Comptes Rendus Mathématique

Nicolas Fournier

On gelation for the Smoluchowski coagulation equation

Idriss Mazari-Fouquer

Another look at qualitative properties of eigenvalues using effective Hamiltonians

Fan Kang

On Berger’s isoperimetric problem

Qiaonan Gu

Kan Jiang

Lifeng Xi

Comptes Rendus
Mathématique
Qiaonan Gu, Kan Jiang and Lifeng Xi
Bedford–McMullen carpets meet nonlinear curves
Volume 363 (2025), p. 845-851
Online since: 15 July 2025
https://doi.org/10.5802/crmath.752
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 845-851
https://doi.org/10.5802/crmath.752
Research article / Article de recherche
Dynamical systems / Systèmes dynamiques
Bedford–McMullen carpets meet nonlinear
curves
Les tapis de Bedford–McMullen intersectent des courbes
non linéaires
Qiaonan Gu a, Kan Jiang a and Lifeng Xi a
a Department of Mathematics, Ningbo University, Ningbo, People’s Republic of China
E-mails: 13586815362@163.com (Qiaonan Gu), jiangkan@nbu.edu.cn, xilifeng@nbu.edu.cn
Abstract. Let f be a regular function (the precise definition will be provided in the paper). Then, for any
α ∈ (0,2) and any ϵ> 0, there exists a Bedford–McMullen carpet K with convex hull [0,1]2 such that
α−ϵ< dimH (K ) <α and
{
(x, y) : y = f (x)}∩K = {(0,1)},
where dimH denotes the Hausdorff dimension. In particular, this result holds for y = cos(x). Unlike classical
methods for analyzing slicing sets, our proofs rely exclusively on mathematical induction.
Résumé. Soit f une fonction régulière (la définition précise sera donnée dans l’article). Alors, pour tout
α ∈ (0,2) et tout ϵ> 0, il existe un tapis de Bedford–McMullen K d’enveloppe convexe [0,1]2 tel que
α−ϵ< dimH (K ) <α et
{
(x, y) : y = f (x)}∩K = {(0,1)},
où dimH désigne la dimension de Hausdorff. En particulier, ce résultat est valable pour y = cos(x). Contrai-
rement aux méthodes classiques d’analyse des ensembles de coupe, notre démonstration repose exclusive-
ment sur un raisonnement par récurrence.
2020 Mathematics Subject Classification. 28A80, 28A78.
Funding. Kan Jiang was supported by Zhejiang Provincial Natural Science Foundation of China under Grant
No. LMS25A010009. Lifeng Xi was supported by National Natural Science Foundation of China under Grant
Nos. 12171259, 11831007, and 11771226.
Manuscript received 13 February 2025, revised 9 April 2025, accepted 18 April 2025.
1. Introduction
Let K ⊂ R2 be a fractal set and Γ ⊂ R2 be a curve. In geometric measure theory, determining the
Hausdorff dimension of K ∩Γ is a challenging problem [2]. Classical results, such as Marstrand’s
theorem [2], provide upper bounds for the dimension of linear slices. Roughly speaking, for
a given fractal set K ⊂ R2 and a line Γ ⊂ R2, generically, the Hausdorff dimension of K ∩ Γ is
dimH (K )−1. The reader may refer to Mattila’s book [2, Theorem 10.11]. However, the situation
becomes significantly more complicated when dealing with nonlinear slicing sets. For instance,
when K is a Bedford–McMullen carpet and Γ is a nonlinear curve, the analysis of K ∩ Γ is
generally non-trivial. To date, there has been limited references in this area. In this paper, we
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
846 Qiaonan Gu, Kan Jiang and Lifeng Xi
focus on the case where K is a Bedford–McMullen carpet and Γ is a nonlinear curve satisfying
specific conditions. We prove that although some Bedford–McMullen carpet has large Hausdorff
dimension, it simultaneously intersects many smooth curves with only finite points.
We begin by introducing the following definition.
Definition 1. Let f ∈C∞(R) be a function. Suppose that s is the smallest positive integer such that
f (s)(0) ̸= 0, where f (s) denotes the s-th order of the derivative and s ∈ N≥2. Let m = ns ,n ≥ 10100.
We say that f is a regular function if it satisfies the following properties:
(1) f (0) = 1, 1−1/m > f (1−1/n) > f (1) > 1/m;
(2) f is strictly decreasing on [0,1];
(3) (n −1)−s ≤ − f (s)(x)s! ≤ 1−1/m, ∀ x ∈ [0,1/3].
Remark 2. We give some remarks on the condition:
1−1/m > f (1−1/n) > f (1) > 1/m.
Since f is strictly decreasing, f (1−1/n) > f (1) is clear. If we let n →∞, then the above inequalities
become
f (0) = 1 ≥ f (1) ≥ 0.
The first inequality is clear as f is strictly decreasing. If
0 ≤ a ≤ n/3−1, 0 ≤ b ≤ m/3,
then the third condition implies that∣∣∣∣ f (s)(x)s! (a +1)sns
∣∣∣∣≤ ∣∣∣∣ f (s)(x)s! 13s
∣∣∣∣≤ m −b −1m , x ∈ [0,1/3].
We shall use this inequality in the proof of Theorem 5.
We present some essential definitions that will be used throughout the paper. Let N+ denote
the set of all positive integers. For a given integer k ∈ N+, we define N≥k =
{
x ∈ N+ : x ≥ k}. For
any positive real number α ∈R+, we denote by [α] the integer part of α. For instance, [2.025] = 2.
We recall the definition of Bedford–McMullen carpet. Let n,m ∈ N≥2, m > n, and let A be a
subset of {0,1, . . . ,n−1}×{0,1, . . . ,m−1}. Without loss of generality, we assume that the cardinality
of A is at least 2, and call A the digit set. We define a class of self-affine functions as follows:
ga(x, y) =
(
x +xi
n
,
y + yi
m
)
, a = (xi , yi ) ∈A .
The collection of {ga}a∈A is called an iterated function system (IFS). Hutchinson [1] proved that
there exists a unique non-empty compact set K satisfying
K = ⋃
a∈A
ga(K ).
Such a set K is referred to as a Bedford–McMullen carpet. For any k ∈N+ and ai ∈A with 1 ≤ i ≤ k,
we define
ga1a2···ak
(
[0,1]2
)= ga1 ◦ ga2 ◦ · · · ◦ gak ([0,1]2)
as a basic rectangle at the k-th level. If the curve y = f (x) intersects a basic rectangle I , we denote
by (x, y) f ∈ I .
The following result, due to McMullen [3], provides the Hausdorff dimension of a Bedford–
McMullen carpet.
Theorem 3. Let K be a Bedford–McMullen carpet associated with the IFS {gai }ai∈A . Then the
Hausdorff dimension of K is given by
dimH (K ) =
ln
(∑n−1
j=0 t
logm n
j
)
lnn
,
Qiaonan Gu, Kan Jiang and Lifeng Xi 847
where t j denotes the number of indices i such that ( j , i ) ∈A .
We state the main result of this paper.
Theorem 4. Let f be a regular function defined in Definition 1. Then for any α ∈ (0,2) and any
ϵ> 0, there exists a Bedford–McMullen carpet K with the convex hull [0,1]2 such that
α−ϵ< dimH (K ) <α,
and that {
(x, y) : y = f (x)}∩K = {(0,1)},
where dimH stands for the Hausdorff dimension.
Theorem 4 immediately follows from the following result.
Theorem 5. Let f be a regular function defined in Definition 1. Let K be a Bedford–McMullen
carpet defined by the digit set
D = {(0,0), (0,m −1), (n −1,0), (n −1,m −1), (ai ,bi )}, 1 ≤ i ≤ u, u ∈N+.
If m = ns with n ≥ 10100, and 0 ≤ ai ≤ n/3−1, 0 ≤ bi ≤ m/3, ai ,bi ∈N+∪ {0}, then{
(x, y) : y = f (x)}∩K = {(0,1)}.
Applying Theorem 4, we obtain the following example.
Example 6. For anyα ∈ (0,2), ϵ> 0 and any closed interval [κ1,κ2] ⊂ (0,1), there exists a Bedford–
McMullen carpet K such that
α−ϵ< dimH (K ) <α,( ⋃
β∈[0.9,1]
{
(x, y) : y = cos(βx)})∩K = {(0,1)},
and ( ⋃
γ∈[κ1,κ2]
{
(x, y) : y = 1−γx2})∩K = {(0,1)}.
Let u(x) = cos(βx), β ∈ [0.9,1] and v(x) = 1−γx2, γ ∈ [κ1,κ2]. It is easy to check that for these
two functions we have s = 2. Hence, when we construct some Bedford–McMullen carpet, we can
assume m = n2. Note that κ1 and κ2 are two fixed numbers. Hence, if n, m = n2 are sufficiently
large, v(x) and u(x) are regular functions. So, Example 6 follows from Theorem 4 and the proof of
Theorem 5.
By Theorem 4, numerous other functions can be readily identified. We do not give more
examples. The key insight of this example lies in the existence of a uniform Bedford–McMullen
set that intersects a variety of distinct functions at a single unique point.
This paper is organized as follows. In Section 2, we prove Theorem 4 via Theorem 5. Section 3
is entirely devoted to the detailed proof of Theorem 5.
2. Proof of Theorem 4 by using Theorem 5
Proof. For any α ∈ (0,1), if m is sufficiently large, then [mα] ∈ (0,m/3). Thus, we may select [mα]
basic rectangles within the vertical strip with the form [0,1/n]×[i /m, (i +1)/m], 0 ≤ i ≤ [m/3]−1,
m = ns . Additionally, we include the four basic rectangles at the corners of the unit square at the
first level. We denote the resulting Bedford–McMullen carpet by K1. In light of Theorem 3, the
Hausdorff dimension of K1 is given by
log
((
[mα]+1)1/s +21/s)
logn
.
848 Qiaonan Gu, Kan Jiang and Lifeng Xi
Since m = ns , the above dimension converges to α as n → ∞. Furthermore, by means of
Theorem 5, the curve y = f (x) meets the Bedford–McMullen carpet K1 with a unique point, i.e.
(0,1).
For α ∈ (1,2), we have α−1 ∈ (0,1). Thus, for sufficiently large m,
[mα−1] ∈ [0,m/3].
We can then choose [n/3] distinct vertical strips (each of the form
[
i /n, (i +1)/n]× [0,1], where
0 ≤ i ≤ [ n3 ]−2) in the unit square at the first level. For each strip, we select [mα−1] basic rectangles
of the form: [
i /n, (i +1)/n]× [ j /m, ( j +1)/m], 0 ≤ i ≤ [n/3]−2, 0 ≤ j ≤ [m/3]−1.
We also include the four-cornered basic rectangles at the first level. Let K2 be the resulting
Bedford–McMullen carpet. Applying McMullen’s result once again, the Hausdorff dimension of
K2 is
log
((
[mα−1]+1)1/s + ([mα−1])1/s × ([ n3 ]−2)+21/s)
logn
,
which converges to α as n →∞. By virtue of Theorem 5, we conclude that
K2 ∩
{
(x, y) : y = f (x)}= {(0,1)}.
3. Proof of Theorem 5
Proof. The digit set D consists of four digits{
(0,0), (0,m −1), (n −1,0), (n −1,m −1)}.
Define the following functions:
f1(x, y) = (x/n, y/m),
f3(x, y) =
(
(x +n −1)/n, y/m),
f4(x, y) =
(
(x +n −1)/n, (y +m −1)/m),
f5(x, y) =
(
x/n, (y +m −1)/m).
To represent u digits (ai ,bi ), 1 ≤ i ≤ u, we use a uniform function:
f2(x, y) =
(
(x +a)/n, (y +b)/m), 0 ≤ a ≤ n/3−1, 0 ≤ b ≤ m/3, a,b ∈N+∪ {0}.
For instance, let n = 5, m = 25, a = 1, b = 11, y = cos(x), the reader may refer to Figure 1 for the
first iteration.
In what follows, we let F (x1, x2) = x2 − f (x1). Suppose y = f (x) and (x, y) ∈ K . Since f (0) = 1, it
follows that the curve y = f (x) intersects f5
(
[0,1]2
)
. Moreover, it is straightforward to verify that:
F
(
f1(1,1)
)≤ F ( f3(1,1))= 1/m − f (1) < 0,
as f (1) > 1/m. Since f is strictly decreasing and
∂x1 F =− f ′(x1) ≥ 0, ∂x2 F = 1 > 0,
by the above inequalities, it follows that the four corners of the rectangles, i.e. f3
(
[0,1]2
)
and
f1
(
[0,1]2
)
, are lying below the curve y = f (x). Thus, the curve y = f (x) does not intersect
f1
(
[0,1]2
)
or f3
(
[0,1]2
)
. We shall use similar ideas to prove other cases. Next, observe that:
F
(
f2(1,1)
)= b +1
m
− f
(
a +1
n
)
.
Qiaonan Gu, Kan Jiang and Lifeng Xi 849
Figure 1. y = cos(x) intersects basic rectangles at the first level.
By Taylor’s formula and the definition of s, there exists ξ ∈ (0,(a +1)/n)) such that
F
(
f2(1,1)
)= b +1
m
− f (0)− f
(s)(ξ)
s!
(a +1)s
ns
.
To prove
F
(
f2(1,1)
)< 0,
it suffices to show ∣∣∣∣ f (s)(ξ)s! (a +1)sns
∣∣∣∣≤ m −b −1m .
Since 0 ≤ a ≤ n/3−1, 0 ≤ b ≤ m/3, we only need:∣∣∣∣ f (s)(ξ)s! (a +1)sns
∣∣∣∣≤ ∣∣∣∣ f (s)(ξ)s! 13s
∣∣∣∣≤ m −b −1m .
The last inequality holds due to Remark 2. Hence, the curve y = f (x) does not intersect f2
(
[0,1]2
)
.
Using the conditions of the regular function, we have
F
(
f4(0,0)
)= 1−1/m − f (1−1/n) > 0,
and
1/m < f (1) < 1−1/m.
Hence, at the first level, the curve y = f (x) intersects only f5
(
[0,1]2
)
.
If (x, y) f ∈ f5
(
[0,1]2
)
, we can show via induction that for any k ∈N+,
(x, y) f ∈ f5k
(
[0,1]2
)
.
Taking k → ∞, we obtain (x, y) = (0,1). To prove this, assume that at the k-th level, (x, y) f ∈
f5k
(
[0,1]2
)
, we aim to show (x, y) f ∈ f5k+1
(
[0,1]2
)
. Since f is strictly decreasing, it suffices to verify:
F
(
f5k 3(1,1)
)< 0, F ( f5k 4(0,0))> 0, F ( f5k 2(1,1))< 0,
and
1− (m −1)/mk+1 < f (1/nk ) < 1−1/mk+1.
850 Qiaonan Gu, Kan Jiang and Lifeng Xi
Here f5k i (x, y) = f5 ◦ · · · ◦ f5︸ ︷︷ ︸
k
◦ fi (x, y), i = 2,3,4. Note that
F
(
f5k 3(1,1)
)= 1−1/mk +1/mk+1 − f ( 1
nk
)
.
By Taylor’s theorem, there exists some ς ∈ (0,1/nk ) such that
F
(
f5k 3(1,1)
)= 1−1/mk +1/mk+1 −1− f (s)(ς)
s!
1
nks
.
Hence, to show F
(
f5k 3(1,1)
)< 0, it suffices to prove
− f
(s)(ς)
s!
< 1−1/m, ς ∈ (0,1/nk ) ⊂ (0,1/3).
However, this is one of the conditions for the regular function.
For the second inequality, we have
F
(
f5k 4(0,0)
)= 1−1/mk+1 − f (n −1
nk+1
)
.
Employing Taylor’s theorem:
F
(
f5k 4(0,0)
)= 1−1/mk+1 −(1+ f (s)(η)
s!
(
n −1
nk+1
)s)
,
where η ∈ (0, n−1
nk+1
)
. To show F
(
f5k 4(0,0)
)> 0, it suffices to have
− f (s)(x)/s! > (n −1)−s
for any x ∈ [0,1/3]. This is guaranteed by the conditions of the regular function. For the third
inequality:
F
(
f5k 2(1,1)
)= b +1
mk+1
+1− 1
mk
− f
(
a +1
nk+1
)
.
Utilizing Taylor’s theorem:
b +1
mk+1
+1− 1
mk
− f
(
a +1
nk+1
)
= b +1
mk+1
+1− 1
mk
−
(
1+ f
(s)(κ)
s!
(
a +1
nk+1
)s)
,
where κ ∈ (0,(a +1)/nk+1). However, by the following conditions, i.e.
m = ns , 0 ≤ a ≤ n/3−1 and 0 ≤ b ≤ m/3,
the inequality F
(
f5k 2(1,1)
)< 0 holds from∣∣ f (s)(x)/s!∣∣≤ 3s((m −b −1)/m), ∀ x ∈ [0,1/3].
This is a consequence of the regular function, as noted in Remark 2. Finally, we verify:
1− (m −1)/mk+1 < f (1/nk ) < 1−1/mk+1.
In terms of Taylor’s theorem again,
f (1/nk ) = f (0)+ f
(s)(ξ)
s!
1
nks
= 1+ f
(s)(ξ)
s!
1
nks
,
where ξ ∈ (0,1/nk ). Assuming:
1
ns
= 1/m < − f
(s)(x)
s!
< 1−1/m
for any x ∈ [0,1/3], the inequalities hold. This is ensured by the conditions of the regular function.
Qiaonan Gu, Kan Jiang and Lifeng Xi 851
Acknowledgments
The authors would like to express their gratitude to the referee for the helpful comments.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] J. E. Hutchinson, “Fractals and self-similarity”, Indiana Univ. Math. J. 30 (1981), no. 5,
pp. 713–747.
[2] P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability.,
1st paperback edition, Cambridge Studies in Advanced Mathematics, Cambridge University
Press, 1999, xii+343 pages.
[3] C. McMullen, “The Hausdorff dimension of general Sierpiński carpets”, Nagoya Math. J. 96
(1984), pp. 1–9.


Bedford–McMullen carpets meet nonlinear curves

Khaled Basdouri

Sami Benabdelhafidh

Cohomology and deformations of modified Rota–Baxter Lie–Yamaguti algebras

Shoutao Guo

Li Liang

Xiaoyan Yang

Balancedness of complete cohomology with respect to subcategories

Bruno Chiarellotto

Fabien Trihan

Comptes Rendus
Mathématique
Bruno Chiarellotto and Fabien Trihan
Isogeny and overconvergence
Volume 363 (2025), p. 867-871
Online since: 15 July 2025
https://doi.org/10.5802/crmath.774
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 867-871
https://doi.org/10.5802/crmath.774
Research article / Article de recherche
Algebraic geometry, Number theory / Géométrie algébrique, Théorie des nombres
Isogeny and overconvergence
Isogénie et surconvergence
Bruno Chiarellotto a and Fabien Trihan b
a Dipartimento di Matematica “Tullio Levi-Civita” (DM), Univ. Padova, Via Trieste 63, 35121 Padova, Italy
b Sophia University, Department of Information and Communication Sciences 7-1 Kioicho, Chiyoda-ku,
Tokyo 102-8554, Japan
E-mails: chiarbru@math.unipd.it, f-trihan-52m@sophia.ac.jp
Abstract. In this paper, we apply Tsuzuki’s main theorem in [12] to establish a criterion for when two
abelian varieties over a function field K of characteristic p are isogenous. Specifically, assuming that their
endomorphism algebras tensored with Qp are division algebras, we prove that if the maximal quotients of
minimal slope (i.e., the unique maximal isoclinic quotient corresponding to the minimal slope, defined up
to isogeny) of their associated p-divisible groups are isogenous, then the abelian varieties themselves are
isogenous over K . We also extend this result to certain p-divisible groups, highlighting the deep connection
between isogenies of abelian varieties and the structure of overconvergent F -isocrystals.
Résumé. Dans cet article, nous appliquons le théorème principal de Tsuzuki dans [12] afin d’établir un critère
permettant de déterminer quand deux variétés abéliennes sur un corps de fonctions K de caractéristique p
sont isogènes. Plus précisément, en supposant que leurs algèbres d’endomorphismes tensorisées par Qp
sont des algèbres à division, nous démontrons que si les quotients maximaux de pente minimale (c’est-à-
dire le quotient isocline maximal unique correspondant à la pente minimale, défini à isogénie près) de leurs
groupes p-divisibles associés sont isogènes, alors les variétés abéliennes elles-mêmes sont isogènes sur K .
Nous étendons également ce résultat à certains groupes p-divisibles, mettant en évidence le lien profond
entre les isogénies de variétés abéliennes et la structure des F -isocristaux surconvergents.
Funding. The second named author (F. T.) was supported by the Japanese Society for Promotion of Sciences
(JSPS) (Research grant C/21K03186). The first author by European Union – Next Generation EU under the
National Recovery and Resilience Plan (PNRR), Mission 4 Component 2 Investment 1.1 – Call PRIN 2022,
MUR Project 20222B24AY “The arithmetic of motives and L-functions”.
Manuscript received 18 November 2024, revised 20 March 2025 and 21 June 2025, accepted 22 June 2025.
The aim of this note is to establish a new criterion on detecting isogenies of abelian varieties
over function fields of characteristic p > 0. Namely, abelian varieties A/K over function fields
induce, via their Néron models, abelian schemes on an open subset of a connected proper
smooth curve C /Fq (with function field K ). By the theory of Oort–Zink (see [8]), the associated
p-divisible group of an abelian scheme admits a slope filtration up to isogeny—that is, while the
filtration itself is not unique, its isogeny class is uniquely determined by the Newton polygon. Our
main result is that one can detect the isogeny class of the abelian variety by looking at only one
of its slopes. More precisely, under the assumption that the endomorphism algebras tensored
withQp are division algebras, if the maximal quotient of minimal slope (i.e., the unique maximal
isoclinic quotient corresponding to the minimal slope, defined up to isogeny) of the p-divisible
groups of A and B are isogenous, then the abelian varieties themselves are isogenous over K .
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
868 Bruno Chiarellotto and Fabien Trihan
Our proof is using the “minimal slope conjecture” proved by [3,12] and the fully faithfulness
of various functors between p-divisible groups and overconvergent F -isocrystals. Finally, we
extend our result to the case of p-divisible groups over varieties over finite fields and to higher
transcendence degree fields of characteristic p.
First, let us recall the “minimal slope conjecture” proved by [3,12]. We consider X a smooth
connected scheme over a finite field and M † an overconvergent F -isocrystal on X , and denote
the convergent F -isocrystal on X associated to M † by M . We say M admits a slope filtration if
there exists an increasing filtration
0 = M0 ⊂ M1 ⊂ ·· · ⊂ Mr−1 ⊂ Mr = M
of M as convergent F -isocrystals on X /K such that:
(i) Mi /Mi−1 is nonzero and isoclinic of slope si ;
(ii) s1 < s2 < ·· · < sr .
We call s1 (resp. sr ) the minimal slope (resp. the maximal slope) of M when M ̸= 0. Such a slope
filtration exists if the Newton polygons of the Frobenius structure of a convergent F -isocrystal M
are constant, which is always true after a certain shrinking of X (see [12, 1.1]). Hence M always
admits a unique slope filtration as convergent F -isocrystals after a certain shrinking of X .
We can now recall the theorem of [3,12]:
Theorem 1. Assume that M † and N † are two irreducible overconvergent isocrystals over X with
slope filtrations. After renumbering the slope filtration as follows:
M = M0 ⊃ M1 ⊃ M2 ⊃ ·· · ⊃ Mr−1 ⊃ Mr = 0
with the sequence of slopes s0 > s1 > ·· · > sr−1, suppose that there is a nontrivial morphism
h : N /N1 → M/M1 between the minimal slope sub-objects (or, by duality between the maximal
slope quotients) as convergent F -isocrystals. Then there exists a unique isomorphism g † : N † → M †
of overconvergent F -isocrystals compatible with the map g .
We will need the following lemma:
Lemma 2. Let A be an abelian variety over a function field of characteristic p. We denote A its
Néron model over the projective smooth connected curve C with function field K . Let U be an open
subset of C where A/K has good reduction. Let D†(A) be the overconvergent F -isocrystal over U /Qp
associated to the abelian scheme A
∣∣
U . Then D
†(A) is semisimple and it will be simple if and only
if End(A)⊗Qp is a division algebra.
Proof. Note that by [9, Theorem 1.2], D†(A) is semisimple. Now a semisimple module is simple if
and only if its endomorphism ring is a division ring. But again by [9, Theorem 1.1], End
(
D†(A)
)≃
End
(
A
∣∣
U
)⊗Qp ≃ End(A)⊗Qp , where the second isomorphism is deduced by universal properties
of the Néron model and smoothness of A
∣∣
U /U . □
Remark 3.
(1) Assume End(A)⊗Q to be a division algebra. If it is equal toQ, then obviously End(A)⊗Qp
is still a division algebra. If F := End(A)⊗Q is strictly larger than Q and a commutative
field, then FQp is a division algebra if and only if (p) is a primary ideal of OF .
(2) Note that if End(A)⊗Qp is a division algebra, then so is End(A)⊗Q, in particular, since
abelian varieties are up to isogeny semisimple, this implies that A/K is simple. However,
this does not imply that the p-divisible group A
∣∣
U [p
∞] is simple. In particular, it can
have multiple slopes.
We can now state the main application to abelian varieties over function fields of characteristic
p that we have in mind.
Bruno Chiarellotto and Fabien Trihan 869
Theorem 4. Let A and B be abelian varieties over a function field K of characteristic p. Suppose
that:
(1) End(A)⊗Qp and End(B)⊗Qp are (not necessarily identical) division algebras;
(2) the maximal quotients of minimal slope (i.e., the unique maximal isoclinic quotients
corresponding to the minimal slope, defined up to isogeny) of A[p∞] and B [p∞] are
isogenous as p-divisible groups.
Then A and B are isogenous over K .
Proof. Let A and B be the Néron models of A and B over a smooth projective curve C with
function field K . Choose a nonempty open subset U ⊂ C over which both A and B have good
reduction and such that the associated p-divisible groups A
∣∣
U [p
∞] and B
∣∣
U [p
∞] have constant
Newton polygons. In this setting, the canonical slope filtration exists on the p-divisible groups,
and we denote by Q A and QB the maximal quotients of minimal slope of A[p∞] and B [p∞],
respectively. By the functoriality of the slope filtration and the fact that it is defined by a universal
property, these quotients extend canonically to the p-divisible groups associated to the Néron
models over U ; denote these extensions by QUA and Q
U
B . By hypothesis, there exists an isogeny
φη : Q A −→QB ,
defined on the generic fiber (over the field K ). Then, by applying de Jong’s local result [4,
Theorem 1.1] to each Spec(Ov ) (v a closed point in U ), one obtains local extensions of the iso-
geny φηv to isogenies φv . These φv are compatible at the generic point Spec(K ) ⊂ Spec(Ov ), as
they all restrict toφη. By normality of U and the fact that an isogeny is determined by its behavior
on codimension-1 points (see the proof of [10, Proposition 4]), these local pieces {φv } glue to a
single isogeny
φ : QUA −→QUB
over U .
The slope filtration at the level of p-divisible groups over U induces a slope filtration on both
D†(A) and D†(B) as convergent F -isocrystals with therefore same minimal slope. Moreover, by
our first hypothesis and Lemma 2, D†(A) and D†(B) are irreducible so that by the minimal slope
conjecture (see [12, Theorem 1.3] and its final form [3, Corollary 1.1.4]), this implies that the
overconvergent F -isocrystals themselves are isomorphic:
D†(A) ∼= D†(B).
We could again use [9, Theorem 1.1] to deduce that A and B are isogenous, but let us consider
another proof that can apply to p-divisible groups in the next corollary.
Since the functor from F -crystals up to isogeny to convergent F -isocrystals, and from con-
vergent F -isocrystals to overconvergent F -isocrystals, is fully faithful (see [2, 2.4] and [6, Theo-
rem 1.1]), we deduce that the Dieudonné modules up to isogeny associated with A and B are
isomorphic.
By the fully faithfulness of the Dieudonné functor (established by [5]), we conclude that the
p-divisible groups A [p∞]
∣∣
U and B[p
∞]
∣∣
U are isomorphic up to isogeny. Consequently, their
generic fibers A[p∞] and B [p∞] over K are isogenous.
Finally, by the fully faithfulness of the functor from abelian varieties over K to their p-divisible
groups (proved by de Jong in [4, Theorem 2.6]), we deduce that A and B are isogenous over K . □
Remark 5. We note that, as the referee pointed out, our Theorem 4 can be extended to higher
transcendence degree fields of characteristic p using Tsuzuki’s general form of the minimal slope
conjecture in higher dimensions (see [12, Theorem 1.4]). Indeed, the proof of Theorem 4 carries
through: the overconvergent F -isocrystals D†(A) and D†(B) are semisimple, as noted in the
proof of [3, Theorem 5.2.2] (and hence irreducible by Lemma 2 under our hypotheses), and
870 Bruno Chiarellotto and Fabien Trihan
the spreading of the isogeny can be established by reducing the question to [4, Corollary 1.2],
following the same reasoning as in [10, Proposition 4].
Corollary 6. Let G and H be two p-divisible groups over X , a smooth variety over a finite field.
Assume that both have constant Newton polygon on X , and that their associated Dieudonné
crystals are overconvergent and irreducible. Then if G and H have their slope filtration with same
maximal slope quotients (or minimal slope quotients) isomorphic. Then G and H are isogenous.
Proof. Under our hypothesis D†(G) ≃ D†(H), implying as in the proof of the previous theorem
G ≃ H up to isogeny. □
Remark 7. Assume that X is a smooth, geometrically connected curve, and let G be a p-
divisible group over X that is potentially semistable in the sense of [11, Theorem 4.5], so that
its associated Dieudonné crystal D(G)† is overconvergent. Suppose further that G is principally
polarized in such a way that the associated Rosati involution is positive, which in turn implies
that the Frobenius linear endomorphism on the Dieudonné module D(Gx ) is self-adjoint and
hence semisimple for a closed point x of X . In addition, assume that D(G)† is ι-pure. Then, by [9,
Proposition 7.9], D(G)† is semisimple in F -Isoc†. Finally, if we further assume that
End(G)⊗Qp
is a division algebra, it follows that
EndF -Isoc† (D(G)
†) = End(G)⊗Qp
is also a division algebra, and since D(G)† is semisimple, it is in fact simple. Note that the
condition on End(G)⊗Qp does not imply that G itself is simple.
As a concluding remark, we hope that this result will contribute to a better understanding
of moduli spaces of abelian varieties over function fields or more generally of finitely generated
fields of characteristic p. It might also be interesting to compare our result with the notion of
minimal p-divisible groups introduced by Oort [7]. Finally, we mention another application of
the “minimal slope conjecture” by Ambrosi–D’Addezio [1] in deducing finiteness results on the
torsion of abelian varieties.
Acknowledgments
We would like to thank the referee for his careful reading as well as for suggesting a better
formulation (even an improvement) of our main result. We also would like to thank O. Brinon,
W. Gajda, K. Kedlaya, I. Longhi, A. Pal, T. Suzuki, and K. S. Tan for enlightening conversations.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] E. Ambrosi and M. D’Addezio, “Maximal tori of monodromy groups of F -isocrystals and an
application to abelian varieties”, Algebr. Geom. 9 (2022), no. 5, pp. 633–650.
[2] P. Berthelot, “Cohomologie rigide et cohomologie rigide à supports propres”, 1996. Online
at https : / / wstein . org / people / berthelo / publis / Cohomologie _ Rigide _ I . pdf. Preprint
IRMAR 96-03 (1996).
Bruno Chiarellotto and Fabien Trihan 871
[3] M. D’Addezio, “Parabolicity conjecture of F -isocrystals”, Ann. Math. (2) 198 (2023), no. 2,
pp. 619–656.
[4] A. J. de Jong, “Homomorphisms of Barsotti–Tate groups and crystals in positive character-
istic”, Invent. Math. 134 (1998), no. 2, pp. 301–333.
[5] A. J. de Jong and W. Messing, “Crystalline Dieudonné theory over excellent schemes”, Bull.
Soc. Math. Fr. 127 (1999), no. 2, pp. 333–348.
[6] K. S. Kedlaya, “Full faithfulness for overconvergent F -isocrystals”, in Geometric aspects of
Dwork theory. Vol. I, II , Walter de Gruyter, 2004, pp. 819–835.
[7] F. Oort, “Minimal p-divisible groups”, Ann. Math. (2) 161 (2005), no. 2, pp. 1021–1036.
[8] F. Oort and T. Zink, “Families of p-divisible groups with constant Newton polygon”, Doc.
Math. 7 (2002), pp. 183–201.
[9] A. Pál, “The p-adic monodromy group of abelian varieties over global function fields of
characteristic p”, Doc. Math. 27 (2022), pp. 1509–1579.
[10] J. T. Tate, “p-divisible groups”, in Proc. Conf. Local Fields (Driebergen, 1966), Springer, 1967,
pp. 158–183.
[11] F. Trihan, “A note on semistable Barsotti–Tate groups”, J. Math. Sci., Tokyo 15 (2008), no. 3,
pp. 411–425.
[12] N. Tsuzuki, “Minimal slope conjecture of F -isocrystals”, Invent. Math. 231 (2023), no. 1,
pp. 39–109.


Isogeny and overconvergence

José Matias

Pedro Miguel Santos

Elvira Zappale

Lower semicontinuity of nonlocal $L^\infty $ energies on $SBV_0(I)$

Théophile Chaumont-Frelet

Alexandre Ern

Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell’s equations

Faouzi Triki

Mirza Karamehmedović

On the series solutions of integral equations in scattering

Not a member yet