Comptes Rendus Mathématique

Alexander Keimer

Lukas Pflug

Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity

Aissa Aibèche

Cherif Amrouche

Bassem Bahouli

Trace Operator’s Range Characterization for Sobolev Spaces on Lipschitz Domains of $\protect \mathbb{R}^2$

Laure Cardoulis

An inverse problem for a hyperbolic system in a bounded domain

Pradeep K. Rai

Comptes Rendus
Mathématique
Pradeep K. Rai
On the occurrence of elementary abelian p-groups as the Schur multiplier of
non-abelian p-groups
Volume 361 (2023), p. 803-806
Published online: 11 May 2023
https://doi.org/10.5802/crmath.445
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 803-806
https://doi.org/10.5802/crmath.445
Group theory / Théorie des groupes
On the occurrence of elementary abelian
p-groups as the Schur multiplier of
non-abelian p-groups
Pradeep K. Raia
a Mahindra University, Hyderabad, Telangana, India
E-mail: pradeepkumar.rai@mahindrauniversity.edu.in
Abstract. We prove that every elementary abelian p-group, for odd primes p, occurs as the Schur multiplier
of some non-abelian finite p-group.
Keywords. Schur multiplier, finite p-group.
Mathematical subject classification (2010). 20J99, 20D15.
Funding. This research is supported by INSPIRE Faculty Research Grant DST/INSPIRE/04/2016/001594.
Manuscript received 25 August 2022, revised and accepted 21 November 2022.
1. Introduction
The Schur multiplier M(G) of a finite group G is defined as the second cohomology group of G
with coefficients in C∗. It plays an important role in the theory of extensions of groups. Finding
bounds on the order, exponents, and ranks of the Schur multiplier of prime power groups, as well
as classifying groups with trivial Schur multipliers, have been main lines of investigation in the
past. Another problem where it appears that not much progress has been made is the following:
Question 1. Is it true that every finite abelian p-group is isomorphic to the Schur multiplier of
some non-abelian finite p-group? [5, Question 15.30].
More generally,
Question 2. Which abelian groups occur as the Schur multipliers of non-abelian finite
groups? [6].
The immediate questions that arise from these questions are the following:
Question 3. For all natural numbers n does there exist a non-abelian p-group G such that
|M(G)| = pn?
Question 4. For all natural numbers e does there exist a non-abelian p-group G such that
exponent of M(G) is pe ?
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
804 Pradeep K. Rai
Question 5. For all natural numbers d does there exist a non-abelian p-group G such that the
minimal number of generators of M(G) is d?
It is easy to see that the answers to the Questions 3 and 4 are affirmative. To see this, consider
G = Ep × (Zpm+1 ×Zpm ),
where Ep is the extra-special p-group of order p3 and exponent p, and Zn denote the cyclic
group of order n. For a group G , let γ2(G) denote the derived subgroup of G . Applying [4,
Theorem 2.2.10] we get that
M(G) ∼= M(Ep )×M(Zpm+1 ×Zpm )×
(
Ep
γ2(Ep )
⊗ (Zpm+1 ×Zpm )
)
.
Therefore |M(G)| = p2pm p4 = pm+6 and exponent of M(G) is pm . See [2, 3] for the existence of
non-abelian groups having Schur multiplier of order pn for each n ≤ 6.
We now state the main result of this article, which gives an affirmative answer to the Ques-
tion 5:
Theorem 6. Given a natural number n and an odd prime p, there exists a non-abelian p-group
whose Schur multiplier is elementary abelian of order pn .
2. Proof
Before proceeding to the proof of Theorem 6, we set some notations that are mostly standard.
Let G be a group and x, y ∈G . Then [x, y] denotes the commutator x−1 y−1x y . By d(G) we denote
the minimal number of generators of G . The exponent of the group G is denoted by exp(G). We
write γi (G) for the i -th term in the lower central series of G . By Zpn we denote the cyclic group of
order pn .
The proof is founded on the following two results of Blackburn and Evens [1].
Proposition 7 ([4, Corollary 3.2.4]). Let G be a finite nilpotent group of class c ≥ 2 and let G = F /R
be a presentation of G as a factor group of a free group of finite rank. Then
(1) upon identifying G/γ2(G) with F /γ2(F )R and γc (G) with γc (F )R/R, the map
G/γ2(G)⊗γc (G) λ7−→ γc+1(F )/([F,R]∩γc+1(F ))
f γ2(F )R ⊗xR 7−→ [ f , x]([F,R]∩γc+1(F ))
is a surjective homomorphism.
(2) Let X = Kerλ. Then there is an exact sequence
1 7−→ X 7−→G/γ2(G)⊗γc (G) 7−→ M(G) 7−→ M(G/γc (G)) 7−→ γc (G) 7−→ 1.
Lemma 8 ([1, Remark, Section 3]). Let G be a group of nilpotency class 2 with G/γ2(G) elementary
abelian. Then X , as defined in Proposition 7, is generated by all elements of the form xγ2(G)⊗[y, z]+
yγ2(G)⊗ [z, x]+ zγ2(G)⊗ [x, y] and all elements of the form xγ2(G)⊗xp .
The proof of the Theorem 6 also makes use of the following lemma. The proof of the lemma
can be extracted from the proof of [7, Proposition 3.2].
Lemma 9. Let G be a special p-group minimally generated by x1, x2, . . . , xd , and WG be the
subgroup of G/γ2(G)⊗γ2(G) generated by all elements of the form xγ2(G)⊗[y, z]+ yγ2(G)⊗[z, x]+
zγ2(G)⊗ [x, y], x, y, z ∈G. Then WG is generated by BG where
BG = {xiγ2(G)⊗ [x j , xk ]+x jγ2(G)⊗ [xk , xi ]+xkγ2(G)⊗ [xi , x j ] | 1 ≤ i < j < k ≤ d}.
Pradeep K. Rai 805
For a special p-group G notice that G/γ2(G) and γ2(G) are elementary abelian and therefore
can be regraded as vector spaces over Zp . With this in mind, we are now ready to present the
following two propositions, that are the main ingredients of the proof of the Theorem 6.
Proposition 10. For an odd prime p and natural numbers d and j , where d ≥ 4 and 3 ≤
j +2 ≤ d, let G j be the special p-group of exponent p minimally generated by x1, x2, . . . , xd , such
that all commutators [xr , xs ], other than [x1, x2], [x2, x3], . . . [x j+1, x j+2], are identity and the set
{[x1, x2], [x2, x3], . . . [x j+1, x j+2]} is linearly independent. Then M(G j ) is elementary abelian of order
p
1
2 d(d−1)+2 j+1.
Proof. Since G j is a group of nilpotency class 2, exp(M(G j )) divides exp(G). Therefore M(G j ) is
an elementary abelian group. Now, using the exact sequence given in Proposition 7, we get that
∣∣M(G j )∣∣=
∣∣∣M( G jγ2(G j )| )∣∣∣∣∣γ2(G j )∣∣
∣∣∣ G jγ2(G j ) ⊗γ2(G j )∣∣∣∣∣X ∣∣ . (1)
Let WG j and BG j be the sets defined in Lemma 9. To obtain |X |, note from Lemma 8 that
|X | = |WG j | because the group G j is of exponent p. Let (x, y, z) denotes the element xγ2(G j )⊗
[y, z]+yγ2(G j )⊗[z, x]+zγ2(G j )⊗[x, y] ∈G j /γ2(G j )⊗γ2(G j ) for x, y, z ∈G j . Consider the following
subset DG j of BG j
DG j =
{
(xi , xi+1, xk )
∣∣ i = 1,2, . . . j +1, k = 1,2, . . .d and k ̸= i −1, i , i +1},
and notice that all the elements of BG j can be generated by elements of DG j . As a result DG j
generates WG j . We further claim that DG j is a basis for WG j . To see this, let
j+1∑
i=1
d∑
k=1
k ̸=i−1,i ,i+1
αi k (xi , xi+1, xk ) = 0.
This gives
j+1∑
i=1
(( d∑
k=1
k ̸=i−1,i ,i+1
αi k xk
)
+α(i−1)(i+1)xi−1
)
⊗ [xi , xi+1] = 0.
Now, since {x1γ2(G), x2γ2(G), . . . xdγ2(G)} and {[x1, x2], [x2, x3], . . . , [x j+1, x j+2]} are linearly inde-
pendent, it follows that αi k = 0 for all i = 1,2, . . . j +1, k = 1,2, . . .d ,k ̸= i −1, i , i +1. Next, notice
that |DG j | = (d − 3) j + 1, and hence |WG j | = p(d−3) j+1. From [4, Corollary 2.2.12] we also have
that |M(G/γ2(G j ))| = p 12 d(d−1), because G j /γ2(G j ) is an elementary abelian p-group of order pd .
Putting these values and |γ2(G j )| = p j+1 in Equation (2.1) we get that
|M(G j )| = p
1
2 d(d−1)+2 j+1. □
The following Proposition can be proved along the same lines.
Proposition 11. For an odd prime p and a natural number j , where 4 ≤ j + 3 ≤ d, let G j
be the special p-group of exponent p minimally generated by x1, x2, . . . , xd , d ≥ 4, such that all
commutators [xr , xs ], other than [x1, x2], [x2, x3], . . . [x j , x j+1], [x j+2, x j+3], are identity and the
set {[x1, x2], [x2, x3], . . . [x j , x j+1], [x j+2, x j+3]} is linearly independent. Then M(G j ) is elementary
abelian of order p
1
2 d(d−1)+2 j .
We are now ready to prove the Theorem 6.
Proof of Theorem 6. Let m be a natural number such that m ≥ 10. Then there exist unique non-
negative integers k ≥ 5 and l ≤ k −1 such that m = 12 k(k −1)+ l . If l is a non-zero even number,
i.e., l = 2 j for some j ≥ 1, consider the group G j constructed in Proposition 11 with d = k so
that |M(G j )| = pm . In case l is an odd number greater than or equal to 3, i.e., l = 2 j + 1 for
806 Pradeep K. Rai
some j ≥ 1, consider the group G j constructed in Proposition 10 with d = k. If l = 0 and k is
an even number, consider the group G j constructed in 10 with d = k − 1 and j = k2 − 1, so that
|M(G j )| = p 12 (k−1)(k−2)+k−1 = p 12 k(k−1). If l = 0 and k is an odd number, consider the group G j
constructed in 11 with d = k−1 and j = k−12 , so that |M(G j )| = p
1
2 (k−1)(k−2)+k−1 = p 12 k(k−1). If l = 1
and k is an even number, consider the group G j constructed in 11 with d = k −1 and j = k2 , so
that |M(G j )| = p 12 (k−1)(k−2)+k = p 12 k(k−1)+1. If l = 1 and k is an odd number, consider the group G j
constructed in 10 with d = k−1 and j = k−12 , so that |M(G j )| = p
1
2 (k−1)(k−2)+k−1+1 = p 12 k(k−1)+1. For
the existence of groups with elementary abelian Schur multiplier of order pm ,m ≤ 9, see [2,3]. □
References
[1] N. Blackburn, L. Evens, “Schur multipliers of p-groups”, J. Reine Angew. Math. 309 (1979), p. 100-113.
[2] B. Eick, T. J. Ghorbanzadeh, “Computing the Schur multipliers of the Lie p-rings in the family defined by a symbolic
Lie p-ring presentation”, J. Symb. Comput. 106 (2021), p. 68-77.
[3] S. Hatui, V. Kakkar, M. K. Yadav, “The Schur multiplier of groups of order p5”, J. Group Theory 22 (2019), no. 4, p. 647-
687.
[4] G. Karpilovsky, The Schur multiplier, London Mathematical Society Monographs. New Series, vol. 2, Clarendon Press,
1987.
[5] E. I. Khukhro, V. D. Mazurov (eds.), The Kourovka notebook. Unsolved problems in group theory, 20th ed., Sobolev
Institute of Mathematics, 2022.
[6] P. Moravec, “A talk in Groups St Andrews”, 2009, https://mathshistory.st-andrews.ac.uk/Groups/2009/slides/
moravec.pdf.
[7] P. K. Rai, “On the Schur multiplier of the Special p-groups”, J. Pure Appl. Algebra 222 (2018), no. 2, p. 316-322.


On the occurrence of elementary abelian $p$-groups as the Schur multiplier of non-abelian $p$-groups

Stefanos Aivazidis

Thomas Müller

Congruences associated with families of nilpotent subgroups and a theorem of Hirsch

Mourad Choulli

Stable determination of the nonlinear term in a quasilinear elliptic equation by boundary measurements

Paul Manns

Thomas M. Surowiec

On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$

Rodolphe Richard

Andrei Yafaev

Comptes Rendus
Mathématique
Rodolphe Richard and Andrei Yafaev
On the generalised André–Pink–Zannier conjecture.
Volume 361 (2023), p. 1717-1722
Published online: 21 December 2023
https://doi.org/10.5802/crmath.491
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1717-1722
https://doi.org/10.5802/crmath.491
Algebraic geometry, Number theory / Géométrie algébrique, Théorie des nombres
On the generalised André–Pink–Zannier
conjecture.
Rodolphe Richard a and Andrei Yafaev ∗,a
a UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK
E-mails: r.richard@ucl.ac.uk, yafaev@ucl.ac.uk
Abstract. We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a
height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits
of points in a generalised Hecke orbit in terms of this height function, assuming the “weakly adelic Mumford–
Tate hypothesis” and prove the generalised André–Pink–Zannier conjecture (a special case of the Zilber-Pink
conjecture) under this assumption using the Pila–Zannier strategy.
Résumé. On introduit et étudie la notion d’orbite de Hecke généralisée dans une variété de Shimura. On
définit une notion de hauteur sur une telle orbite et étudie ses properiétés. On obtient une borne inférieure
pour la taille des orbites Galoisiennes de points dans une orbite de Hecke généralisee en termes de cette
fonction hauteur en admettant « la conjecture de Mumford–Tate faiblement adélique » et on démontre la
conjecture de André–Pink–Zannier généralisée (un cas particulier de la conjecture de Zilber-Pink) en utilisant
la stratégie de Pila–Zannier.
Keywords. Shimura varieties, Hecke orbits, Zilber-Pink, Heights, Siegel sets, Mumford–Tate conjecture, Adelic
linear groups.
2020 Mathematics Subject Classification. 03C64, 11G18, 11G50, 11F80, 14L30, 20G35, 15A16, 14G35.
Funding. This project was funded by Leverhulme Trust grant RPG-2019-180.
Manuscript received 27 May 2022, revised 28 September 2022 and 18 February 2023, accepted 6 March 2023.
This note describes the results of [21]. We refer to [5] for the language of Shimura varieties, and
to [23, §3] for the notion of weakly special subvarieties.
1. Generalised André–Pink–Zannier conjecture and Main results
Let (G , X ) be a Shimura datum, let K ≤ G(A f ) be a compact open subgroup, and let S =
ShK (G , X ) = G(Q)\X ×G(A f )/K be the associated Shimura variety. Let x0 ∈ X (which we view
as a morphism S→GR) and let M ≤G be its Mumford–Tate group.
In the following definition Hom(M ,G)(Q) denotes the set of algebraic group morphisms
defined overQ.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1718 Rodolphe Richard and Andrei Yafaev
Definition 1. We define the Generalised Hecke orbit H (x0) of x0 in X as
H (x0) := X ∩ {φ◦x0 :φ ∈ Hom(M ,G)(Q)}
and the Generalised Hecke orbit H ([x0, g0]) of [x0, g0] in ShK (G , X ) as
H ([x0, g0]) := {[x, g ] : x ∈H (x0), g ∈G(A f )}.
This notion of Hecke orbit is the most general; we refer to [21, §2.5] for a comparison with
some other notions of Hecke orbits. For instance, if A is an abelian variety with Mumford–Tate
group M ≤ G := GSp(2dim(A)), every abelian variety B isogenous to A can be obtained from a
morphism M → G ∈ H (x0). Definition 1 is also readily1 at least as general than the notion of
generalised Hecke orbit defined by [16, §3].
The following conjecture extends questions of [1] of André, [25] of the second author and [16]
of Pink. Similar questions have been considered by Zannier [26] in the context of abelian
schemes. An argument of Orr ([12]) shows that it is a special case of the Zilber-Pink conjecture
for pure Shimura varieties.
Conjecture 2 (Generalised André–Pink–Zannier, [21, Conj. 1.1]). Let S be a Shimura variety and
Σ a subset of a generalised Hecke orbit in S. Then the irreducible components of the Zariski closure
of Σ are weakly special subvarieties.
For a sufficiently large field E of finite type over Q we have the following (cf. [24]): S and s0 =
[x0,1] are defined over E and there exists a Galois representation ρx0 : Gal(E/E) → M(A f )∩K such
that
∀σ ∈ Gal(E/E), g ∈G(A f ), σ([x0, g ]) = [x0,ρx0 (σ) · g ].
Our main result is the following.
Theorem 3 ([21, Thm. 1.2]). In the above situation, let U = ρx0 (Gal(E/E)). We assume:
∃C > 0,∀ p, [K ∩M(Qp ) : U ∩M(Qp )] ≤C . (1)
Then, for any Σ⊆H (x0), every irreducible component of ΣZ ar is weakly special.
We call (1) the weakly adelic Mumford–Tate hypothesis. It is related to a weak form of the
conjecture [22, 11.4?] due to Serre. The assumption (1) is known in many cases. This leads to
the following unconditional results.
1.1.
Combining Theorem 3 with [4, Thm. A(i)] we have the following, which strictly contains a 2005
result of Pink [16, §7] (and [3, Thm. B]).
Theorem 4 ([21]). The conjecture 2 is true if S is of abelian type, and Σ contains a point s which
satisfies the Mumford–Tate conjecture (at some ℓ, in the sense of [24]).
The assumptions of Theorem 4 are satisfied in the case where S = Ag and Σ contains a
point [A], where the abelian variety A satisfies the Mumford–Tate conjecture (at some prime ℓ).
Examples of such abelian varieties include: dim(A) ≤ 3; or when dim(A) is odd and End(A) ≃ Z.
More examples are given in [15], and many examples are mentioned in [10, §2.4].
1Or combine [16, Prop. 3.6] and [21, Prop. 2.15].
Rodolphe Richard and Andrei Yafaev 1719
1.2.
The following is not restricted to Shimura varieties of abelian type.
Theorem 5 ([2, Thm. 1.2]). We decompose the adjoint datum (M ad , XM ad ) of (M , XM ) :=
(M , M(R) · x0) as a product
(p1, . . . , p f ) : (M
ad , XM ad ) ≃ (M1, X1)× . . .× (M f , X f )
with respect to theQ-simple factors Mi of M ad .
Assume that for some compact open subgroups Ki ≤ Mi (A f )
∀ i ∈ {1; . . . ; f }, [pi ◦adM (x0)] ∈ ShKi (Mi , Xi )(C)∖ShKi (Mi , Xi )(Q).
Then U = ρx0 (Gal(E/E)) satisfies (1).
Theorems 6 and 7 follow from the combination of Theorem 3 with Theorem 5.
Theorem 6 ([21]). The conjecture 2 is true if Σ contains a Q-Zariski generic point s of a special
subvariety Z ⊆ S, namely: for every proper subvariety V ⊊ Z defined overQ, we have s ̸∈V (C).
Theorem 7 ([21]). The conjecture 2 is true if M ad is Q-simple and Σ contains a point s in S(C)∖
S(Q).
In the case M ad = {1} we recover a result of [6] and [9].
Theorem 8 ([6] and [9]). The conjecture 2 is true if Σ contains a special point.
1.3. Previous results towards Conjecture 2
For the history of the Conjecture and previous results, see the introduction of [21].
The results [16, 17] towards Conjecture 2, based on equidistribution of Hecke points, are
limited to the case where s is Hodge generic, and to an assumption similar to (1) ([16, Def. 6.3],
[17, §6-7, pp. 57–59]). The case of general S, in the Hodge generic case Z = S, can be treated using
the results of [17] and an extension of [18, Prop 3.5].
For general Z and S, Conjecture 2 was obtained, under an assumption substantially weaker
than (1), but for a much more restrictive notion of Hecke orbits (the “S -Hecke orbit” for a finite
set of primes S ). See [20] for an approach of based on Ratner’s theorems, and see [11] for an
approach based on the Pila–Zannier strategy.
2. Some useful results
The proof of Theorem 3 in Section 3 relies on the following results, which are of independent
interest.
2.1. Geometric Hecke orbits
We define W =G ·φ0 ⊆ Hom(M ,G) the conjugacy class of the injectionφ0 : M ,→G , as an algebraic
variety. We define the geometric Hecke orbit as
H g (x0) = X ∩ {φ◦x0 :φ ∈W (Q)} ⊂H (x0).
The following is an essential tool in reduction steps in the proof of Theorem 3.
Theorem 9 ([21, §2.4]). The generalised Hecke orbit H (x0) is a union of finitely many geometric
Hecke orbits.
1720 Rodolphe Richard and Andrei Yafaev
2.2. Height functions and estimates
Letm and gbe Lie algebras overQ, and chooseZ-structures onm and g. We denote bymẐ ≤m⊗A f
and gẐ ≤ g⊗A f the corresponding Ẑ-structures and define
HA f : Hom(m,g)⊗A f →Z≥1 by Φ 7→ min{n ∈Z≥1 : n ·Φ(mẐ) ⊆ gẐ}.
2.2.1.
We choose Z-structures in such a way that gẐ is invariant under the adjoint action of K
and mẐ =m⊗A f ∩gẐ.
Proposition 10 ([21, §4.3]). There is a well defined function H[x0,1] : H ([x0,1]) →Z≥1 given by
[φ◦x0, g ] 7→ HA f (d(g−1 ·φ · g )).
This function is constant on the orbits of Gal(E/E) in H ([x0,1]).
For a set A and two functions f , g : A →R≥0, we write f ≼ g , resp. f ≈ g when
∃ b,c,d ∈R>0,∀ a ∈ A, f (a) ≤ d +b · g (a)c , resp. f ≼ g and g ≼ f .
The following estimate requires the assumption (1).
Theorem 11 ([21, Thm. 6.4 and Prop. 3.6]). As s = [φ◦x0, g ] ranges through H ([x0,1]), we have
|Gal(E/E) · s| ≈ HA f (d(g−1 ·φ · g )). (2)
2.2.2.
Only the lower bound HA f (d(g
−1 ·φ · g ))≼ |Gal(E/E) · s| is needed for the proof of Theorem 3.
This lower bound is derived from the following.
Theorem 12 ([21, Thm. B.1]). Let M ≤ GL(N ) be a linear algebraic subgroup defined over Q,
denote by φ0 : M → GL(N ) the identity morphism and W the GL(N )-conjugacy class of φ0. We
choose a basis of m such that mẐ =m⊗A f ∩gl(N , Ẑ), and define M(Ẑ) = M(A f )∩GL(N , Ẑ).
There exists c = c(φ0) ∈R>0 such that, as φ ranges through W (A f ), we have
[φ(M(Ẑ)) :φ(M(Ẑ))∩GL(N , Ẑ)] ≥ 1
c
ω(HA f (dφ))
·HA f (dφ). (3)
(Where ω(n) is the number of prime factors of n.)
A main tool in proving (3) is Theorem 13. We establish Theorem 13 using estimates [21,
Prop. A.1] on p-adic norms of exponentials of a matrix X ∈ Md (Qp ).
Theorem 13 (Lemma of the exponentials, [21, Thm. A.3]). Let X ∈ Md (Qp ) be such that
exp(X ) converges and denote by exp(X )Z the subgroup generated by exp(X ) in GLd (Qp ). We de-
fine Hp (X ) = max{1;∥X ∥}. Then we have
[exp(X )Z : exp(X )Z∩GLd (Zp )] ≥ Hp (X )/d .
If p > d, we have [exp(X )Z : exp(X )Z∩GLd (Zp )] ≥ Hp (X ).
2.3. Comparison with the global height
Let S⊆G(R)+ be a finite union of Siegel sets, and denote by S ·φ0 ⊆W (R) its image.
There exists a closed affine embedding W → An defined over Q, say ι : φ 7→ dφ : W →
Hom(m,g). For (q1, . . . , qn) ∈Qn , we denote by HQ(q1, . . . , qn) ∈Z≥1 the usual height of (q1, . . . , qn)
and we denote by H f (q1, . . . , qn) ∈Z≥1 the lowest common multiple of the denominators of the qi .
We recover HA f (dφ) = H f (ι(φ)) on W (Q).
Rodolphe Richard and Andrei Yafaev 1721
Theorem 14 ([21, Thm. 5.16]). As φ ranges through S ·φ0 ∩W (Q), we have
H f (ι(φ)) ≈ HQ(ι(φ)). (4)
Theorem 14 was [13, Thm. 1.1] obtained by M. Orr, in the case W = G . This is insufficient for
us. We replaced the “block version of Pila–Wilkie” (used by Orr) by Theorem 15.
The above (4) is used to relate the height used in (2) with the height used in Theorem 15. We
deduce Theorem 15 from [14, Thm. 1.7]. The referee informed us that Theorem 15 can also be
deduced from [7, Cor. 7.2]. Theorem 15 and its proof in [21] are both much simpler.
Theorem 15 ([21, Thm. 7.1]). Let W ⊆AN be a closed subvariety defined over Q, let X be a semi-
algebraic set, and let p : W (R) → X be a semialgebraic map.
Let Z ⊆ X be a definable subset, and denote Z alg be the union of the semialgebraic subsets of X
which are contained in Z and of non-zero dimension.
Then, for every ϵ ∈R>0, there exists C (ϵ, Z ) ∈R>0, such that
∀ T ≫ 0, |(Z ∖Z alg)∩p({w ∈W (Q) : HQ(w) ≤ T })| ≤C (ϵ, Z ) ·T ϵ.
3. Outline of the proof of Theorem 3
We reduce the Conjecture 2 to the case where V :=Σ= {s0; s1; . . .} is irreducible, G is adjoint and V
is Hodge generic in S. We rely on functoriality properties of geometric and generalised Hecke
orbits.2 Theorem 9 allows us to use geometric and generalised Hecke orbits interchangeably. We
also prove, cf. [21, §6.3], functoriality properties of the assumption (1).
The final objective of the proof is to apply the geometric part of the André–Oort conjecture [23]
(or [19]), and use induction on the number of simple factors of M ad . For every n large enough,
we construct a weakly special subvariety Zn ⊆ V of non-zero dimension such that sn ∈ Zn .
Then [19, 23] describes
⋃
Zn , and we deduce Conjecture 2.
In order to construct the non-zero dimensional Zn , we use the Pila–Zannier strategy. The
generalised Hecke orbit is naturally related to W (Q) where W = G · φ0 ≃ G/ZG (M) (cf. [21,
Lem. 2.5].)
The goal is to apply the variant Theorem 15 of the Pila–Wilkie theorem, after constructing
many rational points of small height in some set definable in an o-minimal structure. This
definable set is
Ṽ =
(
−1
π (V )∩S
)
/ZG(R)(M) ⊆W (R)
where π : G(R) → X → S is the uniformisation map, and S ⊆ G(R) is a finite union of Siegel sets
such that S =π(S).
Let E be field of definition of V . Then V contains the Galois orbits Gal(E/E)·sn . Each point s′ ∈
Gal(E/E) · sn lifts to a rational point s̃′ ∈ Ṽ ∩W (Q).
By Proposition 10, the value of H[x0,1] is constant as s
′ ranges through Gal(E/E) · sn . It follows
from (2) that there are #Gal(E/E) · sn ≈ H[x0,1](sn) such points.3 By (4), we have H[x0,1](sn) ≈
HQ(s̃n). We introduce
Qn := {φ ∈S ·φ0 ∩W (Q) : [φ◦x0 : 1] ∈ Gal(E/E) · sn} ⊆ Ṽ .
For φ ∈Qn , we have HA f (dφ) = H[x0,1]([φ◦x0,1]) = H[x0,1](sn) ≈ HQ(s̃n).
Denote by p the map G(R) ·φ0 → X with G(R) ·φ0 ⊆W (R). We have surjections Qn → p(Qn) →
Gal(E/E) · sn . Thus #Qn ≥ #Gal(E/E) · sn ≈ HQ(s̃n).
Thus Ṽ contains #Qn ≈ HQ(s̃n) points of height ≈ HQ(s̃n).
2This avoids one difficulty in the approach [12] of Orr.
3This is where the assumption (1) is needed.
1722 Rodolphe Richard and Andrei Yafaev
By Theorem 15, for sufficiently large n, there exist φn in Qn such that p(φn) ∈ Z alg, with Z =
p(Ṽ ). By Ax–Lindemann–Weierstrass theorem [8], it follows that s′n = [φn ,1] ∈ Zn ⊆ V , for a non-
zero dimensional weakly special subvariety Zn . Using Galois action, we may assume s′n = sn .
This concludes the proof of Theorem 3.
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