Comptes Rendus Mathématique

Tatsuki Kuwagaki

Vivek Shende

Adjoints, wrapping, and morphisms at infinity

Masaharu Ishikawa

Tat-Thang Nguyen

Atypical values at infinity of real polynomial maps with $2$-dimensional fibers

Ricardo García López

Claude Sabbah

Comptes RendusMathématiqueRicardo García López and Claude SabbahCorrigendum to “Hodge–Lyubeznik numbers”Volume 363 (2025), p. 843-844Online since: 15 July 2025https://doi.org/10.5802/crmath.773This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 843-844https://doi.org/10.5802/crmath.773Erratum / ErratumAlgebraic geometry, Singularity theory / Géométrie algébrique, Théorie dessingularitésCorrigendum to “Hodge–Lyubeznik numbers”Corrigendum à « Nombres de Hodge–Lyubeznik »Ricardo García López a and Claude Sabbah ba Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Vía 585,08007 Barcelona, Spainb CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau cedex, FranceE-mails: ricardogarcia@ub.edu, claude.sabbah@polytechnique.eduAbstract. A Tate twist is missing in the exact sequence of Example (1), leading to incorrect formulas for someHodge–Lyubeznik numbers, as noticed by Bradley Dirks. We provide the correct formulas.Résumé. Un twist de Tate est manquant dans la suite exacte de l’Exemple (1), induisant des formulesincorrectes pour certains nombres de Hodge–Lyubeznik, comme remarqué par Bradley Dirks. Nous donnonsles formules correctes.Keywords. Hodge numbers, Lyubeznik numbers, mixed Hodge modules.Mots-clés. Nombres de Hodge, nombres de Lyubeznik, modules de Hodge mixtes.2020 Mathematics Subject Classification. 13D45, 14B15, 32S35.Funding. The first author received partial support from grant PID2022-137283NB-C22 funded byMCIN/AEI/10.13039/501100011033.Manuscript received 4 June 2025, accepted 21 June 2025.The exact sequence of [1, Example (1)] should readH s−3(M) L−→ H s−1(M)(1) −→ H s{0}(X )(1) −→ H s−2(M)L−→ H s (M)(1).The Tate twist in the middle term was mistakenly omitted, leading to incorrect formulas forthe Hodge–Lyubeznik numbers computed by means of it. The consequences of this error werenoticed by Bradley Dirks. With the exact same reasoning, the formulas for the Hodge–Lyubezniknumbers are:• In Example (1):(1) λp,q0,s (OX ,0) = h−p,−q(H s−1(M)prim)for 2 ≤ s ≤ d −1;(2) λp,qr,d (OX ,0) = hp+d−1,q+d−1(H d+r−2(M)coprim)for 2 ≤ r ≤ d ;(3) all other λp,qr,s (OX ,0) vanish.• In Example (2):λ−p,−q0,4 (Ai ) = hp,q(H 3(Mi )prim)= hp,q (H 3(Mi ))−hp−1,q−1(H 1(Mi )).In particular, if N1 = X3(70,16,16,14,7,6) we obtain, according to [2],λ−1,−20,4 (A1) = h1,2(H 3(N1))= 3365330286081,ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/844 Ricardo García López and Claude Sabbahwhile if N2 = X3(56,49,8,6,5,4,4),λ−1,−20,4 (A2) = h1,2(H 3(N2))= 3343868254721.• In Remark (1) on page 220,hp,qK (−n) = hp,q (H d{0}(X ))= hp,q (H d−1(M)prim),and K (−n) is pure of weight d − 1. It follows that Q(n) is pure of weight 1 − d andhp,qQ(n) = h−p,−q (H d−1(M)prim).We take this opportunity to correct a typo: on page 218, two lines after (6), one should read “itsrestriction to U isQHU [d ](n)”.AcknowledgmentsWe thank Bradley Dirks for noticing erroneous formulas and the typo.Declaration of interestsThe authors do not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and have declared no affiliations other than their researchorganizations.References[1] R. García López and C. Sabbah, “Hodge–Lyubeznik numbers”, C. R. Math. 363 (2025),pp. 213–221.[2] J. Wang, Z. Yu and Y. Wang, “Examples of diffeomorphic complete intersections with differ-ent Hodge numbers”, Topology Appl. 339 (2023), article no. 108690 (6 pages).

Corrigendum to “Hodge–Lyubeznik numbers”

Aleksey Zinger

The (co)homology of the Deligne–Mumford moduli spaces of marked rational curves

Guillaume Rialland

Multi-solitary waves for the one-dimensional Zakharov system

Shalom Eliahou

Jean-Louis Verger-Gaugry

Comptes RendusMathématiqueShalom Eliahou and Jean-Louis Verger-GaugryThe number system in rational base 3/2 and the 3x +1 problemVolume 363 (2025), p. 329-336Online since: 20 May 2025https://doi.org/10.5802/crmath.662This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 329-336https://doi.org/10.5802/crmath.662Research article / Article de rechercheCombinatorics, Number theory / Combinatoire, Théorie des nombresThe number system in rational base 3/2 andthe 3x +1 problemLe système de numération en base rationnelle 3/2 et leproblème 3x +1Shalom Eliahou ,a,b and Jean-Louis Verger-Gaugry ,ca Univ. Littoral Côte d’Opale, UR 2597 – LMPA – Laboratoire de Mathématiques Pures et AppliquéesJoseph Liouville, F-62100 Calais, Franceb CNRS FR 2037, Francec Université-Grenoble-Alpes, Université Savoie Mont Blanc, F-73000 Chambéry, FranceE-mails: eliahou@univ-littoral.fr, verger-gaugry.jean-louis@unsechat-math.frAbstract. The representation of numbers in rational base p/q was introduced in 2008 by Akiyama, Frougny& Sakarovitch, with a special focus on the case p/q = 3/2. Unnoticed since then, natural questions related torepresentations in that specific base turn out to intimately involve the Collatz 3x +1 function. Our purposein this note is to expose these links and motivate further research into them.Résumé. La représentation des nombres dans la base rationnelle p/q a été introduite en 2008 par Akiyama,Frougny & Sakarovitch, avec un accent particulier sur le cas p/q = 3/2. Des questions naturelles liées auxreprésentations dans cette base spécifique, passées inaperçues jusqu’ici, s’avèrent impliquer intimement lafonction de Collatz 3x + 1. Le but de cette note est d’exposer ces liens et de motiver des recherches plusapprofondies sur ceux-ci.Keywords. Collatz conjecture, iteration, numeration, odometer.Mots-clés. Conjecture de Collatz, itération, numération, odomètre.2020 Mathematics Subject Classification. 11B83, 68R15.Manuscript received 24 April 2024, revised 22 May 2024 and 21 June 2024, accepted 29 June 2024.1. IntroductionThe representation of numbers in rational base p/q with p > q coprime positive integers wasintroduced in 2008 in [2], with a focus on the case p/q = 3/2 to study Mahler’s Z -number problemand the Josephus problem.Given n ∈N, its representation in rational base 3/2 is the unique expression of the formn =k∑i=0ai2(32)iISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/330 Shalom Eliahou and Jean-Louis Verger-Gaugrywith digits ai ∈ {0,1,2} for all i ≥ 0 and ak ̸= 0 if n ≥ 1. We then denote the corresponding word in{0,1,2}∗ by〈n〉 = ak ak−1 · · ·a0.We set 〈0〉 = ϵ, the empty word, and we identify any word 0w ∈ {0,1,2}∗ with w .Denoting L = {〈n〉 | n ∈N} ⊂ {0,1,2}∗, here are two natural questions about L :(1) If w ∈L , which extensions among w0, w1, w2 still belong to L ?(2) Given k ≥ 1, what is the lexicographically largest word of length k in L ? Equivalently,what is the largest n ∈N such that 〈n〉 is of length k?Apparently unnoticed so far, answers to these questions turn out to intimately involve the Collatz3x +1 function, namely T : N→N defined byT (n) ={n/2 if n ≡ 0 mod 2,(3n +1)/2 if n ≡ 1 mod 2. (1)The purpose of this note is to expose these links and motivate further research into them. Itscontents are as follows. In Section 2, we recall basic properties of the numeration system inrational base 3/2, in particular the recursive construction of the word 〈n〉 in Section 2.1 and theodometer allowing to construct 〈n + 1〉 from 〈n〉 in Section 2.2. We also propose a variant ofthe Collatz conjecture in Section 2.3. In Sections 3.1 and 3.2, we address the above questions (1)and (2), respectively. Section 4 concludes the note with two questions and two conjectures closelyrelated to the Collatz conjecture and possibly of the same order of difficulty.2. BasicsFollowing [2], every n ∈N has a unique expression of the formn = 12(ak (3/2)k +ak−1(3/2)k−1 +·· ·+a0)with ai ∈ {0,1,2} for all i and ak ̸= 0 if n ≥ 1. Accordingly, we associate to n the word〈n〉 = ak ak−1 · · ·a0 ∈ {0,1,2}∗,called the representation of n in rational base 3/2. We set 〈0〉 = ϵ, the empty word, and we identifyany word 0w ∈ {0,1,2}∗ with w .By [5, Lemma 4.1], the length of the word 〈n〉 is given by |〈n〉| = log3/2(n)+O(1).Definition 1. We say that a word w ∈ {0,1,2}∗ is admissible if w = 〈n〉 for some n ∈N. We denoteby L ⊂ {0,1,2}∗ the language of admissible words, i.e.L = {〈n〉 | n ∈N}.As observed in [2], the language L is prefix-closed, i.e. stable under taking prefixes. That is,every prefix w ′ of an admissible word w = w ′w ′′ is itself admissible.Conversely, given an admissible word w ∈ L , which extensions among w0, w1, w2 remainadmissible? The answer involves the Collatz 3x +1 function. See Proposition 7.2.1. Construction of 〈n〉Given n ∈ N, a recursive construction of 〈n〉 ∈ {0,1,2}∗ may be described as follows. Set 〈0〉 = ϵ,the empty word.Shalom Eliahou and Jean-Louis Verger-Gaugry 331Proposition 2. Let n0 ∈N\ {0}. Denote 〈n0〉 = ak · · ·a0 with ai ∈ {0,1,2} for all i . Then:{a0 ≡ 2n0 mod 3;〈n0〉 = 〈n1〉a0, where n1 = (2n0 −a0)/3.Moreover, we have a0 = 2n0 −3n1 and a0 ≡ n1 mod 2.Proof. As 〈n0〉 = ak · · ·a0, we haven0 = 12(ak (3/2)k +·· ·+a1(3/2)+a0)=(12(ak (3/2)k−1 +·· ·+a1))32+ a02.Hence a0 ≡ 2n0 mod 3, and setting n1 = (2n0 − a0)/3, we have 〈n1〉 = ak · · ·a1. The claimsfollow. □2.2. The odometerThe odometer constructs the word 〈n+1〉 from the word 〈n〉. It is described as a finite automatonin [2]. Here is an equivalent description.Let σ = (2 1 0) be the cyclic permutation 2 7→ 1 7→ 0 7→ 2. For a word u ∈ {0,1,2}∗, let uσ ∈{0,1,2}∗ denote the word obtained by applyingσ to each letter of u. For instance, 12012σ = 01201,then identified to 1201.Proposition 3. Let n ∈N. If 〈n〉 = u0v ∈ {0,1,2}∗ with v ∈ {1,2}∗, then 〈n +1〉 = u2vσ.Proof. See [2, Section 3.2.3, pp. 66–67]. □The rule applies even if u or v is the empty word. In particular, if 〈n〉 = v ∈ {1,2}∗, thenidentifying v with 0v , we get 〈n + 1〉 = 2vσ. Table 1, listing 〈n〉 for 1 ≤ n ≤ 27, was constructedby hand using this description of the odometer. The interested reader will effortlessly extend itmuch farther.Table 1. The words 〈n〉 for 1 ≤ n ≤ 27.n 〈n〉1 22 213 2104 2125 21016 21207 21228 210119 21200n 〈n〉10 2120211 2122112 21011013 21011214 21200115 21202016 21202217 21221118 2101100n 〈n〉19 210110220 210112121 212001022 212001223 212020124 212022025 212022226 212211127 210110002.3. The Collatz conjectureDating back to the 1930’s, this conjecture deals with iterates T (k) of the Collatz function T : N→Ndefined in (1).Conjecture 4. Let n0 ∈N, n0 ≥ 1. Then there exists k ≥ 1 such that T (k)(n0) = 1.332 Shalom Eliahou and Jean-Louis Verger-GaugryHere is an equivalent version, as shown below.Conjecture 5. Let n0 ∈N, n0 ≥ 1. Then the proportion of even numbers in the sequence of iteratesT (k)(n0) tends to 1/2 as k tends to infinity.Proposition 6. Conjectures 4 and 5 are equivalent.Proof. If Conjecture 4 holds, then at some point the sequence of iterates T (k)(n0) collapses to1,2,1,2,1,2, . . . , whence Conjecture 5.Conversely, assume Conjecture 5 holds. Given an integer n ≥ 1, letΩ(n) = (n,T (n),T (2)(n), . . .)denote the trajectory of n under iteration of T . Moreover, given k ≥ 1, letΩk (n) =(n,T (n),T (2)(n), . . . ,T (k−1)(n)),and let k1 denote the number of odd integers in Ωk (n). Assume for a contradiction that Conjec-ture 4 fails. Then there is an integer m ≥ 10 such that minΩ(m) = m. (Note that minΩ(n) = 1 forn ≤ 9.) We claim that for all k ≥ 1, we haveT (k)(m) ≤ (3.1)k12km. (2)The proof is straightforward, by induction on k. Indeed, (2) holds for k = 1, since m is oddby minimality in Ω(m) and T (m) = (3m + 1)/2 = (3 + 1/m)m/2 ≤ (3.1)m/2 since m ≥ 10. Letnow k ≥ 1 and assume that T (k)(m) ≤ (3.1)k1 /2k m. Denote n = T (k)(m). If n is even, thenT (k+1)(m) = T (n) = n/2 ≤ (3.1)k1 /2k+1m, as desired. If n is odd, thenT (k+1)(m) = T (n) = (3+1/n)n/2 ≤ (3+1/m)n/2 ≤ (3.1)n/2since n ≥ m ≥ 10. Hence T (k+1)(m) ≤ (3.1)k1+1/2k+1m, as desired. The claim is proved.Now (3.1)k1 /2k ≥ 1 if and only if k1/k ≥ ln(2)/ln(3.1), and ln(2)/ln(3.1) > 0.6. By Conjecture 5,for k large enough, the proportion k1/k will fall below 0.6. Hence T (k)(m) ≤ (3.1)k1 /2k m < m,contradicting the minimality of m. This completes the proof of the proposition. □At the time of writing, Conjecture 4 has been verified up to n0 ≤ 268. See [3]. With this bound,the results of [4] and its Table 2 on page 54 imply that, besides the trivial cycle {1,2} under T , noother cycle, if any, can have length less than 114,208,327,604. See [1] for a general reference onthe conjecture, and [6] for recent strong results about it.3. Links with the Collatz functionWe now address the two questions about L = {〈n〉 | n ∈ N} asked in the Introduction, and showthat in both cases, the answer naturally involves the Collatz function T .3.1. Extending admissible wordsHere is the first question. Given w ∈L , which extensions among w0, w1, w2 still belong to L ?Proposition 7. Let w ∈L , say w = 〈n〉 for some n ∈N. Then:{w1 ∈L ⇐⇒ n ≡ 1 mod 2, and then w1 = 〈(3n +1)/2〉 = 〈T (n)〉;w0 ∈L ⇐⇒ n ≡ 0 mod 2 ⇐⇒ w2 ∈L , and then w0 = 〈3n/2〉 and w2 = 〈(3n +2)/2〉.Proof. If 〈n〉1 ∈L , Proposition 2 implies 1 ≡ n mod 2 and w1 = 〈(3n+1)/2〉. Similarly, if 〈n〉0 ∈Lthen 0 ≡ n mod 2 and w0 = 〈3n/2〉. Finally, if 〈n〉2 ∈ L then 2 ≡ n mod 2 and w2 = 〈(3n +2)/2〉.The stated equivalences follow. □Corollary 8. Let n ∈N be odd. Then 〈T (n)〉 = 〈n〉1.Shalom Eliahou and Jean-Louis Verger-Gaugry 333Proof. Immediate from the above proposition. □Remark 9. The map n 7→ n/2 for n even is straightforward in the base 2 representation of n:just remove the last digit, 0. Symmetrically, Corollary 8 shows that the map n 7→ (3n + 1)/2 forn odd is straightforward in the base 3/2 representation of n: just add 1 as rightmost digit. Byany chance, could it happen that considering the map n 7→ T (n) simultaneously in base 2 and inrational base 3/2 may lead to new insights on the Collatz conjecture?3.2. Saturated wordsGiven a number system B over a finite set of digits, we say that a positive integer n ≥ 1 is saturatedif its representation 〈n〉B in base B is of length strictly less than that of 〈m〉B for all m ≥ n +1.Accordingly, we say that 〈n〉B is a saturated word. For instance in base 10, the saturated integersand corresponding words are the n = 10ℓ−1 = 9 · · ·9.The digits 1,2 of the saturated words in rational base 3/2 turn out to be unpredictable in somesense and closely related to the 3x + 1 problem. Recall that |w | denotes the length of the finiteword w .Notation 10. Given k ≥ 1, we denote by sat(k) the largest positive integer whose representationin base 3/2 is of length k. That is,sat(k) = max{n ∈N, |〈n〉| = k}.Proposition 11. For all k ≥ 1, 〈sat(k)〉 is the unique admissible word of length k with digits in{1,2}.Proof. By the odometer as described in Proposition 3, if 〈n〉 is of length k and contains a 0 digit,then 〈n+1〉 is of the same length k. This shows that the inequality |〈n+1〉| > |〈n〉| happens exactlywhen 〈n〉 contains no 0 digit except possibly as an irrelevant prefix. □For instance, a look at 〈n〉 for 1 ≤ n ≤ 27 in Table 1 yields sat(k) and the corresponding wordfor k ≤ 7 in Table 2. Observe in Table 2 that 〈sat(k −1)〉 is a prefix of 〈sat(k)〉. Indeed, this is truefor all k ≥ 1.Proposition 12. For all k ≥ 2, we have 〈sat(k)〉 = 〈sat(k − 1)〉a0 where a0 ∈ {1,2} and a0 ≡sat(k −1) mod 2.Proof. Since L is prefix-closed, it follows from Proposition 11 that the length k − 1 prefix of〈sat(k)〉 equals 〈sat(k − 1)〉. Thus 〈sat(k)〉 = 〈sat(k − 1)〉a0 where a0 ∈ {1,2}, and a0 ≡ sat(k −1) mod 2 by Proposition 2. □Table 2. The first seven saturated numbers and words.k sat(k) 〈sat(k)〉1 1 22 2 213 4 2124 7 21225 11 212216 17 2122117 26 2122111334 Shalom Eliahou and Jean-Louis Verger-Gaugry3.3. The function UIt turns out that the sequence(sat(k))k≥1 coincides with the sequence of iterates U(k)(0) of acertain function U closely related to the Collatz function T . See Proposition 16 below.Notation 13. Let U : N→N be defined asU (n) ={(3n +2)/2 = T (n)+n +1 if n ≡ 0 mod 2,(3n +1)/2 = T (n) if n ≡ 1 mod 2.Remark 14. We have U (n) ≡ T (n)+n +1 mod 2 for all n ∈N.Starting from any n0 ∈ N, let us consider the sequence of iterates U (k)(n0) for k ≥ 1. Thefollowing proposition shows that, for the sequence of corresponding words 〈U (k)(n0)〉 in rationalbase 3/2, each term is a prefix of the next one and the added digit is either 1 or 2.Proposition 15. Let n ∈N. Then:〈U (n)〉 ={〈n〉1 if n ≡ 1 mod 2,〈n〉2 if n ≡ 0 mod 2.Proof. Straightforward from Proposition 7. □A particularly interesting sequence of iterates U (k)(n0) is for n0 = 0.Proposition 16. For all k ≥ 1, we have sat(k) =U (k)(0).Proof. For k = 1, we have sat(1) = 2 and U (1) = T (1) = 2. For k ≥ 2, we only need showsat(k) =U (sat(k −1)). Indeed, it follows from Proposition 12 thatsat(k) = 3sat(k −1)+a02where a0 ∈ {1,2} and a0 ≡ sat(k − 1) mod 2. Hence sat(k) = U(sat(k − 1)) by definition of U , asdesired. □Consequently, the growth factor of the sequence sat(k) tends to 3/2.4. Open questionsWe conclude this note with two questions and two conjectures.Question 17. Is the Collatz conjecture equivalent to some behavior of the iteration of the func-tion U ? For instance, starting with any n0 ∈ N, is it true that the proportion of 1’s in 〈U (k)(n0)〉tends to 1/2 as k goes to infinity, as supported by numerical evidence? Note that by Proposition 15,the number of digits “0” remains constant along the trajectory of n0.More specifically, for n0 = 0, we have U (k)(0) = sat(k) for all k ≥ 1 by Proposition 16. Havingcomputed sat(k) up to k = 220 = 1,048,576, we find that the word w = 〈sat(220)〉, of length k = 220,counts exactly 525,067 digits “1” and 523,509 digits “2”. Moreover, for all 1 ≤ ℓ < 17, the word wcontains all 2ℓ words in {1,2}ℓ as subwords. See Table 3 for the word 〈sat(200)〉, with exactly 100digits “1”. This numerical evidence leads us to the following conjecture.Conjecture 18. The proportion of digits “1” in the word 〈sat(k)〉 tends to 1/2 as k tends to infinity.Shalom Eliahou and Jean-Louis Verger-Gaugry 335Is Conjecture 18 equivalent to the Collatz conjecture? The question arises from the closerelationship between the functions T and U , Remark 14 and the analogy with Conjecture 5. Atany rate, even if not equivalent to Conjecture 5, Conjecture 18 might well be of the same order ofdifficulty.In fact, the numerical evidence below Question 17 leads to an even stronger conjecture. Since〈sat(k −1)〉 is a prefix of 〈sat(k)〉 for all k ≥ 2, we may define the limit〈sat(∞)〉 = limk→∞〈sat(k)〉,an infinite word over {1,2}. Informally, recall that an infinite word w over a finite alphabet B ofcardinality |B| = b is said to be normal if for all ℓ ≥ 1, all bℓ words in Bℓ occur as subwords inw[n], the length n prefix of w , with frequency tending to 1/bℓ as n →∞. Our stronger conjectureruns as follows.Conjecture 19. The word 〈sat(∞)〉 is normal.Table 3. The word 〈sat(200)〉 = 〈U 200(0)〉.21221112212112212121122121121211111221211111112122222112212122121211211222212212222221211111221221212211211222211122111222122221211212122122212122212121211111111212122222111112211121212222122111112112Our second question is about fully understanding the Collatz map T in rational base 3/2.Question 20. For n odd, we have seen that 〈T (n)〉 = 〈n〉1. For n even, how does one determine〈T (n)〉 from 〈n〉? Is any answer susceptible to lead to new insights on the Collatz conjecture?Note that determining the parity of n ∈ N from the digits ai of the word 〈n〉 = ak · · ·a0 ∈ L ismore cumbersome than in base 2 or 10. Indeed, the following criterion is straightforward but notquite practical for calculations by hand.Proposition 21. Let n ∈N\ {0} and 〈n〉 = ak · · ·a0 ∈L . Thenn ≡ 0 mod 2 ⇐⇒k∑i=02k−i 3i ai ≡ 0 mod 2k+2.Proof. We have n = 12(∑ki=0(3/2)i ai) = 2−(k+1)(∑ki=0 2k−i 3i ai ). Hence n ≡ 0 mod 2 ⇐⇒ 2k+1n ≡0 mod 2k+2 ⇐⇒ ∑ki=0 2k−i 3i ai ≡ 0 mod 2k+2. □AcknowledgmentsWe thank the anonymous referee for their careful reading and constructive comments, in partic-ular for insisting to formally state Proposition 6 and carefully prove it, as opposed to just a remarkwith informal arguments as in the original version of this article.Declaration of interestsThe authors do not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and have declared no affiliations other than their researchorganizations.336 Shalom Eliahou and Jean-Louis Verger-GaugryReferences[1] J. C. Lagarias (ed.), The ultimate challenge: the 3x + 1 problem, American MathematicalSociety, 2010, pp. xiv+344.[2] S. Akiyama, C. Frougny and J. Sakarovitch, “Powers of rationals modulo 1 and rational basenumber systems”, Isr. J. Math. 168 (2008), pp. 53–91.[3] D. Barina, “Convergence verification of the Collatz problem”, J. Supercomput. 77 (2021),pp. 2681–2688.[4] S. Eliahou, “The 3x + 1 problem: new lower bounds on nontrivial cycle lengths”, DiscreteMath. 118 (1993), no. 1-3, pp. 45–56.[5] J. F. Morgenbesser, W. Steiner and J. M. Thuswaldner, “Patterns in rational base numbersystems”, J. Fourier Anal. Appl. 19 (2013), no. 2, pp. 225–250.[6] T. Tao, “Almost all orbits of the Collatz map attain almost bounded values”, Forum Math. Pi10 (2022), article no. E12 (56 pages).

The number system in rational base $3/2$ and the $3x+1$ problem

Didier Henrion

Jean-Bernard Lasserre

An infinite-dimensional Christoffel function and detection of abnormal trajectories

Bruno Chiarellotto

Fabien Trihan

Comptes RendusMathématiqueBruno Chiarellotto and Fabien TrihanIsogeny and overconvergenceVolume 363 (2025), p. 867-871Online since: 15 July 2025https://doi.org/10.5802/crmath.774This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 867-871https://doi.org/10.5802/crmath.774Research article / Article de rechercheAlgebraic geometry, Number theory / Géométrie algébrique, Théorie des nombresIsogeny and overconvergenceIsogénie et surconvergenceBruno Chiarellotto a and Fabien Trihan ba Dipartimento di Matematica “Tullio Levi-Civita” (DM), Univ. Padova, Via Trieste 63, 35121 Padova, Italyb Sophia University, Department of Information and Communication Sciences 7-1 Kioicho, Chiyoda-ku,Tokyo 102-8554, JapanE-mails: chiarbru@math.unipd.it, f-trihan-52m@sophia.ac.jpAbstract. In this paper, we apply Tsuzuki’s main theorem in [12] to establish a criterion for when twoabelian varieties over a function field K of characteristic p are isogenous. Specifically, assuming that theirendomorphism algebras tensored with Qp are division algebras, we prove that if the maximal quotients ofminimal slope (i.e., the unique maximal isoclinic quotient corresponding to the minimal slope, defined upto isogeny) of their associated p-divisible groups are isogenous, then the abelian varieties themselves areisogenous over K . We also extend this result to certain p-divisible groups, highlighting the deep connectionbetween 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 critèrepermettant de déterminer quand deux variétés abéliennes sur un corps de fonctions K de caractéristique psont isogènes. Plus précisément, en supposant que leurs algèbres d’endomorphismes tensorisées par Qpsont 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 leursgroupes 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 profondentre 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 theNational 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 varietiesover function fields of characteristic p > 0. Namely, abelian varieties A/K over function fieldsinduce, via their Néron models, abelian schemes on an open subset of a connected propersmooth curve C /Fq (with function field K ). By the theory of Oort–Zink (see [8]), the associatedp-divisible group of an abelian scheme admits a slope filtration up to isogeny—that is, while thefiltration itself is not unique, its isogeny class is uniquely determined by the Newton polygon. Ourmain result is that one can detect the isogeny class of the abelian variety by looking at only oneof its slopes. More precisely, under the assumption that the endomorphism algebras tensoredwithQp are division algebras, if the maximal quotient of minimal slope (i.e., the unique maximalisoclinic quotient corresponding to the minimal slope, defined up to isogeny) of the p-divisiblegroups 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 TrihanOur proof is using the “minimal slope conjecture” proved by [3,12] and the fully faithfulnessof various functors between p-divisible groups and overconvergent F -isocrystals. Finally, weextend our result to the case of p-divisible groups over varieties over finite fields and to highertranscendence degree fields of characteristic p.First, let us recall the “minimal slope conjecture” proved by [3,12]. We consider X a smoothconnected scheme over a finite field and M † an overconvergent F -isocrystal on X , and denotethe convergent F -isocrystal on X associated to M † by M . We say M admits a slope filtration ifthere exists an increasing filtration0 = M0 ⊂ M1 ⊂ ·· · ⊂ Mr−1 ⊂ Mr = Mof 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 slopefiltration exists if the Newton polygons of the Frobenius structure of a convergent F -isocrystal Mare constant, which is always true after a certain shrinking of X (see [12, 1.1]). Hence M alwaysadmits 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 withslope filtrations. After renumbering the slope filtration as follows:M = M0 ⊃ M1 ⊃ M2 ⊃ ·· · ⊃ Mr−1 ⊃ Mr = 0with the sequence of slopes s0 > s1 > ·· · > sr−1, suppose that there is a nontrivial morphismh : N /N1 → M/M1 between the minimal slope sub-objects (or, by duality between the maximalslope 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 itsNéron model over the projective smooth connected curve C with function field K . Let U be an opensubset of C where A/K has good reduction. Let D†(A) be the overconvergent F -isocrystal over U /Qpassociated to the abelian scheme A∣∣U . Then D†(A) is semisimple and it will be simple if and onlyif 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 ifand 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 propertiesof 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)⊗Qpis still a division algebra. If F := End(A)⊗Q is strictly larger than Q and a commutativefield, 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, sinceabelian 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 canhave multiple slopes.We can now state the main application to abelian varieties over function fields of characteristicp that we have in mind.Bruno Chiarellotto and Fabien Trihan 869Theorem 4. Let A and B be abelian varieties over a function field K of characteristic p. Supposethat:(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 quotientscorresponding to the minimal slope, defined up to isogeny) of A[p∞] and B [p∞] areisogenous 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 withfunction field K . Choose a nonempty open subset U ⊂ C over which both A and B have goodreduction and such that the associated p-divisible groups A∣∣U [p∞] and B∣∣U [p∞] have constantNewton 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 universalproperty, these quotients extend canonically to the p-divisible groups associated to the Néronmodels over U ; denote these extensions by QUA and QUB . By hypothesis, there exists an isogenyφη : 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 ), asthey all restrict toφη. By normality of U and the fact that an isogeny is determined by its behavioron codimension-1 points (see the proof of [10, Proposition 4]), these local pieces {φv } glue to asingle isogenyφ : QUA −→QUBover U .The slope filtration at the level of p-divisible groups over U induces a slope filtration on bothD†(A) and D†(B) as convergent F -isocrystals with therefore same minimal slope. Moreover, byour first hypothesis and Lemma 2, D†(A) and D†(B) are irreducible so that by the minimal slopeconjecture (see [12, Theorem 1.3] and its final form [3, Corollary 1.1.4]), this implies that theoverconvergent 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 consideranother 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 areisomorphic.By the fully faithfulness of the Dieudonné functor (established by [5]), we conclude that thep-divisible groups A [p∞]∣∣U and B[p∞]∣∣U are isomorphic up to isogeny. Consequently, theirgeneric 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-divisiblegroups (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 highertranscendence degree fields of characteristic p using Tsuzuki’s general form of the minimal slopeconjecture in higher dimensions (see [12, Theorem 1.4]). Indeed, the proof of Theorem 4 carriesthrough: the overconvergent F -isocrystals D†(A) and D†(B) are semisimple, as noted in theproof of [3, Theorem 5.2.2] (and hence irreducible by Lemma 2 under our hypotheses), and870 Bruno Chiarellotto and Fabien Trihanthe 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 Dieudonnécrystals are overconvergent and irreducible. Then if G and H have their slope filtration with samemaximal 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 theoremG ≃ 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 thatits associated Dieudonné crystal D(G)† is overconvergent. Suppose further that G is principallypolarized in such a way that the associated Rosati involution is positive, which in turn impliesthat the Frobenius linear endomorphism on the Dieudonné module D(Gx ) is self-adjoint andhence 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 thatEnd(G)⊗Qpis a division algebra, it follows thatEndF -Isoc† (D(G)†) = End(G)⊗Qpis also a division algebra, and since D(G)† is semisimple, it is in fact simple. Note that thecondition 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 understandingof moduli spaces of abelian varieties over function fields or more generally of finitely generatedfields of characteristic p. It might also be interesting to compare our result with the notion ofminimal p-divisible groups introduced by Oort [7]. Finally, we mention another application ofthe “minimal slope conjecture” by Ambrosi–D’Addezio [1] in deducing finiteness results on thetorsion of abelian varieties.AcknowledgmentsWe would like to thank the referee for his careful reading as well as for suggesting a betterformulation (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 interestsThe authors do not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and have declared no affiliations other than their researchorganizations.References[1] E. Ambrosi and M. D’Addezio, “Maximal tori of monodromy groups of F -isocrystals and anapplication to abelian varieties”, Algebr. Geom. 9 (2022), no. 5, pp. 633–650.[2] P. Berthelot, “Cohomologie rigide et cohomologie rigide à supports propres”, 1996. Onlineat https : / / wstein . org / people / berthelo / publis / Cohomologie _ Rigide _ I . pdf. PreprintIRMAR 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 ofDwork 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 ofcharacteristic 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.

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