbb{C}[X]$ be a monic polynomial of degree $d$ with roots $\alpha_{1},\,\ldots,\,\alpha_{d}\in\mathbb{C}$. The Mahler measure of $P$ is

In the following, let $\alpha_{1},\dots,\alpha_{s(P)}$ be an enumeration of the roots of $P$ in $\mathbb{C}$ which have absolute value strictly greater than 1, repeated according to their multiplicity, so that $s(P)$ denotes their count (with multiplicity) and we may rewrite $\operatorname{\mathcal{M}}(P)=\prod_{i=1}^{s(P)}|\alpha_{i}|.$ If $\alpha\in\overline{\mathbb{Q}}$ is an algebraic integer, the Mahler measure $\operatorname{\mathcal{M}}(\alpha)$ of $\alpha$ will be defined as the Mahler measure $\operatorname{\mathcal{M}}(P_{\alpha})$ of its minimal polynomial $P_{\alpha}$ over $\mathbb{Z}$. The Mahler measure on algebraic integers is obviously invariant under the action of $\operatorname{Ga\ell}(\overline{\mathbb{Q}}/\mathbb{Q})$.

The Mahler measure is multiplicative. By virtue of Kronecker’s theorem, for $P$ a monic, irreducible polynomial with integer coefficients, we have

In 1933, Lehmer [Leh33] asked whether one could find irreducible polynomials with integer coefficients whose Mahler measure gets arbitrarily close to 1 (but is not 1). It is conjectured that this is not possible:

In fact, Lehmer’s polynomial

for which $\operatorname{\mathcal{M}}(P_{\mathrm{Lehmer}})=1.17628...$, is suspected to attain the smallest Mahler measure greater than 1. Partial results towards Lehmer’s conjecture are known; we list some of them below.

We will be concerned with the following weaker version of Lehmer’s conjecture.

For a given $s$, we will call the statement in Conjecture (C) Lehmer’s conjecture at level $s$. In this way, Conjecture (C) could be described as “Lehmer’s conjecture at all levels”, and Lehmer’s conjecture (B) as “Lehmer’s conjecture at all levels uniformly”.

Fix an absolutely (almost) simple isotropic $\mathbb{R}$-group $\mathbf{F}$ and consider, for each integer $s\geq 1$, the family of semisimple $\mathbb{R}$-groups

The following theorem is the main result of this paper.

In view of the theorem, let us emphasize that Margulis’ conjecture for products of simple groups of any fixed $\mathbb{R}$-type and their extensions to $\mathbb{C}$ is already sufficient to imply the weak Lehmer conjecture, which in turn implies Margulis’ conjecture for all semisimple groups (see §3). For the sake of clarity, we will first carry the argument out in detail for type $\mathsf{A}_{n}$ split in §4. To complete the proof of the theorem, we then give a comprehensive treatment of all other $\mathbb{R}$-types in §5. It is noteworthy that the argument works for every $\mathbb{R}$-type in a similar way.

It would be interesting to know whether Margulis’ conjecture has implications for the full Lehmer conjecture (B), beyond Conjecture (C). Connections between the full Lehmer conjecture and linear groups have already been brought to light, for example in the work of Breuillard [Bre07] and of Breuillard–Varju [BV20].

Throughout the paper, we adopt the conventions of Bourbaki. In particular, $\mathbb{N}=\{0,1,2,\dots\}$ and we denote $\mathbb{N}^{*}=\mathbb{N}\backslash\{0\}$.

• •

  If $P\in\mathbb{C}[x]$, $s(P)$ denotes the number of roots of $P$ in $\mathbb{C}$ which have absolute value $>1$, counted with multiplicity.

• •

  $\operatorname{\mathcal{M}}(P)$ denotes the Mahler measure of $P\in\mathbb{C}[x]$.

• •

  If $\alpha\in\overline{\mathbb{Q}}$ is an algebraic integer, $P_{\alpha}$ denotes its minimal polynomial over $\mathbb{Z}$.

• •

  $\mathcal{O}_{K}$ denotes the ring of integers of a number field $K$.

• •

  $\mathrm{M}_{n}(A)$ denotes the set of $n\times n$ matrices with entries in an algebraic structure $A$, endowed with whichever structure is inherited from $A$.

The authors would like to thank the anonymous referee for his helpful comments, which improved the clarity of the paper and led to a more thorough treatment of the exceptional groups. The authors also thank E. Breuillard and G. Margulis for interesting conversations.

Let $\alpha\in\overline{\mathbb{Q}}$ be an algebraic integer which is not an integer nor a root of unity, and let $d\geq 2$ denote the degree of $P_{\alpha}$. Then, the following lower bound for $M(\alpha)$ is known:

This was proved by Voutier [Vou96, Theorem], improving the bound of Dobrowolski [Dob79, Theorem 1] (who obtained a factor of $1/1200$ instead of $1/4$). Another bound due to Laurent⁽³⁾⁽³⁾(3)In a private communication, Michel Laurent informed us that this bound was not published and that Schinzel [Sch73] proved the sharp inequality $\operatorname{\mathcal{M}}(\alpha)\geq\big{(}\tfrac{1+\sqrt{5}}{2}\big{)}^{d/2}$, for every $\alpha$ which is a totally real algebraic integer of degree $d\geq 2$. (1983), emphasizing the number of real roots, is given by Margulis [Mar91, p. 322]: if $P$ is a non-cyclotomic polynomial of degree $d\geq 2$ with $r$ real roots, then

where $c>0$ is an absolute constant.

Recall that a polynomial $P\in\mathbb{Z}[X]$ is called palindromic if

Every palindromic polynomial $P$ of odd degree is divisible by $X+1$, hence irreducible palindromic polynomials of degree $>1$ must have even degree.

Smyth [Smy71] proved that the polynomial $P_{\mathrm{Smyth}}(X)=X^{3}-X-1$ had the smallest Mahler measure among non-palindromic polynomials.

Although we will not use it, we note in passing another instance for which the Lehmer conjecture is known to hold. Borwein, Dobrowolski, and Mossinghoff [BDM07] proved that if $P$ is a polynomial of degree $d$ without cyclotomic factors, all of whose coefficients are odd integers, then

Recall that a Salem number is an algebraic integer $u\in\mathbb{R}$ which is $>1$ and all of whose Galois conjugates in $\mathbb{C}$ have absolute value $\leq 1$, with at least one of absolute value $=1$. Hence, any Salem number $\alpha$ satisfies $s(P_{\alpha})=1$ where $P_{\alpha}$ is the minimal polynomial of $\alpha$, and it is clear that $P_{\alpha}$ is palindromic. Conversely, any irreducible palindromic polynomial $P\in\mathbb{Z}[x]$ with $s(P)=1$ and $\deg P\geq 4$ is the minimal polynomial of a Salem number. The following conjecture is thus equivalent to Lehmer’s conjecture at level $s=1$ (see the paragraph after Conjecture (C)).

Salem numbers constitute an important family of algebraic numbers. For example, the polynomial $P_{\text{Lehmer}}$ with the smallest known Mahler measure (see §1.2) turns out to be the minimal polynomial of a Salem number.

A beautiful connection with discrete subgroups of Lie groups was established by Sury [Sur92]: he proved that Conjecture (D) was equivalent to the following conjecture.

Extending the picture to Kleinian groups, Neumann and Reid [NR92] formulated the following conjecture.

The short geodesic conjecture for hyperbolic 2-orbifolds is equivalent to Conjecture (E), hence to (D). On the other hand, the short geodesic conjecture for hyperbolic 3-orbifolds is equivalent to the following complex Salem conjecture. Call an algebraic integer $\alpha\in\mathbb{C}$ a complex Salem number if $\alpha$ is not real, $\alpha$ and its complex conjugate $\overline{\alpha}$ are the only two Galois-conjugates of $\alpha$ in $\mathbb{C}$ of absolute value $>1$, and $\alpha$ has at least one conjugate of absolute value $=1$.

In fact, the short geodesic conjecture for hyperbolic $3$-orbifolds implies the conjecture for $2$-orbifolds. This can be seen arithmetically, as the complex Salem conjecture implies the traditional Salem conjecture. Indeed, if $\alpha$ is a Salem number, then $P_{\alpha}(-x^{2})$ is the minimal polynomial of a complex Salem number with the same Mahler measure as $\alpha$.

For a detailed introduction to Conjecture (F), we refer the reader to the book of Maclachlan and Reid [MR03].

Salem numbers can also be used to obtain a lower bound for the length of closed geodesics in noncompact arithmetic hyperbolic orbifolds of even dimension $n$. This was done by Emery, Ratcliffe and Tschantz [ERT19]. More precisely, for any integer $n\geq 2$, let $\mathcal{H}^{n}$ denote hyperbolic $n$-space, and define

For any even dimension $n$, if $\Gamma\subset\operatorname{Isom}(\mathcal{H}^{n})$ is a non-uniform arithmetic lattice, then the length of any closed geodesic in $\mathcal{H}^{n}/\Gamma$ is at least $\beta_{n}$ [ERT19, Corollary 1.3]. It follows that Conjecture (D) is equivalent to the existence of a uniform lower bound on the length of closed geodesics in non-compact arithmetic hyperbolic orbifolds of even dimension.

Let us conclude this section with a consequence of Conjecture (A) in the context of locally symmetric spaces. Let $S$ be a symmetric space (e.g., $S=G/K$ where $G$ is a semisimple Lie group, $K\subset G$ a maximal compact subgroup). Following Gelander [Gel04], an $S$-manifold is a complete Riemannian manifold locally isometric to $S$, i.e., a manifold of the form $\Gamma\backslash S$, where $\Gamma\subset\operatorname{Isom}(S)$ is a discrete torsion-free subgroup. It is irreducible if $\Gamma$ is an irreducible lattice. For $d,v\in\mathbb{N}$, a $(d,v)$-simplicial complex is a simplicial complex with at most $v$ vertices, all of degree at most $d$. An interesting consequence of Conjecture (A) is the following conjecture of Gelander.

Conjecture (H) was recently proved by Fraczyk [Fra21, Theorem 1.16] for arithmetic 3-manifolds.

We can now state Margulis’ celebrated arithmeticity theorem.

In the remainder of this section, we will assume that $\operatorname{rank}_{\mathbb{R}}(\mathbf{G})\geq 2$, that $G$ has no compact factors (i.e. $\mathbf{G}^{\operatorname{is}}=\mathbf{G}$) and has trivial center. Let $\Gamma\subset G$ be an irreducible lattice. Then the following facts are among the key steps of the proof of the arithmeticity theorem. We refer the reader to [Zim84, §6.1] or [Ben08, §11.5] for proofs and details.

1. (3.1.1)

   The trace field $K=\mathbb{Q}(\operatorname{tr}\operatorname{Ad}\Gamma)$ of $\Gamma$, the field generated by the set $\{\operatorname{tr}(\operatorname{Ad}(\gamma))\,|\,\gamma\in\Gamma\}$, is a number field. This follows from Margulis’ superrigidity theorem, together with the fact that $\Gamma$ is finitely generated.

2. (3.1.2)

   Since $\Gamma$ is Zariski-dense in $\mathbf{G}(\mathbb{R})$, $\mathbf{G}$ can be defined over $K$. That is, there is a $K$-group $\mathbf{G}^{\prime}$ and a place $v_{0}$ of $K$ such that $K_{v_{0}}=\mathbb{R}$, $\mathbf{G}\cong\mathbf{G}^{\prime}$ as $\mathbb{R}$-groups, and the image of $\Gamma$ under this isomorphism lies in $\mathbf{G}^{\prime}(K)$. In the following, we will identify $\mathbf{G}$ and $\mathbf{G}^{\prime}$, and simply assume that $\Gamma\subset\mathbf{G}(K)$.

3. (3.1.3)

   There is a semisimple $\mathbb{Q}$-group $\mathbf{H}$ (endowed with a $\mathbb{Z}$-structure) with trivial center, an $\mathbb{R}$-epimorphism $\tau:\mathbf{H}\to\mathbf{G}$ with $(\ker\tau)(\mathbb{R})$ compact, and a homomorphism $\iota:\Gamma\to\mathbf{H}(\mathbb{Q})$ such that $\tau\circ\iota$ is the identity, $\iota(\Gamma)$ is Zariski dense in $\mathbf{H}$ and $\iota(\Gamma)$ is commensurable with $\mathbf{H}(\mathbb{Z})$. The group $\mathbf{H}$ can be constructed as the restriction of scalars $\operatorname{Res}_{K/\mathbb{Q}}(\mathbf{G})$ of $\mathbf{G}$ from $K$ to $\mathbb{Q}$.

We start by indicating how Conjecture (C) implies Conjecture (A). Note that as long as the lattice $\Gamma$ is arithmetic, the argument given below works equally well for groups of rank one. It shows in fact that Conjecture (C) implies Margulis’ conjecture for all semisimple $\mathbb{R}$-groups (in the sense defined below the statement of Conjecture (A)). Let thus $\mathbf{G}$ be a semisimple group and $\Gamma$ an irreducible arithmetic lattice in $\mathbf{G}(\mathbb{R})$.

Without loss of generality, we may assume that $\mathbf{G}$ has trivial center and is without anisotropic factors. Indeed, if $\mathbf{G}$ has center $\mathbf{C}$, and $U^{\prime}$ is a neighborhood of $(\mathbf{G}/\mathbf{C})(\mathbb{R})$ as in (A), then the preimage $U$ of $U^{\prime}$ under the canonical map $\pi:\mathbf{G}(\mathbb{R})\to(\mathbf{G}/\mathbf{C})(\mathbb{R})$ has the required property: if $\Gamma$ is an irreducible cocompact lattice in $\mathbf{G}(\mathbb{R})$, then $\pi(\Gamma)$ is an irreducible cocompact lattice in $(\mathbf{G}/\mathbf{C})(\mathbb{R})$; hence if $\gamma\in U$, $\pi(\gamma)$ must have finite order, and since $\mathbf{C}(\mathbb{R})$ is finite, so does $\gamma$. Similarly, if $\mathbf{G}=\mathbf{G}^{\operatorname{is}}\times\mathbf{G}^{\mathrm{anis}}$ and $U^{\prime}$ is a neighborhood of $\mathbf{G}^{\operatorname{is}}$ as in (A), then the preimage $U$ of $U^{\prime}$ under the canonical map $\pi:\mathbf{G}(\mathbb{R})\to\mathbf{G}^{\operatorname{is}}(\mathbb{R})$ has the required property: if $\Gamma$ is an irreducible cocompact lattice in $\mathbf{G}(\mathbb{R})$, then $\pi(\Gamma)$ is an irreducible cocompact lattice in $\mathbf{G}^{\operatorname{is}}(\mathbb{R})$; hence if $\gamma\in U$, $\pi(\gamma)$ has finite order $m$, and $\gamma^{m}\in\mathbf{G}^{\mathrm{anis}}(\mathbb{R})\cap\Gamma$. As $\mathbf{G}^{\mathrm{anis}}(\mathbb{R})$ is compact and $\Gamma$ is discrete, the latter is a finite group and $\gamma$ has finite order.

Let $\mathbf{H}$ be the group obtained in (3.1.3), so that we have the following diagram.

Recall also from (3.1.3) that $\tau$ has compact kernel and that $\iota(\Gamma)$ is commensurable with $\mathbf{H}(\mathbb{Z})$.

Since the adjoint representation $\mathbf{H}\to\operatorname{Ad}\mathbf{H}$ is defined over $\mathbb{Q}$, we can find a finite-index subgroup $\Lambda$ of $\mathbf{H}(\mathbb{Z})$ for which $\operatorname{Ad}(\Lambda)\subset\operatorname{Ad}(\mathbf{H})(\mathbb{Z})$ (see for example [Mar91, I, (3.1.1)]); in particular, $\operatorname{Ad}(\Lambda)$ preserves a lattice in $\operatorname{Lie}(\mathbf{H}(\mathbb{R}))$. Since $\Lambda$ and $\iota(\Gamma)$ are commensurable, $\operatorname{Ad}(\iota(\Gamma))$ also stabilizes a lattice in $\operatorname{Lie}(\mathbf{H}(\mathbb{R}))$ [Mar91, IX, (4.19)]. Hence the characteristic polynomials of the elements of $\operatorname{Ad}(\iota(\Gamma))$ have integer coefficients.

Let us write $\mathbf{H}(\mathbb{R})=F\times K$ as a direct product where $K$ is compact and $F$ is without compact factors. The morphism $\tau$ then induces an isogeny $F\to\mathbf{G}(\mathbb{R})$ and $\mathrm{d}\tau$ restricts to an isomorphism $\operatorname{Lie}(F)\to\operatorname{Lie}(\mathbf{G}(\mathbb{R}))$. Let $x\in\mathbf{H}(\mathbb{R})$ and write $\operatorname{Ad}_{\mathbf{H}(\mathbb{R})}(x)=\operatorname{Ad}_{F}(y)\oplus\operatorname{Ad}_{K}(z)$ for some $y\in F$, $z\in K$. If we denote $P_{x}$, $P_{y}$, $P_{z}$ the respective characteristic polynomials of $\operatorname{Ad}_{\mathbf{H}(\mathbb{R})}(x)$, $\operatorname{Ad}_{F}(y)$, $\operatorname{Ad}_{K}(z)$, we have that $P_{x}=P_{y}\cdot P_{z}$. Because $K$ is compact, all roots of $P_{z}$ in $\mathbb{C}$ must have absolute value 1; in consequence, $\operatorname{\mathcal{M}}(P_{z})=1$, hence $\operatorname{\mathcal{M}}(P_{x})=\operatorname{\mathcal{M}}(P_{y})$. Moreover, $P_{x}$ satisfies $s(P_{x})=s(P_{y})\leq\dim F=\dim\mathbf{G}$.

Now pick $\gamma\in\Gamma$ and apply the last paragraph to $x=\iota(\gamma)$. Since $\tau(\iota(\gamma))=\gamma$ and $\mathrm{d}\tau\circ\operatorname{Ad}(\iota(\gamma))=\operatorname{Ad}(\gamma)\circ\mathrm{d}\tau$, we obtain that the characteristic polynomial $P_{\gamma}$ of $\operatorname{Ad}_{\mathbf{G}(\mathbb{R})}(\gamma)$ equals $P_{y}$. We have in turn

Let $f:\mathbf{G}(\mathbb{R})\to[1,\infty[$ be defined by $f(g)=\operatorname{\mathcal{M}}(P_{g})$, where as before $P_{g}$ denotes the characteristic polynomial of $\operatorname{Ad}_{\mathbf{G}(\mathbb{R})}(g)$; note that $f$ is a continuous function. If the weak version of Lehmer’s conjecture holds at level $s=\dim\mathbf{G}$, we can find $\varepsilon>0$ such that any polynomial $P$ with integer coefficients and $s(P)\leq\dim\mathbf{G}$ satisfies either $\operatorname{\mathcal{M}}(P)=1$ or $\operatorname{\mathcal{M}}(P)>1+\varepsilon$. This applies to $P_{\iota(\gamma)}$ for any $\gamma\in\Gamma$: as we observed, $P_{\iota(\gamma)}$ has integer coefficients and $s(P_{\iota(\gamma)})\leq\dim\mathbf{G}$. Thus, by the above, the open neighborhood $U=f^{-1}([1,\varepsilon[)$ of $1$ in $\mathbf{G}(\mathbb{R})$ is such that $U\cap\Gamma$ consists of elements $\gamma$ for which $\operatorname{\mathcal{M}}(P_{\gamma})=\operatorname{\mathcal{M}}(P_{\iota(\gamma)})=1$. This means that for $\gamma\in U$, $P_{\gamma}$ is a product of cyclotomic polynomials. Since $\Gamma$ is cocompact, $\operatorname{Ad}(\gamma)$ is semisimple and thus $\operatorname{Ad}(\gamma)$, hence also $\gamma$, have finite order.

Given $P\in\mathbb{Z}[x]$, let us denote as before $\alpha_{1},\dots\alpha_{s(P)}$ the roots of $P$ in $\mathbb{C}$ of absolute value $>1$, and label them in such a way $\alpha_{1},\dots,\alpha_{r(P)}$ lie in $\mathbb{R}$ and $\alpha_{r(P)+1},\dots,\alpha_{s(P)}$ do not. In addition, let us order the latter roots so that $\overline{\alpha}_{r(P)+i}=\alpha_{(r(P)+s(P))/2+i}$ for $1\leq i\leq(s(P)-r(P))/2$. For each $s^{\prime}\in\mathbb{N}$, we will consider the set of polynomials

and for $r\leq s\leq s^{\prime}$, its subsets

By construction, $\mathscr{P}_{\leq s^{\prime}}$ is the disjoint union $\bigcup_{s=0}^{s^{\prime}}\bigcup_{r=0}^{s}\mathscr{P}_{s,r}$.

Suppose that Conjecture (C) does not hold. That is, there are $s^{\prime}\in\mathbb{N}$ and a sequence $(P_{m})_{m\in\mathbb{N}}$ of monic, irreducible polynomials with integer coefficients such that $s(P_{m})\leq s^{\prime}$ and $\operatorname{\mathcal{M}}(P_{m})\to 1$ while $\operatorname{\mathcal{M}}(P_{m})>1$. By virtue of Smyth’s theorem (§2.1), we may assume that each $P_{m}$ is palindromic, i.e. that $P_{m}\in\mathscr{P}_{\leq s^{\prime}}$. Moreover, up to extracting an appropriate subsequence, we may assume that $P_{m}\in\mathscr{P}_{s,r}$ for some fixed integers $r\leq s\leq s^{\prime}$.⁽⁴⁾⁽⁴⁾(4)If $s^{\prime}$ were to be the smallest integer for which Lehmer’s conjecture at level $s^{\prime}$ fails, then obviously $s=s^{\prime}$. Of course, $s>0$. In view of the bound (2.1.1), it must be that the sequence $(\deg P_{m})_{m}$ is unbounded. We may thus additionally assume that $\deg P_{m}>2s$. This implies the following important feature: $P_{m}$ must have a root of absolute value $1$. Indeed, $P_{m}$ has $s$ roots of absolute value $>1$, and because $P_{m}$ is palindromic, as many roots of absolute value $<1$.

The discussion above shows that the negation of Lehmer’s conjecture at level $s^{\prime}$ (cf. the paragraph following Conjecture (C)) amounts to the following statement.

Using statement (L_s,r) as the main ingredient, for each such pair $(s,r)$, we will construct in the semisimple $\mathbb{R}$-group

a sequence of cocompact lattices $\Gamma_{m}<\mathbf{G}_{s,r}(\mathbb{R})$ and a sequence of elements $\gamma_{m}\in\Gamma_{m}$ of infinite order, such that $\gamma_{m}\to e\in\mathbf{G}_{s,r}(\mathbb{R})$ as $m\to\infty$. This shall provide a counterexample to Margulis’ conjecture for $\mathbf{G}_{s,r}$, thus proving the first part of theorem 1.3. The equivalence of Margulis’ conjecture (A) and the weak version (C) of Lehmer’s conjecture then follows immediately by combining this with §3.2.

Let $r\leq s$ be as above. Pick a polynomial $P\in\mathscr{P}_{s,r}$, set $2d=\deg P$, and let $L=\mathbb{Q}(\alpha)$ denote the number field generated over $\mathbb{Q}$ by a root $\alpha$ of $P$. Since $P$ is palindromic, $\alpha^{-1}$ is also a root of $P$. In particular, $\alpha^{-1}$ is integral over $\mathbb{Z}$, and the assignment $\tau:\alpha\mapsto\alpha^{-1}$ defines a non-trivial automorphism of $L$ (which restricts to an automorphism of its ring of integers $\mathcal{O}_{L}$). Let $K=\mathbb{Q}(\alpha+{\alpha}^{-1})$ denote the subfield of $L$ generated over $\mathbb{Q}$ by $\alpha+{\alpha}^{-1}$. Since $K$ is the fixed field of $\tau$, $L$ is a quadratic extension of $K$ whose Galois group is $\{\operatorname{id},\tau\}$. Note that the minimal polynomial of $\alpha$ over $K$ is $X^{2}-(\alpha+{\alpha}^{-1})X+1$.

For $1\leq i\leq s$, let $\sigma_{i}$ denote the embedding $L\to\mathbb{C}$ defined by $\sigma_{i}(\alpha)=\alpha_{i}$. By definition, $L$ has $r$ real embeddings $\sigma_{1},\,\dots,\,\sigma_{r}$ and $s-r$ complex embeddings $\sigma_{r+1},\,\dots,\,\sigma_{s}$ for which the image of $\alpha$ has absolute value $>1$. $L$ also has ${d}-s$ pairs of conjugate complex embeddings $\sigma_{s+1},\,\overline{\sigma}_{s+1},\,\dots,\,\sigma_{d},\,\overline{\sigma}_{d}$ for which $\alpha$ maps to an element of absolute value $1$. Observe that precomposition with $\tau$ sends $\{\sigma_{1},\dots,\sigma_{r}\}$ and $\{\sigma_{r+1},\dots,\sigma_{s}\}$ respectively to the set of real and complex embeddings of $L$ for which the image of $\alpha$ has absolute value $<1$. Similarly, $\sigma_{i}\circ\tau=\overline{\sigma}_{i}$ for $s+1\leq i\leq d$. By construction, the embeddings $\sigma_{i}$ and $\sigma_{i}\circ\tau$ for $1\leq i\leq d$ agree on $K$. This shows that $\{\sigma_{1},\dots,\sigma_{d}\}$ is the complete set of embeddings of $K$ into $\mathbb{C}$ (we omit the restriction to $K$ from the notation). The image $\sigma_{i}(\alpha+{\alpha}^{-1})$ of the generator of $K$ belongs to $\mathbb{R}$ if and only if $\sigma_{i}(\alpha)\in\mathbb{R}$ or $|\sigma_{i}(\alpha)|=1$. In consequence, restricted to $K$, the embeddings $\sigma_{1},\dots,\sigma_{r}$ and $\sigma_{s+1},\dots,\sigma_{d}$ are real, $\sigma_{r+1},\dots,\sigma_{s}$ are complex (coming in conjugate pairs), and $K$ has signature $(r-s+d,(s-r)/2)$.

Let $h:L^{n}\times L^{n}\to L$ be the $\tau$-hermitian form given by

and let $\mathbf{G}=\operatorname{\mathbf{SU}}_{h}$ be the special unitary group associated to the form $h$. $\mathbf{G}$ is a linear algebraic $K$-group, whose group of $K$-points is (isomorphic to) the group of $h$-unitary matrices in $\mathrm{M}_{n}(L)$ of determinant $1$.⁽⁵⁾⁽⁵⁾(5)In fact, since the equations defining $\mathbf{G}$ can be taken with coefficients in $\mathcal{O}_{K}$, $\mathbf{G}$ can also be viewed as an $\mathcal{O}_{K}$-group scheme. The $K\otimes_{\mathbb{Q}}\mathbb{R}$-points of $\mathbf{G}$ can be computed easily by studying the behavior of the extension $L/K$ and of the form $h$ under the different embeddings $\{\sigma_{1},\dots,\sigma_{d}\}$ of $K$.

In the present setting, the extension $L/K$ splits at the places $\sigma_{1},\dots,\sigma_{r}$ (since these extend to real places of $L$) and also at the places $\sigma_{r+1},\dots,\sigma_{s}$ (since these are complex places of $K$), but not at $\sigma_{s+1},\dots,\sigma_{d}$ (since there we have $L\otimes_{K}K_{\sigma_{i}}\cong\mathbb{C}$). At these last places, the hermitian form $h$ becomes the standard hermitian form on $\mathbb{C}^{n}\times\mathbb{C}^{n}$ after identifying the completion of $L/K$ with $\mathbb{C}/\mathbb{R}$ via $\sigma_{i}$ ($s+1\leq i\leq d$). Altogether, we have

where $\operatorname{\mathbf{SU}}_{n}(\mathbb{R})$ denotes the standard anisotropic special unitary group over $\mathbb{R}$.

Let $\Gamma$ be the group of $h$-unitary $n\times n$ matrices of determinant $1$ with entries in $\mathcal{O}_{L}$.⁽⁶⁾⁽⁶⁾(6)With this definition, $\Gamma$ is (commensurable to) the $\mathcal{O}_{K}$-points of $\mathbf{G}$ when it is viewed as an $\mathcal{O}_{K}$-group scheme as above. A classical theorem of Borel and Harish–Chandra [BHC62] states that $\Gamma$ is a lattice in $\mathbf{G}(K\otimes_{\mathbb{Q}}\mathbb{R})$ when embedded diagonally using the inequivalent archimedean places of $K$ among $\sigma_{1},\dots,\sigma_{d}$. (Apply [BHC62, Theorem 7.8] to $\operatorname{Res}_{K/\mathbb{Q}}(\mathbf{G})$ and observe that $\operatorname{Res}_{K/\mathbb{Q}}(\mathbf{G})(\mathbb{R})\cong\mathbf{G}(K\otimes_{\mathbb{Q}}\mathbb{R})$; under this isomorphism, $\operatorname{Res}_{K/\mathbb{Q}}(\mathbf{G})(\mathbb{Z})$ is commensurable to $\Gamma$.) By virtue of the strong approximation theorem (for number fields), $\Gamma$ is an irreducible lattice. Moreover, if $P$ has at least one root in $\mathbb{C}$ of absolute value $1$ (i.e. if $s<d$), we claim that $\Gamma$ is a cocompact lattice in $\mathbf{G}(K\otimes_{\mathbb{Q}}\mathbb{R})$. Indeed, for any embedding $\sigma:K\to\mathbb{C}$ which extends to $L$ in such a way $|\sigma(\alpha)|=1$, the group $\mathbf{G}(K_{\sigma})\cong\operatorname{\mathbf{SU}}_{n}(\mathbb{R})$ is compact, and hence $\mathbf{G}$ must be $K$-anisotropic (as it is anisotropic over $K_{\sigma}$). The claim then follows from Godement’s criterion [BHC62, Theorem 11.8].

Let $\mathbf{G}_{s,r}$ be the $\mathbb{R}$-group defined in  4.1($\star$ ‣ 4.1). The canonical surjection $\pi:\mathbf{G}(K\otimes_{\mathbb{Q}}\mathbb{R})\to\mathbf{G}_{s,r}(\mathbb{R})$ has compact kernel $\prod_{i=1}^{d-s}\operatorname{\mathbf{SU}}_{n}(\mathbb{R})$; this implies that the image $\pi(\Gamma)$ of $\Gamma$ under $\pi$ is an irreducible, cocompact (provided $s<d$), lattice in $\mathbf{G}_{s,r}(\mathbb{R})$.

We set $\gamma$ to be the diagonal $h$-unitary $n\times n$-matrix $\operatorname{diag}(\alpha,\alpha^{-1},1,\dots,1)$ in $\Gamma=\mathbf{G}(\mathcal{O}_{K})$. Viewed as an element of $\mathbf{G}_{s,r}(\mathbb{R})$, the non-trivial block of $\pi(\gamma)$ is

where we labelled the roots $\alpha_{i}$ of $P$ in $\mathbb{C}$ as in §4.1.

Assume (L_s,r) holds, set $t=(s-r)/2$ and pick a sequence $(P_{m})_{m\in\mathbb{N}^{*}}$ of polynomials in $\mathscr{P}_{s,r}$ with at least one root of absolute value 1, satisfying

For each $m\in\mathbb{N}^{*}$, let us denote

all the objects stemming from the construction in §§4.2–4.4 applied to the polynomial $P_{m}$. As above, the non-trivial block of $\pi_{m}(\gamma_{m})\in\mathbf{G}_{s,r}(\mathbb{R})$ is

where $\alpha_{m,i}$ are roots of $P_{m}$ in $\mathbb{C}$ labelled according to §4.1.

Let $U_{m}$ denote the inversion-invariant neighborhood of $1$ in $\mathbb{C}$ given by

By construction, we have $1<|\alpha_{m,i}|\leq\operatorname{\mathcal{M}}(P_{m})<\exp(\eta_{m,t})$ for $1\leq i\leq r+t$. Unfortunately, it may very well be that $\alpha_{m,i}\notin U_{m}$. To remedy this, we will use Dirichlet’s simultaneous approximation theorem (see e.g. [Sch80, II, §1, Theorem 1A]). Let us recall it under the form we will use.