Skip to main content Accessibility help
×
Home
Hostname: page-component-768dbb666b-x9ds4 Total loading time: 2.665 Render date: 2023-02-02T15:57:03.366Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Effective bounds on differences of singular moduli that are S-units

Published online by Cambridge University Press:  22 September 2022

FRANCESCO CAMPAGNA*
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany e-mail: campagna@mpim-bonn.mpg.de
Rights & Permissions[Opens in a new window]

Abstract

Given a singular modulus $j_0$ and a set of rational primes S, we study the problem of effectively determining the set of singular moduli j such that $j-j_0$ is an S-unit. For every $j_0 \neq 0$ , we provide an effective way of finding this set for infinitely many choices of S. The same is true if $j_0=0$ and we assume the Generalised Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a “uniformity conjecture” for singular S-units.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

1. Introduction

This paper is devoted to the study of some diophantine properties of j-invariants of elliptic curves with complex multiplication defined over $\mathbb{C}$ . These numbers, which are classically known by the name of singular moduli, have been studied since the time of Kronecker and Weber, who were interested in explicit generation of class fields relative to imaginary quadratic fields [ Reference Frei16 ]. In this respect, singular moduli prove to be a useful tool, since they are indeed algebraic integers which can be used to generate ring class fields of imaginary quadratic fields [ Reference Cox11 , Theorem 11.1].

During the last decade, there has been an increasing interest in understanding more diophantine properties of these invariants. One of the questions that, for instance, has been addressed is the following: given a set S of rational primes, is the set of singular moduli that are S-units (singular S-units) finite? In case of an affirmative answer, is it possible to provide an effective method to explicitly compute this set? This question, which has been originally motivated by the proof of some effective results of André–Oort type (see [ 3] and [ Reference Kühne29 ]), does not have at present a complete answer. Several partial results have nonetheless been achieved.

In [ Reference Bilu, Habegger and Kuhne2 ] it is proved, building on the previous ineffective result of Habegger [ Reference Habegger23 ], that no singular modulus can be a unit in the ring of algebraic integers. This settles the case $S=\emptyset$ of the question. With different techniques, Li generalises this theorem and proves in [ Reference Li34 ] that for every pair $j_1,j_2 \in \overline{\mathbb{Q}}$ of singular moduli, the algebraic integer $\Phi_N(\,j_1,j_2)$ can never be a unit. Here $\Phi_N(X,Y) \in \mathbb{Z}[X,Y]$ denotes the classical modular polynomial of level N, so we recover the main result of [ Reference Bilu, Habegger and Kuhne2 ] by setting $j_2=0$ and $N=1$ . In a different direction, the fact that no singular modulus is a unit has been used by the author of this manuscript to prove that, if $S_0$ is the infinite set of primes congruent to 1 mod 3, then the set of singular moduli that are $S_0$ -units is empty [ Reference Campagna8 ]. Moreover, very recently Herrero, Menares and Rivera–Letelier gave an ineffective proof of the fact that for every fixed singular modulus $j_0 \in \overline{\mathbb{Q}}$ and for every finite set of primes S, the set of singular moduli j such that $j-j_0$ is an S-unit is finite, see [ Reference Herrero, Menares and Rivera–Letelier25 ] [ Reference Herrero, Menares and Rivera–Letelier26 ] and [ Reference Herrero, Menares and Rivera–Letelier27 ].

In this paper we explore the possibility of providing, for a given singular modulus $j_0$ and for specific sets of primes S, an effective procedure to determine the set of all singular moduli j such that $j-j_0$ is an S-unit. In order to better state our main results, we introduce some notation. First of all, we say that a singular modulus has discriminant $\Delta \in \mathbb{Z}$ if it is the j-invariant of an elliptic curve $E_{/\mathbb{C}}$ with complex multiplication by an order of discriminant $\Delta$ . Let $j \in \overline{\mathbb{Q}}$ be a singular modulus of discriminant $\Delta$ and let $S \subseteq \mathbb{N}$ be a finite set of prime numbers. We call the pair (j,S) a nice $\Delta$ -pair if the following two conditions hold:

  1. (1) every prime $\ell \in S$ splits completely in $\mathbb{Q}(\,j)$ ;

  2. (2) we have $\ell \nmid N_{\mathbb{Q}(\,j)/\mathbb{Q}}(\,j) N_{\mathbb{Q}(\,j)/\mathbb{Q}}(\,j-1728) \Delta$ for all $\ell \in S$ , where $N_{\mathbb{Q}(\,j)/\mathbb{Q}}(\!\cdot\!)$ denotes the norm map from $\mathbb{Q}(\,j)$ to $\mathbb{Q}$ .

The first main result of the paper is the following.

Theorem 1·1. Let $(\,j_0,S)$ be a nice $\Delta_0$ -pair with $\Delta_0 < -4$ and $\#S \leq 2$ . Then there exists an effectively computable bound $B=B(\,j_0,S) \in \mathbb{R}_{\geq 0}$ such that the discriminant $\Delta$ of every singular modulus $j \in \overline{\mathbb{Q}}$ for which $j-j_0$ is an S-unit satisfies $|\Delta| \leq B$ . Moreover, if the extension $\mathbb{Q} \subseteq \mathbb{Q}(\,j_0)$ is not Galois, then the discriminant $\Delta$ of any singular modulus j such that $j-j_0$ is an S-unit is of the form $\Delta=p^{2n} \Delta_0$ for some prime $p \in S$ and some non-negative integer n.

The bound $B(\,j_0,S)$ in the statement of Theorem 1·1 can be made explicit from its proof. To give an idea of what kind of bounds one can get, we take $j_0=-3375$ , the j-invariant of any elliptic curve with complex multiplication by $\mathbb{Z}[(1+\sqrt{-7})/2]$ , and choose S to be any subset of at most two elements in $\{13,17,19\}$ . We get the following result.

Theorem 1·2. Let $j \in \overline{\mathbb{Q}}$ be a singular modulus of discriminant $\Delta$ , and let $S\;:\!=\;\{13,17\}$ . If $j+3375$ is an S-unit, then $|\Delta| \leq 10^{81}$ . The same holds with $S'=\{13,19 \}$ and $S''=\{17,19 \}$ .

In general, in order to construct nice $\Delta$ -pairs it suffices to fix a singular modulus j of discriminant $\Delta$ and to choose, among the set of primes splitting completely in $\mathbb{Q} \subseteq \mathbb{Q}(\,j)$ , a finite subset S satisfying condition (2) above. Since the set of rational primes that are totally split in $\mathbb{Q}(\,j)$ is infinite by the Chebotarëv’s density theorem, this gives rise to infinitely many nice $\Delta$ -pairs for a fixed discriminant $\Delta$ . We remark that if $\mathbb{Q} \subseteq \mathbb{Q}(\,j)$ is not Galois, then every prime splitting completely in this extension will be also totally split in $\mathbb{Q}(\sqrt{\Delta})$ (see the end of the proof of Theorem 1·1). Hence, in some cases one could use [ Reference Campagna7 , Theorem 2.2.1] to show that, for appropriate nice $\Delta_0$ -pair $(\,j_0,S)$ with $\mathbb{Q} \subseteq \mathbb{Q}(\,j_0)$ non-Galois, the set of singular moduli $j \in \overline{\mathbb{Q}}$ for which $j-j_0$ is an S-unit is in fact empty. We point out that the set of singular moduli j that generate a Galois extension of $\mathbb{Q}$ is finite, see Proposition 4·2.

The reason why Theorem 1·1 only deals with sets S containing at most two primes will be apparent from its proof, which we now sketch. Our strategy follows the same idea used in [ Reference Bilu, Habegger and Kuhne2 ]: given a singular modulus $j \in \overline{\mathbb{Q}}$ such that $j-j_0$ is an S-unit, we compute the (logarithmic) Weil height $h(\,j-j_0)$ . This is defined, for every $x \in \overline{\mathbb{Q}}$ , as

\begin{equation*}h(x)=\frac{1}{[K\;:\;\mathbb{Q}]} \sum_{v \in \mathcal{M}_K} [K_v\;:\;\mathbb{Q}_v] \log^+ |x|_{v}, \end{equation*}

where $K\;:\!=\;\mathbb{Q}(x)$ is the field generated by x over the rationals, $\mathcal{M}_K$ is the set of all places of K, the integer $[K_v\;:\;\mathbb{Q}_v]$ is the local degree at the place v and $\log^+|x|_v\;:\!=\;\log \max \{1,|x|_v\}$ . Here, for every non-archimedean place v corresponding to the prime ideal $\mathfrak{p}_v$ lying above the rational prime $p_v$ , the absolute value $|\!\cdot\!|_v$ is normalised in such a way that

\begin{equation*}|x|_v=p_v^{- v_{\mathfrak{p}_v}(x)/e_v },\end{equation*}

where $e_v$ is the ramification index of $\mathfrak{p}_v$ over $p_v$ and $v_{\mathfrak{p}_v}(x)$ is the exponent with which $\mathfrak{p}_v$ appears in the prime ideal factorisation of the $\mathcal{O}_K$ -fractional ideal generated by x. Hence the logarithmic Weil height naturally decomposes into an “archimedean” and “non-archimedean” part.

Since $j-j_0$ is an algebraic integer, the non-archimedean part of its Weil height vanishes. In order to exploit the fact that the above difference is an S-unit, we rather compute the height of $(\,j-j_0)^{-1}$ . Using standard properties of the Weil height, we obtain

\begin{equation*}h(\,j-j_0)=h((\,j-j_0)^{-1})= \left( \text{archimedean part} \right) + \left( \text{non-archimedean part} \right)\end{equation*}

with

\begin{equation*}\left( \text{non-archimedean part} \right)=\frac{1}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \sum_{\mathfrak{p}} f_\mathfrak{p} \cdot v_{\mathfrak{p}}(\,j-j_0) \log \ell_\mathfrak{p},\end{equation*}

where the sum is taken over the prime ideals of $\mathbb{Q}(\,j-j_0)$ lying above the rational primes contained in S and, for every such prime $\mathfrak{p}$ , we denote by $f_\mathfrak{p}$ and $\ell_\mathfrak{p}$ respectively the inertia degree and the residue characteristic of $\mathfrak{p}$ . Our goal is to effectively bound this height from above and from below in such a way that the two bounds contradict each other when the absolute value of the discriminant of the singular modulus j becomes large. This will give the desired effective bound.

An upper bound for the archimedean part has been already studied in [ Reference Bilu, Habegger and Kuhne2 ] and [ Reference Cai6 ]. In order to estimate from above the non-archimedean part, we have to understand the valuation of $j-j_0$ at primes above S. This requires the use of some deformation-theoretic arguments involving quaternion algebras, and constitutes the technical core of the paper. We detail this discussion in Section 3, which culminates in the proof of Theorem 3·1, where we obtain the seeked estimates. Concerning the lower bound for the Weil height, we compare it to the stable Faltings height of the elliptic curve with complex multiplication having j as singular invariant. Using work of Colmez [ Reference Colmez9 ] and Nakkajima–Taguchi [ Reference Nakkajima and Taguchi36 ] it is possible to relate this Faltings height to the logarithmic derivative of the L-function corresponding to the CM field evaluated in 1. The known lower bounds on this logarithmic derivative become strong enough for our purposes only if we restrict to sets S containing no more than two primes.

When $\Delta_0 \in \{-3,-4\}$ , i.e. when $j_0 \in \{0,1728\}$ , the same techniques also lead to similar finiteness results, but one has to be more careful in theses cases since the complex elliptic curves having $j_0$ as singular invariant possess non-trivial automorphisms. This is indeed a problem, and will force us to resort to the Generalised Riemann Hypothesis (GRH) in the case $j_0=0$ . Here are the results that we obtain in these two cases.

Theorem 1·3. Let $S_0$ be the set of rational primes congruent to 1 modulo 4, let $\ell \geq 5$ be an arbitrary prime and set $S_\ell\;:\!=\;S_0 \cup \{\ell \}$ . Then there exists an effectively computable bound $B=B(\ell) \in \mathbb{R}_{\geq 0}$ such that the discriminant $\Delta$ of every singular modulus $j \in \overline{\mathbb{Q}}$ for which $j-1728$ is an $S_\ell$ -unit satisfies $|\Delta| \leq B$ .

Theorem 1·4. Let $S_0$ be the set of rational primes congruent to 1 modulo 3, let $\ell \geq 5$ be an arbitrary prime and set $S_\ell\;:\!=\;S_0 \cup \{\ell \}$ . If the Generalised Riemann Hypothesis holds for the Dirichlet L-functions attached to imaginary quadratic number fields, then there exists an effectively computable bound $B=B(\ell) \in \mathbb{R}_{\geq 0}$ such that the discriminant $\Delta$ of every singular $S_\ell$ -unit $j \in \overline{\mathbb{Q}}$ satisfies $|\Delta| \leq B$ .

The statement of Theorem 1·4 has been simplified for the sake of exposition in this introduction. Indeed, one does not need the full strength of GRH to carry out the proof, but only a weaker, more technical assumption on the logarithmic derivative at $s=1$ of the Dirichlet L-functions of imaginary quadratic fields. We refer the reader to Theorem 5·5 for the stronger result that we are actually going to prove.

After performing some numerical computations, one soon realizes that, given a singular modulus $j_0$ and a finite set of primes S, the upper bound for the number of singular moduli j such that $j-j_0$ is an S-unit seems not to depend on the primes contained in S but only on the size of the set S itself. Since being an S-unit is a Galois-invariant property, this would entail a bound, depending only on $\#S$ , on the size of the Galois orbits of such j’s and, by the Brauer–Siegel theorem [ Reference Lang32 , Chapter XIII, Theorem 4], an analogous bound on their discriminants. Choosing $j_0=0$ , this observation leads to the formulation of the following conjecture for singular S-units.

Conjecture 1·5. For every $s \in \mathbb{N}$ , the number of singular moduli that are S-units for some set of rational primes S with $\#S=s$ is finite.

This conjecture, which we will call “uniformity conjecture for singular S-units”, will be discussed in Section 7, where we also provide some numerical data to support it.

The paper is structured as follows. In Section 2 we recall known facts from the theory of complex multiplication and quaternion algebras, and we fix the terminology which will be used in the paper. In Section 3 we prove Theorem 3·1, which allows to bound the $\ell$ -adic absolute value of differences of singular moduli for certain primes $\ell$ . In Section 4 we provide a proof of Theorems 1·1 and 1·2 while in Section 5 we give a proof of Theorems 1·3 and 1·4. Section 6 discusses the optimality of the bounds found in Theorem 3·1 in the case $j_0=0$ . Finally in Section 7 we provide numerical evidence for some uniformity conjectures concerning differences of singular moduli that are S-units.

2. Prelude: CM elliptic curves, quaternion algebras and optimal embeddings

We recall in this section some of the main definitions and results that will be used in the rest of the paper. We fix once and for all an algebraic closure $\overline{\mathbb{Q}} \supseteq \mathbb{Q}$ of the rationals.

A singular modulus is the j-invariant of an elliptic curve defined over $\overline{\mathbb{Q}}$ with complex multiplication. For every imaginary quadratic order $\mathcal{O}$ of discriminant $\Delta \in \mathbb{Z}_{<0}$ there are exactly $C_{\Delta}$ isomorphism classes of elliptic curves over $\overline{\mathbb{Q}}$ with complex multiplication by $\mathcal{O}$ , where $C_{\Delta} \in \mathbb{N}$ denotes the class number of the order $\mathcal{O}$ . Hence, there are $C_{\Delta}$ corresponding singular moduli, which are all algebraic integers and form a full Galois orbit over $\mathbb{Q}$ (see [ Reference Cox11 , Corollary 10.20], [ Reference Cox11 , Theorem 11.1] and [ Reference Cox11 , Proposition 13.2]). We call them singular moduli of discriminant $\Delta$ or singular moduli relative to the order $\mathcal{O}$ . Reversing subject and complements, we will sometimes also speak of discriminant, CM order, CM field, etc… associated to a singular modulus j.

Recall that, given a number field $K \subseteq \overline{\mathbb{Q}}$ and a set $S\subseteq \mathbb{N}$ of rational primes, an element $x \in K$ is called an S-unit if for every prime $\mathfrak{p} \subseteq K$ not lying above any prime $p \in S$ , we have $x \in \mathcal{O}_{K_\mathfrak{p}}^{\times}$ , where $\mathcal{O}_{K_\mathfrak{p}} \subseteq K_\mathfrak{p}$ denotes the ring of integers in the completion $K_\mathfrak{p}$ of the number field K at the prime $\mathfrak{p}$ . Note that this definition does not depend on the particular number field K containing x. Moreover, if x is actually an algebraic integer, then x is an S-unit if and only if its absolute norm $N_{K/\mathbb{Q}}(x)$ is divided only by primes in S. In this paper we are interested in the study of S-units of the form $j-j_0$ with $j,j_0 \in \overline{\mathbb{Q}}$ singular moduli. If $j_0=0$ is the unique singular modulus of discriminant $\Delta_0=-3$ , we speak of singular S-units. As we will see, the study of these singular differences is intimately related to the theory of supersingular elliptic curves and quaternion algebras. We summarize some relevant results from this theory.

Let k be a field of characteristic $\textrm{char}(k)=\ell >0$ with algebraic closure $\overline{k} \supseteq k$ and let $E/k$ be an elliptic curve. We say that E is supersingular if $E[\ell](\overline{k})=\{O\}$ i.e. if the unique $\ell$ -torsion point of E defined over $\overline{k}$ is the identity $O \in E(\overline{k})$ . If this is the case, then the endomorphism ring $\textrm{End}_{\overline{k}}(E)$ is isomorphic to a maximal order in the unique (up to isomorphism) quaternion algebra over $\mathbb{Q}$ ramified only at $\ell$ and $\infty$ (see [ Reference Deuring13 ] or [ Reference Voight44 , Proposition 42.1.7 and Theorem 42.1.9] for a modern exposition). If k is a finite field, then by Deuring’s lifting theorem [ Reference Lang31 , Chapter 13, Theorem 14] every supersingular elliptic curve over $\overline{k}$ arises as the reduction of some elliptic curve with complex multiplication defined over a number field. Finding such a CM elliptic curve is difficult in general. In contrast, it is very easy to see for which primes a CM elliptic curve defined over a number field has good supersingular reduction. Namely, let F be a number field with ring of integers $\mathcal{O}_F$ and let $E_{/F}$ be an elliptic curve with CM by an order in an imaginary quadratic field K. Fix a prime ideal $\mu \subseteq \mathcal{O}_F$ lying above a rational prime $\ell \in \mathbb{Z}$ that does not split in K. Since CM elliptic curves have potential good reduction everywhere (see [ Reference Silverman43 , VII, Proposition 5.5]) we can assume, possibly after enlarging the field of definition F, that E has good reduction at $\mu$ and that all the geometric endomorphisms of E are defined over F. Then the reduced elliptic curve $\widetilde{E}\;:\!=\; E \text{ mod } \mu$ is supersingular by [ Reference Lang31 , Chapter 13, Theorem 12]. Moreover, the natural reduction map modulo $\mu$ induces an injective ring homomorphism

\begin{equation*}\varphi\;:\;\textrm{End}_{F}(E) \hookrightarrow \textrm{End}_{\overline{\mathbb{F}}_\ell}(\widetilde{E})\end{equation*}

between the corresponding endomorphism rings (see [ Reference Silverman42 , II, Proposition 4.4]). As we will see in Theorem 2·4, in many cases (depending on the prime $\ell$ and on the CM order of E) the above embedding will be optimal, in the following sense.

Let $\mathbb{B}$ be a quaternion algebra over $\mathbb{Q}$ and let $R \subseteq \mathbb{B}$ be an order, i.e. a full $\mathbb{Z}$ -lattice which is also a subring of $\mathbb{B}$ . Let $\mathbb{Q} \subseteq K$ be a quadratic field extension and let $\mathcal{O} \subseteq K$ also be an order. Any ring homomorphism $\varphi\;:\;\mathcal{O} \to R$ can be naturally extended, after tensoring with $\mathbb{Q}$ , to a ring homomorphism $K \to \mathbb{B}$ that we still denote by $\varphi$ , with abuse of notation. We say that an injective ring homomorphism $\iota\;:\;\mathcal{O} \hookrightarrow R$ is an optimal embedding if

\begin{equation*}\iota(K) \cap R= \iota(\mathcal{O}),\end{equation*}

where the above intersection takes place in $\mathbb{B}$ . There is a simple criterion which allows to determine whether a given imaginary quadratic order optimally embeds into a quaternionic order. In order to state it, let us denote by $\textrm{trd}, \textrm{nrd}\;:\;\mathbb{B} \to \mathbb{Q}$ respectively the reduced trace and the reduced norm in the quaternion algebra $\mathbb{B}$ , see [ Reference Voight44 , Section 3.3]. This notation will be in force for the rest of the paper.

Lemma 2·1. Let R be an order in a quaternion algebra $\mathbb{B}$ and $\mathcal{O}$ an order of discriminant $\Delta$ in an imaginary quadratic field K. Let $V\subseteq \mathbb{B}$ be the subspace of pure quaternions

\begin{equation*}V\;:\!=\;\{x\in \mathbb{B}\;:\;{{trd}}(x)=0\}.\end{equation*}

Then $\mathcal{O}$ embeds (resp. optimally embeds) in R if and only if $|\Delta|$ is represented (resp. primitively represented) by the ternary quadratic lattice

\begin{equation*}R_0\;:\!=\; V \cap (\mathbb{Z}+2R)\end{equation*}

endowed with the natural scalar product induced by the reduced norm on $\mathbb{B}$ .

Remark 2·2. This lemma has been proved for non-optimal embeddings and for maximal orders R in [ Reference Gross19 , Proposition 12.9]. Probably for this reason, the lattice $R_0$ is sometimes called the Gross lattice associated to R. The argument in loc. cit. easily generalises to our situation. We provide a full proof for completeness.

Proof. We first prove that $\mathcal{O}$ embeds in R if and only if $|\Delta|$ is represented by $R_0$ , and we discuss conditions on the optimality of this embedding at a second stage.

Write $\mathcal{O}=\mathbb{Z}[ ({\Delta+\sqrt{\Delta}})/{2} ]$ and suppose first that $f\;:\;\mathcal{O} \hookrightarrow R$ is an embedding. Let $b\;:\!=\;f(\sqrt{\Delta})$ so that $\textrm{trd}(b)=0$ and $\textrm{nrd}(b)=|\Delta|$ . Since

\begin{equation*}f\left( \frac{\Delta+\sqrt{\Delta}}{2} \right)=\frac{\Delta+b}{2} \in R\end{equation*}

we see that $b\in R_0$ so that $|\Delta|$ is represented by this lattice. Suppose conversely that there exists $b \in R_0$ such that $\textrm{nrd}(b)=|\Delta|$ . Since $\textrm{trd}(b)=0$ , we see that $b^2=\Delta$ . By writing $b=a+2r$ with $a\in \mathbb{Z}$ and $r\in R$ , one has

\begin{equation*}b^2=(a+2r)^2=a^2+4r^2+4ar=\Delta\end{equation*}

and this immediately implies that $a\equiv \Delta \text{ mod } 2$ , so that $\Delta+b \in 2R$ . Hence we have $(\Delta+b)/2 \in R$ and we obtain an embedding $f\;:\;\mathcal{O}\hookrightarrow R$ by setting

(1) \begin{equation} f\left( \frac{\Delta+\sqrt{\Delta}}{2} \right)=\frac{\Delta+b}{2}.\end{equation}

We now discuss optimality. Fix $\{\alpha_1,\alpha_2, \alpha_3 \}$ to be a basis of $R_0$ as a $\mathbb{Z}$ -module and let Q(X, Y, Z) be the ternary quadratic form induced by the reduced norm with respect to this basis.

Assume that $f\;:\;\mathcal{O} \hookrightarrow R$ is an optimal embedding. By the proof above, we know that $b\;:\!=\;f(\sqrt{\Delta}) \in R_0$ is such that $\textrm{nrd}(b)=|\Delta|$ . Suppose by contradiction that $b=a_1 \alpha_1+a_2 \alpha_2+a_3 \alpha_3$ with $a_1,a_2,a_3 \in \mathbb{Z}$ not coprime, so that $c\;:\!=\; \gcd(a_1,a_2,a_3)>1$ (we adopt the convention that the greatest common divisor is always positive). Then $\widetilde{b}\;:\!=\;b/c \in R_0$ satisfies

\begin{equation*}\widetilde{b}^2=\frac{\Delta}{c^2} \in \mathbb{Z}\quad \text{and}\quad \frac{1}{2} \left(\frac{\Delta}{c^2} + \widetilde{b}\right) \in R.\end{equation*}

in the same way as above. Thus $\frac{1}{2} \left( ({\Delta}/{c^2})+({\sqrt{\Delta}}/{c})\right) \in K$ is an algebraic integer and the order $\widetilde{\mathcal{O}}\;:\!=\;\mathbb{Z} \left[ \frac{1}{2} \left(({\Delta}/{c^2})+({\sqrt{\Delta}}/{c})\right) \right]$ , which strictly contains $\mathcal{O}$ , also embeds in R through the extension $f\;:\;K \hookrightarrow \mathbb{B}$ . This contradicts the optimality of $f\;:\;\mathcal{O} \hookrightarrow R$ .

Suppose now that $|\Delta|$ is primitively represented by $R_0$ i.e. that there exist $a_1,a_2,a_3 \in \mathbb{Z}$ coprime such that $\textrm{nrd}(a_1 \alpha_1+a_2 \alpha_2+a_3 \alpha_3)=|\Delta|$ . We want to show that, setting $b\;:\!=\;a_1 \alpha_1+a_2 \alpha_2+a_3 \alpha_3$ , the embedding f defined by (1) is optimal. We will equivalently prove that, if $c\in \mathbb{Z}_{>0}$ is such that $\widetilde{\mathcal{O}}\;:\!=\;\mathbb{Z} \left[ \frac{1}{2} \left(({\sqrt{\Delta}}/{c})+({\Delta}/{c^2})\right) \right]$ is an order, then

(2) \begin{equation} f\!\left( K \right) \cap R = f\left( \widetilde{\mathcal{O}} \right)\end{equation}

implies $\widetilde{\mathcal{O}}=\mathcal{O}$ . Since $b=f(\sqrt{\Delta})$ , equality (2) entails $\frac{1}{2} \left(({b}/{c})+({\Delta}/{c^2})\right) \in R$ so that $b/c \in R_0$ . But now

\begin{equation*}b/c=\frac{a_1}{c} \alpha_1 + \frac{a_2}{c} \alpha_2 + \frac{a_3}{c} \alpha_3 \in R_0\end{equation*}

and all the coefficients $a_i/c$ must be integral since $\{\alpha_1, \alpha_2, \alpha_3 \}$ is a basis of $R_0$ as a $\mathbb{Z}$ -module. By assumption, the $a_i$ ’s are coprime, so we must have $c=1$ . Hence $\widetilde{\mathcal{O}}=\mathcal{O}$ and this concludes the proof.

Remark 2·3. The proof of Lemma 2·1 actually establishes a bijection between the set of embeddings $f\;:\;\mathcal{O} \hookrightarrow R$ and the set of elements $b \in R_0$ such that $\textrm{nrd}(b)=|\Delta|$ . Under this bijection, the embedding f corresponds to the element $f(\sqrt{\Delta}) \in R_0$ .

In order to carry out our study of singular differences that are S-units, it is fundamental to understand what is the biggest exponent with which a prime ideal can appear in the factorisation of such a difference. Roughly speaking, saying that a difference of singular moduli $j-j_0$ has a certain $\mu$ -adic valuation $n=v_{\mu}(\,j-j_0)$ for some prime ideal $\mu \subseteq \mathbb{Q}(\,j-j_0)$ is equivalent to saying that the CM elliptic curve $E_j$ with $j(E_j)=j$ is isomorphic to the elliptic curve $E_{j_0}$ with $j(E_0)=j_0$ when reduced modulo $\mu^n$ . Therefore, in order to understand the exponents appearing in the prime ideal factorisation of a singular difference, it is crucial to determine when such isomorphisms can occur. With this goal in mind, we conclude this section by outlining some aspects of the reduction theory of CM elliptic curves defined over number fields. We refer the reader to [ Reference Conrad10 , Reference Gross20 , Reference Gross and Zagier21 ] and [ Reference Lauter and Viray33 ] for further discussions on the topic.

Let $\mathcal{O}$ be an order of discriminant $\Delta$ in an imaginary quadratic field K and let $\ell \nmid \Delta$ be a prime inert in K. Consider an elliptic curve E with complex multiplication by the order $\mathcal{O}$ and defined over the ring class field $H_\mathcal{O}\;:\!=\;K(\,j(E'))$ . After completing with respect to any prime above $\ell$ , we can consider $H_\mathcal{O}$ as a subfield of the maximal unramified extension $\mathbb{Q}_{\ell}^{\text{unr}}$ of $\mathbb{Q}_\ell$ . This is because the extension $\mathbb{Q} \subseteq H_\mathcal{O}$ is unramified at $\ell$ by the assumption $\ell \nmid \Delta$ , see [ Reference Cox11 , Chapter 9, Section A]. Let $L\;:\!=\;\widehat{\mathbb{Q}^{\text{unr}}_\ell}$ be the completion of $\mathbb{Q}_{\ell}^{\text{unr}}$ with ring of integers W and uniformiser $\pi$ . Then by [ Reference Serre and Tate40 , Theorems 8 and 9] and [ Reference Lang31 , Chapter 13, Theorem 12] there exists an elliptic scheme $\mathcal{E} \to \textrm{Spec}\ W$ such that:

  1. (i) the generic fiber $E\;:\!=\;\mathcal{E} \times_W \textrm{Spec}\ L$ is isomorphic to $E^{\prime} $ over the algebraic closure of L. Since the CM order $\mathcal{O}$ is contained in W, all the geometric endomorphisms of E are defined over L, see [ Reference Shimura41 , Chapter II, Proposition 30];

  2. (ii) the special fiber $E_0\;:\!=\;\mathcal{E} \times_W \textrm{Spec}\ W/\pi$ is a supersingular elliptic curve since, by assumption, $\ell$ does not split in K. Note that $W/\pi \cong \overline{\mathbb{F}}_\ell$ , the algebraic closure of the finite field with $\ell$ elements.

For all $n\in \mathbb{N}$ , set $E_n\;:\!=\;\mathcal{E} \times_W \textrm{Spec}\ W/ \pi^{n+1}$ . We are interested in understanding the endomorphism rings $A_{\ell,n}\;:\!=\;\textrm{End}_{W/\pi^{n+1}}(E_n)$ . When $n=0$ , we have already seen that the ring $A_{\ell,0}$ is isomorphic to a maximal order in $\mathbb{B}_{\ell, \infty}$ , the unique (up to isomorphism) definite quaternion algebra over the rationals which ramifies only at $\ell$ and $\infty$ . All the other rings $A_{\ell,n}$ can be recovered from $A_{\ell,0}$ , as explained in the following theorem.

Theorem 2·4. Let $\mathcal{O}$ be an order of discriminant $\Delta$ in an imaginary quadratic field K and let $\ell \nmid \Delta$ be a prime inert in K. Set $L\;:\!=\;\widehat{\mathbb{Q}^{\text{unr}}_\ell}$ to be the completion of the maximal unramified extension of $\mathbb{Q}_\ell$ , with ring of integers W and uniformiser $\pi$ . Let $\mathcal{E} \to \textrm{Spec} (W)$ be an elliptic scheme whose generic fiber $E\;:\!=\; \mathcal{E} \times_W \textrm{Spec}\ L$ has complex multiplication by $\mathcal{O}$ . For every $n \in \mathbb{N}$ , denote by

\begin{equation*}E_n\;:\!=\;\mathcal{E} \times_W \textrm{Spec}\ W/\pi^{n+1} \text{and}\ A_{\ell,n}\;:\!=\;\textrm{End}_{W/\pi^{n+1}}(E_n)\end{equation*}

respectively the reduction of $\mathcal{E}$ modulo $\pi^{n+1}$ and its endomorphism ring. Then:

  1. (a) for every $n \in \mathbb{N}$ we have

    \begin{equation*} A_{\ell,n} \cong \mathcal{O} + \ell^n A_{\ell,0}, \end{equation*}
    where the sum takes place in $A_{\ell,0}$ in which $\mathcal{O}$ is embedded via the reduction modulo $\pi$ ;
  2. (b) for every $n \in \mathbb{N}$ the ring $\textrm{End}_{W/\pi^{n+1}}(E_n)$ is isomorphic to a quaternion order in $\mathbb{B}_{\ell, \infty}$ and the natural reduction map

    \begin{equation*} \mathcal{O} \cong \textrm{End}_W (\mathcal{E}) \longrightarrow \textrm{End}_{W/\pi^{n+1}}(E_n) \end{equation*}
    induced by the reduction modulo $\pi^{n+1}$ is an optimal embedding.

The above theorem is a combination and a reformulation of various results already appearing in the literature. We give a brief overview of the proof and point out the relevant references.

Proof of Theorem 2·4. Part (a) of the theorem is a special case of [ Reference Lauter and Viray33 , Formula 6.6]. As for part (b): the first statement follows from the fact that $\ell$ is a prime of supersingular reduction for E and from part (a). For the second statement, note first of all that there is a natural isomorphism between $\textrm{End}_L(E) \cong \mathcal{O}$ and $\textrm{End}_W(\mathcal{E})$ , since by assumption $\mathcal{E}$ is a Néron model for E over W (see [ Reference Bosch, Lutkebohmert and Raynaud5 , Propositions 1.2/8 and 1.4/4]). Reductions modulo $\pi$ and $\pi^n$ give the following commutative diagram

in which all the arrows are injective by [ Reference Conrad10 , Theorem 2.1 (2)]. Since $\ell$ does not divide the conductor of the order $\mathcal{O}$ , the embedding $\varphi_0$ is optimal by [ Reference Lauter and Viray33 , Proposition 2.2]. It follows from the commutativity of the diagram above that also the embedding $\varphi_{n-1}$ is optimal, and the theorem is proved.

3. The $\ell$ -adic valuation of differences of singular moduli

In order to bound from above the Weil height of a difference of singular moduli, it is of crucial importance to understand the exponents appearing in the prime factorisation of such a difference. The goal of this section is to prove, under certain conditions, an upper bound for these exponents. In what follows, we will always use $\mathbb{F}_\ell$ to denote the finite field with $\ell$ elements, where $\ell \in \mathbb{N}$ is a prime number, and denote by $\overline{\mathbb{F}}_\ell$ an algebraic closure of this field. Recall also that given an order $\mathcal{O}$ in an imaginary quadratic field K, the ring class field of K relative to the order $\mathcal{O}$ is the field generated over K by any singular modulus relative to $\mathcal{O}$ .

Theorem 3·1. Let $j_0 \in \overline{\mathbb{Q}}$ be a singular modulus relative to an order $\mathcal{O}_{j_0}$ of discriminant $\Delta_0$ and let $\ell \in \mathbb{Z}$ be a prime not dividing $\Delta_0$ . For any singular modulus $j \in \overline{\mathbb{Q}}$ relative to an order $\mathcal{O}_j$ of discriminant $\Delta \neq \Delta_0$ , denote by H the compositum of the ring class fields relative to $\mathcal{O}_{j_0}$ and $\mathcal{O}_j$ . Let $\mu \subseteq H$ be a prime ideal lying above $\ell$ and assume that:

  1. (i) the prime $\mu \cap \mathbb{Q}(\,j_0)$ has residue degree 1 over $\ell$ ;

  2. (ii) there exists an elliptic curve ${E_0}_{/\mathbb{Q}(\,j_0)}$ with $j(E_0)=j_0$ and having good reduction at $\mu \cap \mathbb{Q}(\,j_0)$ .

Then, if $v_\mu(\!\cdot\!)$ denotes the normalised valuation associated to $\mu$ , we have

(3) \begin{equation} v_\mu(\,j-j_0) \leq\begin{cases}\frac{d_0}{2} \left( \frac{\log(\Delta_0^2|\Delta|)}{2 \log \ell}+ \frac{1}{2} \right) &\text{ if } \ell \nmid \Delta \text{ and } \mathcal{O}_{j_0} \not \subseteq \mathcal{O}_j, \\\frac{d_0}{2} &\text{ if } \ell \mid \Delta,\end{cases}\end{equation}

where $d_0$ is the number of automorphisms of any elliptic curve $E_{/\overline{\mathbb{F}}_\ell}$ with $j(E) = j_0 \text{ mod } \mu$ .

Remark 3·2. Note that we have $d_0=2$ in all cases except if $j_0 \equiv 0$ or $j_0 \equiv 1728$ mod $\mu$ . In these two cases, the value of $d_0$ also depends on $\ell$ , see [ Reference Silverman43 , III, Theorem 10.1].

The dichotomy in the conclusion of Theorem 3·1 is reflected by its proof, which we divide according to the conditions displayed in (3). In all cases, everything boils down to the study of optimal embeddings of the order $\mathcal{O}_j$ in a family of nested orders contained in the endomorphism ring of a certain supersingular elliptic curve defined over $\overline{\mathbb{F}}_\ell$ . One of the main issues is that for a supersingular elliptic curve $E_{/\overline{\mathbb{F}}_\ell}$ , explicitely computing its endomorphism ring is a difficult problem in general. An explicit parametrisation of the endomorphism rings of supersingular elliptic curves over $\overline{\mathbb{F}}_\ell$ has been achieved by Lauter and Viray in [ Reference Lauter and Viray33 , Section 6]. However, the author found these parametrizations somehow difficult to use for explicit estimates. Therefore, in order to achieve our results, we adopted a different strategy. The idea is that, since we are only interested in providing estimates for the $\mu$ -adic valuation of singular differences and not in precisely determining their prime ideal factorisation, we do not need the full knowledge of the supersingular endomorphism rings of the elliptic curves involved. We instead “approximate”, when possible, the unknown quaternion orders with quaternion orders whose properties are less mysterious. The next proposition is the cornerstone of this strategy.

Proposition 3·3. Let $j \in \overline{\mathbb{Q}}$ be a singular modulus of discriminant $\Delta$ and let $E_{/\mathbb{Q}(\,j)}$ be an elliptic curve with $j(E)=j$ . Choose a degree 1 prime $\mathfrak{p} \subseteq \mathbb{Q}(\,j)$ lying above a rational prime $p \in \mathbb{Z}$ not dividing $\Delta$ and suppose that E has good supersingular reduction $\widetilde{E}$ modulo $\mathfrak{p}$ . Denote by $\varphi \in \textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E})$ the Frobenius endomorphism $(x,y) \mapsto (x^{\,p},y^{\,p})$ , where the coordinates x,y come from the choice of a Weierstrass model for E. Then there exists a morphism $\psi \in \textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E})$ such that

\begin{equation*}\psi^2+|\Delta| \psi+ \frac{\Delta^2 + |\Delta|}{4}=0 \quad \text{and} \quad \psi \circ \varphi = \varphi \circ \overline{\psi},\end{equation*}

where $\overline{\cdot}\;:\textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E}) \otimes_\mathbb{Z} \mathbb{Q} \to \textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E}) \otimes_\mathbb{Z} \mathbb{Q}$ denotes the standard involution. In fact, the morphism $\psi$ can be taken inside the image of the reduction map $\textrm{End}_{\overline{\mathbb{Q}}}(E) \to \textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E})$ modulo any prime in $\overline{\mathbb{Q}}$ lying above $\mathfrak{p}$ .

Remark 3·4. Recall that the standard involution on the quaternion algebra $\textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E}) \otimes_\mathbb{Z} \mathbb{Q}$ corresponds to taking the dual isogeny when restricted to $\textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E})$ . This essentially follows from the uniqueness of the standard involution on quaternion algebras, see [ Reference Voight44 , Corollary 3.4.4].

Proof. In this proof, we fix for convenience an embedding $\overline{\mathbb{Q}} \hookrightarrow \mathbb{C}$ . Let $\mathcal{O}$ be the order of discriminant $\Delta$ and $K \subseteq \overline{\mathbb{Q}}$ be its field of fractions. For an element $\beta \in K$ , we denote by $\overline{\beta}$ its conjugate through the unique non-trivial automorphism of $K/\mathbb{Q}$ . This will not cause confusion with the standard involution on $\textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E}) \otimes_\mathbb{Z} \mathbb{Q}$ , as we explain below.

By assumption, there exists an elliptic scheme $\mathcal{E}$ over the localisation at $\mathfrak{p}$ of the ring of integers in $\mathbb{Q}(\,j)$ such that the generic fiber of $\mathcal{E}$ is isomorphic to E while its special fiber is a supersingular elliptic curve $\widetilde{E}$ defined over $\mathbb{F}_p$ . Set $H_\mathcal{O}\;:\!=\;K(\,j)$ , which is a degree 2 extension of $\mathbb{Q}(\,j)$ , and fix a prime $\mathcal{P} \subseteq H_\mathcal{O}$ lying above $\mathfrak{p}$ . Since E has supersingular reduction modulo $\mathfrak{p}$ , the latter has degree 1 and p is unramified in K, by [ Reference Lang31 , Chapter 13, Theorem 12] we must have $f(\mathcal{P}/\mathfrak{p})=2$ , where $f(\mathcal{P}/\mathfrak{p})$ denotes the inertia degree of $\mathcal{P}$ over $\mathfrak{p}$ . In particular, we see that the decomposition group of $\mathcal{P}$ over $\mathfrak{p}$ is precisely $\textrm{Gal}(H_\mathcal{O}/\mathbb{Q}(\,j))$ . We fix $\sigma \in \textrm{Gal}(H_\mathcal{O}/\mathbb{Q}(\,j))$ to be the unique non-trivial element. Then $\sigma$ restricts to an automorphism of $R_\mathcal{P}$ , the localization at $\mathcal{P}$ of the ring of integers of $H_\mathcal{O}$ , inducing the Frobenius endomorphism $\tau\;:\;x \mapsto x^p$ on the residue field.

With abuse of notation, we denote again by $\mathcal{E}$ the base-change $\mathcal{E}_{R_\mathcal{P}}$ and by $\widetilde{E}$ the special fiber of $\mathcal{E}_{R_\mathcal{P}}$ (which is isomorphic to the base-change of the special fiber of $\mathcal{E}$ to the residue field of $R_\mathcal{P}$ ). It follows from the Néron mapping property [ Reference Bosch, Lutkebohmert and Raynaud5 , Proposition 1.4/4] that every endomorphism $\lambda \in \textrm{End}_{H_\mathcal{O}}(E)$ induces an endomorphism $\lambda_\mathcal{E}$ of $\mathcal{E}$ . Define $\lambda \text{ mod } \mathcal{P}$ to be the restriction of $\lambda_\mathcal{E}$ to $\widetilde{E}$ . The Galois group $\textrm{Gal}(H_\mathcal{O}/\mathbb{Q}(\,j))$ acts on $R_\mathcal{P}$ and this in turn induces a Galois action on $\textrm{End}_{R_\mathcal{P}}(\mathcal{E})$ . In the same way, there is an action of $\textrm{Gal}(\overline{\mathbb{F}}_p/\mathbb{F}_p)$ on $\textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E})$ . These two actions are compatible, in the sense that for every $\lambda \in \textrm{End}_{H_\mathcal{O}}(E)$ we have

(4) \begin{equation} \sigma(\lambda_\mathcal{E}) \text{ mod } \mathcal{P} = \tau (\lambda \text{ mod } \mathcal{P}), \end{equation}

as one can see using the various functorial properties of fibered products. In what follows, we will often omit the subscript $\mathcal{E}$ when dealing with endomorphism of $\mathcal{E}$ induced by elements in $\textrm{End}_{H_\mathcal{O}}(E)$ . This allows us to ease a bit the notation, since usually elements of $\textrm{End}_{H_\mathcal{O}}(E)$ will already come equipped with their own subscript.

We now fix a normalised isomorphism

\begin{equation*} [\!\cdot\!]_E\;:\;\mathcal{O} \xrightarrow{\sim} \textrm{End}_{\overline{\mathbb{Q}}}(E) \end{equation*}

following [ Reference Silverman42 , II, Proposition 1.1]. Let $\alpha\;:\!=\; ({\Delta+\sqrt{\Delta}})/{2} \in \mathcal{O}$ and note that $[\alpha]_E \in \textrm{End}_{H_\mathcal{O}}(E)$ because, by [ Reference Shimura41 , Chapter II, Proposition 30], all the endomorphisms of E are defined over $H_\mathcal{O}$ . Since $\alpha^2+|\Delta| \alpha+ ({\Delta^2 + |\Delta|})/{4}=0$ , also $[\alpha]_E$ satisfies the same relation. One also has

(5) \begin{equation} \sigma([\alpha]_E)=[\sigma(\alpha)]_{E^{\sigma}}=[\overline{\alpha}]_{E}, \end{equation}

where the first equality follows from [ Reference Silverman42 , II, Theorem 2.2 (a)] and in the second equality we are using the fact that E is defined over $\mathbb{Q}(\,j)$ and $\sigma$ is non-trivial.

Let now $\psi\;:\!=\;\left( [\alpha]_E \text{ mod } \mathcal{P} \right) \in \textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E})$ . For $\beta \in \mathcal{O}$ , the association $[\beta]_E \mapsto [\overline{\beta}]_E$ defines a standard involution on $\textrm{End}_{H_\mathcal{O}}(E)$ , in the sense of [ Reference Voight44 , Definition 3.2.4]. Since reduction mod $\mathcal{P}$ defines an embedding of $\textrm{End}_{H_\mathcal{O}}(E) \hookrightarrow \textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E})$ , by the uniqueness of the standard involution on quadratic $\mathbb{Q}$ -algebras (see [ Reference Voight44 , Lemma 3.4.2]) we have $[\overline{\alpha}]_E \text{ mod } \mathcal{P}= \overline{\psi}$ , where now the conjugation above $\psi$ denotes the usual standard involution on the quaternion algebra $\textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E}) \otimes_\mathbb{Z} \mathbb{Q}$ . We have

\begin{equation*} \overline{\psi} = [\overline{\alpha}]_E \text{ mod } \mathcal{P} = \sigma([\alpha]_E) \text{ mod } \mathcal{P} = \tau ( [\alpha]_E \text{ mod } \mathcal{P} ) = \tau (\psi), \end{equation*}

where we have applied equalities (4) and (5). This yields

\begin{equation*} \varphi \circ \overline{\psi} = \varphi \circ \tau (\psi) = \tau (\tau (\psi)) \circ \varphi = \psi \circ \varphi \end{equation*}

and here we have used the facts that for every $\lambda \in \textrm{End}_{\overline{\mathbb{F}}_p}(\widetilde{E})$ one has $\varphi \circ \lambda = \tau(\lambda) \circ \varphi$ , as can be checked using local coordinates for $\widetilde{E}$ , and that $\tau(\tau(\psi))=\psi$ because $\psi$ is defined over a quadratic extension of $\mathbb{F}_p$ . The proof is concluded.

We are now ready to begin the proof of Theorem 3·1. Let us fix the notation that will be in force during the entire argument. Given the orders $\mathcal{O}_j=\mathbb{Z} \left[ ({\Delta+\sqrt{\Delta}})/{2} \right]$ and $\mathcal{O}_{j_0}=\mathbb{Z} \left[ ({\Delta_0+\sqrt{\Delta_0}})/{2} \right]$ as in the statement of Theorem 3·1, we denote by $K_j$ and $K_{j_0}$ the corresponding imaginary quadratic fields containing them. We then set $H_j$ and $H_{j_0}$ to be the ring class fields of $K_j$ and $K_{j_0}$ relative to the orders $\mathcal{O}_j$ and $\mathcal{O}_{j_0}$ respectively. Using this notation, the field H in the statement of Theorem 3·1 is the compositum in $\overline{\mathbb{Q}}$ of $H_j$ and $H_{j_0}$ .

3·1. First case: $\ell$ does not divide $\Delta$ and $\mathcal{O}_{j_0} \not \subseteq \mathcal{O}_j$

Assume that $E_0$ in the statement of the theorem is given by an integral model over the ring of integers of $\mathbb{Q}(\,j_0)$ with good reduction at $\mu \cap \mathbb{Q}(\,j_0)$ . Let $(E_0)_{/H}$ be the base-change to H of the elliptic curve $(E_0)_{/\mathbb{Q}(\,j_0)}$ , and let $(E_j)_{/H}$ be an elliptic curve with $j(E_j)=j$ and with good reduction at all prime ideals above $\ell$ . Such an elliptic curve $E_j$ exists by [ Reference Serre and Tate40 , Theorems 8 and 9], which we can apply since $\ell \nmid \Delta$ by assumption. In particular, $E_j$ will have good reduction at the prime $\mu$ . We will always identify $\mathcal{O}_j$ and $\mathcal{O}_{j_0}$ with the endomorphism rings of $E_j$ and $E_0$ respectively.

Let $H_\mu$ be the completion of H at the prime $\mu$ . The extension $\mathbb{Q} \subseteq H$ is unramified at $\ell$ because $\ell \nmid \Delta \Delta_0$ (see [ Reference Cox11 , Chapter 9, Section A]), hence $H_\mu$ is contained in $\widehat{\mathbb{Q}^{\text{unr}}_\ell}$ , the completion of the maximal unramified extension of $\mathbb{Q}_\ell$ . Denote by W the ring of integers in $\widehat{\mathbb{Q}^{\text{unr}}_\ell}$ and let $\pi \in W$ be a uniformizer. By abuse of notation, we also use $E_0, E_j$ to denote the elliptic schemes over W with generic fibers isomorphic to the base-changes of $E_0, E_j$ to $\widehat{\mathbb{Q}^{\text{unr}}_\ell}$ respectively. Note that, by our choices, $E_0 \text{ mod } \pi$ is defined over $\mathbb{F}_\ell$ .

Lemma 3·5. In the notation above, we have

\begin{equation*}v_\mu(\,j-j_0) \leq \frac{d_0}{2} \cdot \max \{n \in \mathbb{N}_{\geq 1}\;:\; {Iso}_{W/\pi^n} (E_j, E_0) \neq \emptyset \},\end{equation*}

where, for every $n \in \mathbb{Z}_{\geq 1}$ , we denote by ${Iso}_{W/\pi^n} (E_j, E_0)$ the set of isomorphisms between $E_j \text{ mod } \pi^n$ and $E_0\ \textrm{mod}\ \pi^n$ .

Proof. Notice first of all that the normalised valuation on $\widehat{\mathbb{Q}^{\text{unr}}_\ell}$ , i.e the valuation v satisfying $v(\pi)=1$ , extends the $\mu$ -adic valuation $v_\mu$ on H because $v_\mu(\ell)=1$ . Since W is a complete discrete valuation ring whose quotient field has characteristic 0 and whose residue field $\overline{\mathbb{F}}_\ell$ is algebraically closed of characteristic $\ell >0$ , we can apply [ Reference Gross and Zagier21 , Proposition 2.3] which gives

\begin{equation*}v_\mu(\,j-j_0)= \frac{1}{2} \sum_{n=1}^{\infty} \# \textrm{Iso}_{W/\pi^n} (E_j, E_0).\end{equation*}

Now, certainly $\textrm{Iso}_{W/\pi^{n+1}} (E_j, E_0) \neq \emptyset$ implies $\textrm{Iso}_{W/\pi^{n}} (E_j, E_0) \neq \emptyset$ for every $n \in \mathbb{N}_{>0}$ , since reductions of isomorphisms are isomorphisms. Moreover, whenever the set $\textrm{Iso}_{W/\pi^n} (E_j, E_0)$ is non-empty, its cardinality equals the order of the automorphism group $\textrm{Aut}_{W/\pi^n}(E_0)$ of $E_0 \text{ mod } \pi^n$ . By [ Reference Conrad10 , Theorem 2.1 (2)], we always have the inclusions

\begin{equation*}\textrm{End}_W(E_0) \hookrightarrow \textrm{End}_{W/\pi^n}(E_0) \hookrightarrow \textrm{End}_{W/\pi}(E_0)\end{equation*}

induced respectively by the reduction modulo $\pi^n$ and modulo $\pi$ . This means that

(6) \begin{equation} \# \textrm{Aut}_W(E_0) \leq \# \textrm{Aut}_{W/\pi^n}(E_0) \leq \# \textrm{Aut}_{W/\pi}(E_0)=d_0\end{equation}

so, setting $M\;:\!=\;\max \{n \in \mathbb{N}_{\geq 1}\;:\;\textrm{Iso}_{W/\pi^n} (E_j, E_0) \neq \emptyset \}$ , we obtain

\begin{equation*}v_\mu(\,j-j_0)= \frac{1}{2} \sum_{n=1}^{\infty} \# \textrm{Iso}_{W/\pi^n} (E_j, E_0)=\frac{1}{2} \sum_{n=1}^{M} \# \textrm{Aut}_{W/\pi^n}(E_0) \leq \frac{d_0}{2}\cdot M\end{equation*}

which proves the lemma.

By Lemma 3·5, in order to estimate the valuation at $\mu$ of the difference $j-j_0$ , we need to bound the biggest index n such that the reductions modulo $\pi^n$ of the elliptic curves $E_j$ and $E_0$ are isomorphic. If this maximum is 0, then the two elliptic curves are not even isomorphic over $\overline{\mathbb{F}}_\ell \cong W/\pi$ , so the prime $\mu$ cannot divide $j-j_0$ and there is nothing to prove. Hence, from now on we suppose that $\mu$ divides $j-j_0$ so that $E_0 \text{ mod } \pi \cong E_j \text{ mod } \pi$ over $\overline{\mathbb{F}}_\ell$ . Since $\ell$ does not divide the conductors of the orders $\mathcal{O}_j$ and $\mathcal{O}_{j_0}$ by assumption, and the two orders are different, [ Reference Lang31 , Chapter 13, Theorem 12] ensures that $\ell$ is a prime of supersingular reduction for both $E_j$ and $E_{0}$ . In particular, the ring $R\;:\!=\;\textrm{End}_{W/\pi}(E_0)$ is isomorphic to a maximal order in $\mathbb{B}_{\ell, \infty}\cong R \otimes_\mathbb{Z} \mathbb{Q}$ .

Suppose now that $\textrm{Iso}_{W/\pi^{n+1}}(E_j,E_0)$ is non-empty. Our goal is to find a bound on the exponent $n+1$ . A choice of $f \in \textrm{Iso}_{W/\pi^{n+1}}(E_j,E_0)$ induces an isomorphism

\begin{equation*}\widetilde{f}\;:\;\textrm{End}_{W/\pi^{n+1}}(E_j) \longrightarrow \textrm{End}_{W/\pi^{n+1}}(E_0), \quad \alpha \longmapsto f \circ \alpha \circ f^{-1}\end{equation*}

which, precomposed with the reduction map $\mathcal{O}_j \hookrightarrow \textrm{End}_{W/\pi^{n+1}}(E_j)$ , gives rise to an optimal embedding

(7)

by Theorem 2·4 (b). For growing n, Theorem 2·4 (a) shows that the endomorphism ring of $E_0 \text{ mod } \pi^{n+1}$ becomes more and more “ $\ell$ -adically close” to the order $\mathcal{O}_{j_0}$ . Intuitively, this must imply that having an embedding as in (7) should not be possible for n large enough, yielding the desired bound on $n+1$ . This intuition is correct, as we show below. The main obstacle to making this idea precise is that, as we already said, it is not easy to explicitly compute the endomorphism rings $\textrm{End}_{W/\pi^{n+1}}(E_0)$ for a generic elliptic curve ${E_0}_{/W}$ . To circumvent this problem, we “approximate” the rings $\textrm{End}_{W/\pi^{n+1}}(E_0)$ with smaller orders where we are able to perform the relevant computations. The hypotheses on the prime $\mu$ and on the elliptic curve $E_0$ will make this strategy successful.

Recall that $\mathcal{O}_{j_0}=\mathbb{Z} \left[ ({\Delta_0 + \sqrt{\Delta_0}})/{2} \right]$ and let $\psi \in R$ be the image of $({\Delta_0 + \sqrt{\Delta_0}})/{2}$ via the reduction map modulo $\pi$ . Denote also by $\varphi \in \textrm{End}_{W/\pi}(E_0)$ the Frobenius endomorphism $(x,y) \mapsto (x^\ell, y^\ell)$ . By Proposition 3·3 and using the fact that $E_0 \text{ mod } \pi$ is a supersingular elliptic curve defined over $\mathbb{F}_\ell$ , we have

(8) \begin{equation} \varphi^2+\ell=0,\quad \psi^2+|\Delta_0| \psi+ \frac{\Delta_0^2 + |\Delta_0|}{4}=0 \quad \text{and}\quad \psi \circ \varphi = \varphi \circ \overline{\psi},\end{equation}

where $\overline{\cdot}$ denotes the standard involution on $\textrm{End}_{W/\pi}(E_0) \otimes_\mathbb{Z} \mathbb{Q}$ . Hence, the ring $\widetilde{R}\;:\!=\;\mathbb{Z}[ \psi, \varphi] \subseteq R$ is a rank-4 order inside $\mathbb{B}_{\ell, \infty}$ with basis $\mathcal{B}=\{1, \psi, \varphi, \psi \varphi \}$ satisfying the relations (8). Notice that the reduction map $\mathcal{O}_{j_0} \hookrightarrow R$ identifies $\mathcal{O}_{j_0}$ with the subring $\mathbb{Z}[\psi] \subseteq \mathbb{Z}[\psi, \varphi]$ . The matrix of the bilinear pairing $\langle \alpha, \beta \rangle=\textrm{trd}(\alpha \overline{\beta})$ computed on the basis $\mathcal{B}$ is given by

\begin{equation*}A=\begin{pmatrix}2\;\;\;\;\;\; & \Delta_0\;\;\;\;\;\; & 0\;\;\;\;\;\; & 0 \\\Delta_0\;\;\;\;\;\; & \frac{\Delta_0^2 + |\Delta_0|}{2}\;\;\;\;\;\; & 0\;\;\;\;\;\; & 0 \\0\;\;\;\;\;\; & 0\;\;\;\;\;\; & 2\ell\;\;\;\;\;\; & \Delta_0 \ell \\0\;\;\;\;\;\; & 0\;\;\;\;\;\; & \Delta_0 \ell \;\;\;\;\;\; & \frac{\Delta_0^2 + |\Delta_0|}{2} \ell \end{pmatrix}\end{equation*}

so the discriminant of the order $\widetilde{R}$ equals $\det A=\Delta_0^2 \ell^2$ (see [ Reference Voight44 , Definition 15.2.2 and Exercise 13 in Chapter 15]). Hence, by [ Reference Voight44 , Lemma 15.2.15, Lemma 15.4.7 and Theorem 15.5.5] $\widetilde{R}$ has index $|\Delta_0|$ inside any maximal order containing it, so in particular $|R\;:\;\widetilde{R}|=|\Delta_0|$ . Now, since we are in the hypotheses of Theorem 2·4 (a), we have

\begin{equation*}\textrm{End}_{W/\pi^{n+1}}(E_0) \cong \mathbb{Z}[\psi] + \ell^n R \supseteq \mathbb{Z}[\psi] + \ell^n \widetilde{R}\end{equation*}

and we shall show that the index of the latter inclusion is also bounded by $|\Delta_0|$ .

Lemma 3·6. For all $n\in \mathbb{N}$ the index $\left| (\mathbb{Z}[\psi] + \ell^n R) \;:\;(\mathbb{Z}[\psi] + \ell^n \widetilde{R}) \right|$ divides $|\Delta_0|$ .

Proof. Since $\widetilde{R} \subseteq R$ , we have $\mathbb{Z}[\psi] + \ell^n R = \mathbb{Z}[\psi] + \ell^n R + \ell^n \widetilde{R}$ . Hence

\begin{equation*} \frac{\mathbb{Z}[\psi] + \ell^n R}{\mathbb{Z}[\psi] + \ell^n\vphantom{\hat{\widetilde{R}}} \widetilde{R}} = \frac{\mathbb{Z}[\psi] + \ell^n R + \ell^n \widetilde{R}}{\mathbb{Z}[\psi] + \ell^n \vphantom{\hat{\widetilde{R}}}\widetilde{R}} \cong \frac{\ell^n R}{(\mathbb{Z}[\psi] + \ell^n \vphantom{\hat{\widetilde{R}}}\widetilde{R}) \cap \ell^n R} \end{equation*}

as abelian groups. Now, the containment $\ell^n \widetilde{R} \subseteq (\mathbb{Z}[\psi] + \ell^n \widetilde{R}) \cap \ell^n R$ gives an epimorphism

\begin{equation*} \frac{\ell^n R}{\ell^n \vphantom{\hat{\widetilde{R}}}\widetilde{R}} \twoheadrightarrow \frac{\ell^n R}{(\mathbb{Z}[\psi] + \ell^n \vphantom{\hat{\widetilde{R}}}\widetilde{R}) \cap \ell^n R}, \end{equation*}

and, since R is non-torsion, we have $\ell^n R / \ell^n \widetilde{R} \cong R/\widetilde{R}$ . Since the latter has cardinality $|\Delta_0|$ , the lemma is proved.

Corollary 3·7. The embedding (7) induces an injection

(9) \begin{equation} \mathcal{O}_{j,|\Delta_0|}\;:\!=\;\mathbb{Z} \left[ \frac{\Delta_0^2 \Delta+\sqrt{\Delta_0^2 \Delta}}{2} \right] \hookrightarrow \mathbb{Z}[\psi] + \ell^n \widetilde{R}.\end{equation}

Proof. By Lemma 3·6, for every $x \in \mathbb{Z}[\psi] + \ell^n R$ we have $|\Delta_0| x \in \mathbb{Z}[\psi] + \ell^n \widetilde{R}$ . Since $\mathcal{O}_{j,|\Delta_0|}=\mathbb{Z}+|\Delta_0| \mathcal{O}_j$ , the corollary follows.

Combining Corollary 3·7 with Lemma 2·1, we see that $|\textrm{disc}(\mathcal{O}_{j,|\Delta_0|})|=\Delta_0^2 |\Delta|$ must be represented by the Gross lattice $\Lambda_{\ell,n}$ of the order $\mathbb{Z}[\psi] + \ell^n \widetilde{R}$ . Note that this representation is not necessarily primitive, because the embedding (9) is not necessarily optimal. A computation shows that

\begin{equation*}\Lambda_{\ell,n} = \left \langle |\Delta_0| + 2\psi, 2\ell^n \varphi, 2 \ell^n \psi \varphi \right \rangle_{\mathbb{Z}}\end{equation*}

i.e. $\mathcal{B}'=\{|\Delta_0| + 2\psi, 2\ell^n \varphi, 2 \ell^n \psi \varphi \}$ is a $\mathbb{Z}$ -basis for the Gross lattice of $\mathbb{Z}[\psi] + \ell^n \widetilde{R}$ . The reduced norm restricted to the lattice $\Lambda_{\ell,n}$ induces the ternary quadratic form

(10) \begin{equation} Q_{\ell,n}(X,Y,Z)=|\Delta_0|X^2+ 4\ell^{2n+1} Y^2 + \ell^{2n+1}(\Delta_0^2 + |\Delta_0|) Z^2+ 4\ell^{2n+1}\Delta_0 YZ\end{equation}

written with respect to the basis $\mathcal{B}'$ .

After setting

\begin{equation*}\widetilde{X}=X,\quad \widetilde{Y}=Y+\frac{1}{2}\Delta_0 Z,\quad \widetilde{Z}=Z\end{equation*}

we get the diagonal quadratic form

\begin{equation*}\widetilde{Q}_{\ell,n} (\widetilde{X}, \widetilde{Y}, \widetilde{Z})=|\Delta_0|\widetilde{X}^2+4\ell^{2n+1} \widetilde{Y}^2+\ell^{2n+1} |\Delta_0| \widetilde{Z}^2.\end{equation*}

Suppose now that $Q_{\ell,n}(X,Y,Z)=\Delta_0^2|\Delta|$ has an integral solution $(x,y,z) \in \mathbb{Z}^3$ corresponding to the embedding (9). We first claim that at least one among y and z is non-zero. This follows from our assumptions on $\mathcal{O}_j$ and from the following proposition.

Proposition 3·8. If $y=z=0$ then $\mathcal{O}_{j_0} \subseteq \mathcal{O}_j$ .

Proof. Let $x \in \mathbb{Z}_{>0}$ be such that $Q_{\ell,n}(x,0,0)=\Delta_0^2|\Delta|$ . By Remark 2·3, this equality corresponds to the embedding

(11)

of the order $\mathcal{O}_{j,|\Delta_0|}\subseteq K\;:\!=\;\mathbb{Q}(\sqrt{\Delta})$ into $\mathbb{Z}[\psi] + \ell^n \widetilde{R}$ . The injection (11) is not optimal if $x \neq \pm 1$ . Indeed, using the proof of Lemma 2·1 we get the optimal embedding

determined by the equality $Q_{\ell,n}(1,0,0)=(\Delta_0^2|\Delta|)/x^2$ . Since $Q_{\ell,n}(1,0,0)=|\Delta_0|$ , we see that the above injection is actually the same as

(12)

Recall that we also have embedding (7), which can be rewritten as

(13)

We remind the reader that the above injection (13) is again optimal, and that (11) is originally induced by (13). It is then clear that the injections (11), (12) and (13) are all compatible between each other, meaning that, after tensoring with $\mathbb{Q}$ , one gets the same map $\iota\;:\;K \hookrightarrow \mathbb{B}_{\ell, \infty}$ . In particular, $\mathcal{O}_j$ and $\mathcal{O}_{j_0}$ are contained inside the same imaginary quadratic field $K=\mathbb{Q}(\sqrt{\Delta})=\mathbb{Q}(\sqrt{\Delta_0})$ .

Consider now the order $\mathcal{O}\;:\!=\;\mathcal{O}_{j} + \mathcal{O}_{j_0} \subseteq K$ . We have that $\iota(\mathcal{O}) \subseteq \mathbb{Z}[\psi] + \ell^n R$ , and from the optimality of (13) it follows that $\mathcal{O}=\mathcal{O}_j$ . Hence $\mathcal{O}_{j_0} \subseteq \mathcal{O}_j$ , and this concludes the proof.

Since at least one among y and z is non-zero, we also have that at least one among $\widetilde{y}\;:\!=\;y+ (\Delta_0 z)/2$ and $\widetilde{z}=z$ is non-zero. Note that $\widetilde{y} \in \frac{1}{2}\mathbb{Z}$ and $\widetilde{z} \in \mathbb{Z}$ . Then we have

\begin{align*} \Delta_0^2 |\Delta|=\widetilde{Q}_{\ell,n} (\widetilde{x}, \widetilde{y}, \widetilde{z})&=|\Delta_0|\widetilde{x}^2+4\ell^{2n+1} \widetilde{y}^2+\ell^{2n+1} |\Delta_0| \widetilde{z}^2 \\ & \quad \geq \max \{4\ell^{2n+1} \widetilde{y}^2, \ell^{2n+1} |\Delta_0| \widetilde{z}^2 \} \geq \ell^{2n+1}\end{align*}

which implies

(14) \begin{equation} n+1 \leq \frac{\log(\Delta_0^2 |\Delta|)}{2 \log \ell} + \frac{1}{2}. \end{equation}

Combining now (14) with Lemma 3·5 concludes the first case of the proof of Theorem 3·1.

3·2. Second case: $\ell$ divides $\Delta$

For this part of the proof, we are going to heavily rely on [ Reference Lauter and Viray33 ], of which we have kept the notation. We again assume that the elliptic curve $E_0$ is given by an integral model over the ring of integers of $\mathbb{Q}(\,j_0)$ that has good reduction at $\mu \cap \mathbb{Q}(\,j_0)$ .

Suppose initially that $\ell$ divides the conductor of the order $\mathcal{O}_j$ . Let $H_{j} \subseteq F$ be a minimal extension of the ring class field $H_{j}$ such that there exists an elliptic curve $(E_j)_{/F}$ with $j(E_j)=j$ and having good reduction at all primes of F lying above $\ell$ . Fix such an elliptic curve $E_j$ and base-change it to the compositum $L=F\cdot H_{j_0}$ . Consider also a prime $\mu_L \subseteq L$ lying above $\mu \subseteq H$ and denote by A the ring of integers in the completion of the maximal unramified extension of $L_{\mu_L}$ , with maximal ideal $\mu_L A \subseteq A$ . By abuse of notation, we denote by $E_0, E_j$ the elliptic schemes over A with generic fibers isomorphic to the base-changes of $E_0, E_j$ to the completion of the maximal unramified extension of $L_{\mu_L}$ . The elliptic schemes ${E_j}$ and $E_0$ have good reduction over A and, since A is a complete discrete valuation ring of characteristic 0 with algebraically closed residue field of characteristic $\ell>0$ , we can use the same proof of Lemma 3·5 to see that

(15) \begin{equation} v_{\mu} (\,j-j_0) \leq v_{\mu_L}(\,j-j_0) \leq \frac{d_0}{2} \cdot \max \{n \in \mathbb{N}_{\geq 1}\;:\;\textrm{Iso}_{A/\mu_L^n A} (E_j, E_0) \neq \emptyset \}.\end{equation}

Since $\ell \nmid \Delta_0$ , we can now apply [33, Proposition 4.1] with $E=E_0$ , $\mathcal{O}_{d_1}=\mathcal{O}_{j_0}$ and $\mathcal{O}_{d_2}= \mathcal{O}_j$ . This proposition, used together with the fact that $\ell$ divides the conductor of $\mathcal{O}_j$ , implies that $\textrm{Iso}_{A/\mu_L^n A} (E_j, E_0)=\emptyset$ if $n>1$ . Combined with (15), this gives

\begin{equation*} v_{\mu} (\,j-j_0) \leq \frac{d_0}{2}\end{equation*}

as desired. This yields the theorem in the case that $\ell$ divides the conductor of $\mathcal{O}_j$ .

Assume now that $\ell$ divides $\Delta$ but does not divide the conductor of the order $\mathcal{O}_j$ . Then, if again $E_j$ is an elliptic curve with $j(E_j)=j$ , we can choose $F=H_j$ as a field where $E_j$ has a model with good reduction at all primes dividing $\ell$ . This follows from [ Reference Serre and Tate40 , Theorem 9]. If we complete H at $\mu$ , and we take A to be the ring of integers in the completion of the maximal unramified extension of $H_{\mu}$ and W to be the ring of integers in the completion of the maximal unramified extension of $\mathbb{Q}_\ell$ , then $\text{Frac}(W) \subseteq \textrm{Frac}(A)$ is a ramified degree 2 field extension because the ramification index $e(\mu/\ell) = 2$ by our assumptions. Again by [ Reference Lauter and Viray33 , Proposition 4.1], since we are assuming that $\ell$ does not divide the conductor of $\mathcal{O}_j$ , for every $n\in \mathbb{N}_{>0}$ we have

(16) \begin{equation} \# \textrm{Iso}_{A/\mu^n A}(E_0,E_j) \leq C \cdot \# S_n^{\text{Lie}}(E_0/A),\end{equation}

where $C=C(\,j) \leq 6$ is a positive constant depending on j and $S_n^{\text{Lie}}(E_0/A)$ is the set of all endomorphisms $\varphi \in \textrm{End}_{A/\mu^n A}(E_0)$ satisfying the following three conditions (cfr. [ Reference Lauter and Viray33 , pag. 9218]):

  1. (1) $\varphi^2-\Delta \varphi+\frac{1}{4}(\Delta^2-\Delta)=0$ ;

  2. (2) the inclusion $\mathbb{Z}[\varphi] \hookrightarrow \textrm{End}_{A/\mu A}(E_0)$ is optimal at all primes $p\neq \ell$ . We recall that an embedding of $\mathbb{Z}$ -modules $\mathcal{O} \hookrightarrow R$ is optimal at a prime p if the equality

    \begin{equation*} (\iota (\mathcal{O}) \otimes_\mathbb{Z} \mathbb{Q}_p) \cap (R \otimes_\mathbb{Z} \mathbb{Z}_p) = \iota (\mathcal{O}) \otimes_\mathbb{Z} \mathbb{Z}_p \end{equation*}
    holds (note that the corresponding [ Reference Lauter and Viray33 , Definition 2.1] contains a misprint);
  3. (3) as endomorphism of $\text{Lie}(E_0 \text{ mod } \mu^n A)$ we have $\varphi \equiv \delta \text{ mod } \mu^n$ , where $\delta \in A$ is a fixed root of the polynomial $x^2-\Delta x+\frac{1}{4}(\Delta^2-\Delta)$ .

The set $S_n^{\text{Lie}}(E_0/A)$ can be partitioned as

\begin{equation*}S_n^{\text{Lie}}(E_0/A)= \bigcup_{m \in \mathbb{N}} S_{n,m}^{\text{Lie}}(E_0/A),\end{equation*}

where $S_{n,m}^{\text{Lie}}(E_0/A)$ consists of all the endomorphisms $\varphi \in S_n^{\text{Lie}}(E_0/A)$ such that

\begin{equation*}\textrm{disc} \left(\mathcal{O}_{j_0}[\varphi]\right)=m^2.\end{equation*}

We first claim that, under our assumptions, the sets $S_{n,0}^{\text{Lie}}(E_0/A)$ are empty for all $n \in \mathbb{N}_{>0}$ . Indeed, let $\varphi \in S_{n,0}^{\text{Lie}}(E_0/A)$ so that $\textrm{disc} \left(\mathcal{O}_{j_0}[\varphi]\right)=0$ . Since a division quaternion algebra does not contain suborders of rank 3, this in particular implies that $\mathcal{O}_{j_0}[\varphi]$ has rank 2 as $\mathbb{Z}$ -module, so that $\mathbb{Z}[\varphi]$ is isomorphic to an order in $K_{j_0}$ , not necessarily contained in $\mathcal{O}_{j_0}$ . By the definition of $S_n^{\text{Lie}}(E_0/A)$ , the order $\mathbb{Z}[\varphi]$ has discriminant $\Delta$ , and we deduce that $\mathbb{Z}[\varphi] \cong \mathcal{O}_j \subseteq K_{j_0}$ . However, by assumption $\ell$ divides $\Delta$ but does not divide the conductor of $\mathcal{O}_j$ . Hence $\ell$ must divide the discriminant of $K_{j_0}$ which in turn implies $\ell \mid \Delta_0$ , contradicting our hypotheses. This proves the claim.

On the other hand, in the second paragraph of [ Reference Lauter and Viray33 , pag. 9247] it is proved that, when $\ell$ divides $\Delta$ but does not divide the conductor of $\mathcal{O}_{j}$ , and $\ell \nmid \Delta_0$ , then for every $m>0$ and $n>1$ , the set $S_{n,m}^{\text{Lie}}(E/A)$ is empty. We deduce that $S_n^{\text{Lie}}(E/A)=\emptyset$ for all $n>1$ , and combining this with inequality (16) we obtain $\textrm{Iso}_{A/\mu^n A}(E_0,E_j)=\emptyset$ for all $n>1$ . Finally, using [ Reference Gross and Zagier21 , Proposition 2.3] we obtain

\begin{equation*}v_\mu(\,j-j_0)= \frac{1}{2} \# \textrm{Iso}_{A/\mu A}(E_0,E_j) \leq \frac{d_0}{2}\end{equation*}

and this concludes the proof of Theorem 3·1.

4. Proof of Theorem 1·1

The main scope of this section is to present the proof of Theorem 1·1. At the end of this proof, we will point at the precise estimates that can be used to prove Theorem 1·2 and similar results, and we will provide a proof of the fact (stated in the introduction) that the extension $\mathbb{Q} \subseteq \mathbb{Q}(\,j_0)$ can be Galois for at most a finite number of singular moduli $j_0$ . Before starting, let us recall some notation already used in the introduction. For a number field K we denote by $\mathcal{M}_K$ the set of all places of K and by $\mathcal{M}^{\infty}_K \subseteq \mathcal{M}_K$ the subset of all the infinite ones. For every $w \in \mathcal{M}_K \setminus \mathcal{M}^{\infty}_K$ we indicate by $|\!\cdot\!|_w$ the absolute value in the class of w normalised as follows: if $\mathfrak{p}_w$ denotes the prime ideal corresponding to w and $p_w$ is the rational prime lying below $\mathfrak{p}_w$ , then

\begin{equation*}|x|_w=p_w^{-v_{\mathfrak{p}_w}(x)/e_w}\end{equation*}

for all $x \in K\setminus \{ 0 \}$ , where $v_{\mathfrak{p}_w}(x)$ is the exponent with which the prime $\mathfrak{p}_w$ appears in the factorisation of x, and $e_w$ is the ramification index of $\mathfrak{p}_w$ over $p_w$ .

Proof of Theorem 1·1. Let $(\,j_0,S)$ be a nice $\Delta_0$ -pair with $\Delta_0 < -4$ and $\#S \leq 2$ . We can assume without loss of generality that $\#S=2$ , since if S contains fewer than two elements the statement of the theorem becomes weaker. Hence we can write $S=\{\ell_1, \ell_2 \}$ with $\ell_1, \ell_2 \in \mathbb{N}$ two distinct primes.

In order to prove Theorem 1·1, we follow the strategy used in [ Reference Bilu, Habegger and Kuhne2 ] to prove the emptiness of the set of singular units. Let j be a singular modulus of discriminant $\Delta$ such that $j-j_0$ is an S-unit, and let $h(\!\cdot\!)$ denote the logarithmic Weil height on algebraic numbers. By the usual properties of height functions [ Reference Bombieri and Gubler4 , Lemma 1.5.18], we have

(17) \begin{equation} h(\,j-j_0)=h((\,j-j_0)^{-1})=\frac{1}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \sum_{v \in \mathcal{M}_{\mathbb{Q}(\,j-j_0)}} d_v \log^+ |(\,j-j_0)^{-1}|_v = A + N,\end{equation}

where $d_v\;:\!=\;[\mathbb{Q}(\,j-j_0)_v\;:\;\mathbb{Q}_v]$ is the local degree of the field $\mathbb{Q}(\,j-j_0)$ at the place v and

\begin{align*} &A\;:\!=\;\frac{1}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \sum_{v \in \mathcal{M}^{\infty}_{\mathbb{Q}(\,j-j_0)}} d_v \log^+ |(\,j-j_0)^{-1}|_v, \\ &N\;:\!=\;\frac{1}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \sum_{v \mid \ell_1 \ell_2 } d_v \log |(\,j-j_0)^{-1}|_v\end{align*}

are, respectively, the archimedean and non-archimedean components of the height. Notice that the expression for N follows from our assumption on $j-j_0$ being an S-unit and from the fact that $j-j_0$ is an algebraic integer. We study these two components separately, starting with the archimedean one. From now on, we assume $|\Delta| > \max \{|\Delta_0|, 10^{15} \}$ .

Denote by $C_0$ and $C_\Delta$ the class numbers of the orders associated to $j_0$ and to j respectively. Then by [ Reference Cai6 , Corollary 4.2 (1)] we have

(18) \begin{equation} A \leq \frac{8 F \log |\Delta| \cdot C_0}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} + \log \left( \frac{F \log |\Delta| \cdot C_0 \cdot |\Delta|^{1/2}}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \right) + 4\log |\Delta_0| + 0.33,\end{equation}

where $F\;:\!=\; \max \{2^{\omega(a)}\;:\;a\leq |\Delta|^{1/2} \}$ and $\omega(n)$ denotes the number of prime divisors of an integer $n\in \mathbb{N}$ . Using [ Reference Faye and Riffaut15 , Theorem 4.1] we have

\begin{equation*} [\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}] = [\mathbb{Q}(\,j,j_0)\;:\;\mathbb{Q}] \geq [\mathbb{Q}(\,j)\;:\;\mathbb{Q}]=C_\Delta\end{equation*}

which, combined with (18), gives

(19) \begin{equation} A \leq \frac{8 F \log |\Delta| \cdot C_0}{C_\Delta} + \log \left( \frac{F \log |\Delta| \cdot C_0 \cdot |\Delta|^{1/2}}{C_\Delta} \right) + 4\log |\Delta_0| + 0.33.\end{equation}

As far as the non-archimedean part is concerned, we have

(20) \begin{align} N&=\frac{1}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \sum_{v \mid \ell_1 \ell_2 } d_v \log |(\,j-j_0)^{-1}|_v= \frac{1}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \sum_{i\in \{1,2\}} \sum_{\mathfrak{p} \mid \ell_i} v_{\mathfrak{p}}(\,j-j_0) \log \ell_i^{f_{\mathfrak{p}}} \nonumber\\ \, \, \, \, \, \, \, \, \, \, &=\frac{\log \ell_1}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \sum_{\mathfrak{p} \mid \ell_1} v_{\mathfrak{p}}(\,j-j_0) f_{\mathfrak{p}} + \frac{\log \ell_2}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \sum_{\mathfrak{p} \mid \ell_2} v_{\mathfrak{p}}(\,j-j_0) f_{\mathfrak{p}}, \end{align}

where $f_{\mathfrak{p}}$ denotes the residue degree of the prime $\mathfrak{p} \subseteq \mathbb{Q}(\,j-j_0)$ lying over $\mathfrak{p}\cap \mathbb{Q}$ . For every $\mathfrak{p}\mid \ell_1 \ell_2$ , we choose a prime ideal $\mu \subseteq H$ that divides $\mathfrak{p}$ , where H denotes the compositum inside $\overline{\mathbb{Q}}$ of the ring class fields relative to j and $j_0$ . Note that this makes sense, since we have $\mathbb{Q}(\,j-j_0)\subseteq \mathbb{Q}(\,j,j_0) \subseteq H$ (the first inclusion is actually an equality by [ Reference Faye and Riffaut15 , Theorem 4.1]). We wish now to use Theorem 3·1 to bound $v_\mu(\,j-j_0)$ for all these primes $\mu$ . Let’s check that the hypotheses of the theorem are verified in our context:

  1. (1) since we are assuming $|\Delta_0| < |\Delta|$ , certainly we have $\Delta \neq \Delta_0$ ;

  2. (2) since $(\,j_0,S)$ is a nice $\Delta_0$ -pair, for $i \in \{1,2\}$ the prime $\ell_i$ splits completely in $\mathbb{Q}(\,j_0)$ . In particular, $\mu \cap \mathbb{Q}(\,j_0)$ has residue degree 1, as required;

  3. (3) since $(\,j_0,S)$ is a nice $\Delta_0$ -pair, for $i \in \{1,2\}$ the prime $\ell_i$ does not divide either $\Delta_0$ or $N_{\mathbb{Q}(\,j_0)/\mathbb{Q}}(\,j_0(\,j_0-1728))$ . In particular, this last condition implies that the elliptic curve

    \begin{equation*} {E_0}_{/\mathbb{Q}(\,j_0)}\;:\;y^2+xy=x^3-\frac{36}{j_0-1728} x - \frac{1}{j_0-1728} \end{equation*}
    with $j(E_0)=j_0$ , has good reduction at $\mu$ .

This discussion shows that we can apply Theorem 3·1 to bound $v_\mu(\,j-j_0)$ . Notice that under our assumptions we have, in the notation of the theorem, that $d_0=2$ since $\ell_i \nmid N_{\mathbb{Q}(\,j_0)/\mathbb{Q}}(\,j_0(\,j_0-1728))$ for $i \in \{1,2\}$ . Moreover, the imaginary quadratic order associated to j cannot contain the order associated to $j_0$ because $|\Delta| >|\Delta_0|$ . Thus we obtain

\begin{equation*} v_{\mathfrak{p}}(\,j-j_0) \leq v_{\mu}(\,j-j_0) \leq \max \left \{ \frac{\log(\Delta_0^2|\Delta|)}{2 \log \ell_i}+ \frac{1}{2}, 1 \right \}\end{equation*}

for all primes $\mathfrak{p} \mid \ell_i$ . Combining this with (20) and setting $L\;:\!=\;\max \{ \ell_1, \ell_2 \}$ we obtain

(21) \begin{align} N &\leq \frac{\log \ell_1}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \sum_{\mathfrak{p} \mid \ell_1} \max \left \{ \frac{\log(\Delta_0^2|\Delta|)}{2 \log \ell_1}+ \frac{1}{2}, 1 \right \} f_{\mathfrak{p}} + \frac{\log \ell_2}{[\mathbb{Q}(\,j-j_0)\;:\;\mathbb{Q}]} \end{align}
\begin{align} &\times \sum_{\mathfrak{p} \mid \ell_2} \max \left \{ \frac{\log(\Delta_0^2|\Delta|)}{2 \log \ell_2}+ \frac{1}{2}, 1 \right \} f_{\mathfrak{p}}\nonumber \\ &\leq (\log \ell_1) \max \left \{ \frac{\log(\Delta_0^2|\Delta|)}{2 \log \ell_1}+ \frac{1}{2}, 1 \right \} + (\log \ell_2) \max \left \{ \frac{\log(\Delta_0^2|\Delta|)}{2 \log \ell_2}+ \frac{1}{2}, 1 \right \} \nonumber \\ &= \max \left \{ \frac{\log(\Delta_0^2|\Delta|)}{2}+ \frac{\log \ell_1}{2}, \log \ell_1 \right \} + \max \left \{ \frac{\log(\Delta_0^2|\Delta|)}{2}+ \frac{\log \ell_2}{2}, \log \ell_2 \right \} \nonumber \\ &\leq 2 \max \left \{ \frac{\log(\Delta_0^2|\Delta|)}{2}+ \frac{\log L}{2}, \log L \right \} = \max \{\log(\Delta_0^2|\Delta|) + \log L, 2 \log L \}, \nonumber\end{align}

where in the second inequality we have used the fact that, for every number field K and any prime $q \in \mathbb{N}$ , we always have $\sum_{\mathfrak{q}\mid q} f_{\mathfrak{q}} \leq [K\;:\;\mathbb{Q}]$ (here the sum is taken over the prime ideals of K lying above q). Using now together (17), (19) and (21) we obtain the following upper bound

(22) \begin{align} h(\