Comptes Rendus Mathématique

Xavier Roulleau

Comptes RendusMathématiqueXavier RoulleauThe cuspidal cubic and line arrangements with only triple pointsVolume 363 (2025), p. 593-598Online since: 5 June 2025https://doi.org/10.5802/crmath.741This 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. 593-598https://doi.org/10.5802/crmath.741Research article / Article de rechercheAlgebraic geometry / Géométrie algébriqueThe cuspidal cubic and line arrangementswith only triple pointsLa cubique cuspidale et les arrangements de droites avecseulement des points triplesXavier Roulleau aa Université d’Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, FranceE-mail: xavier.roulleau@univ-angers.frAbstract. We describe a new infinite family of line arrangements in the projective plane with only triple pointsingularities, and we recover previously known examples.Résumé. Nous décrivons une nouvelle famille infinie d’arrangements de droites dans le plan projectif neprésentant que des singularités triples, et nous retrouvons des exemples déjà connus.2020 Mathematics Subject Classification. 05B35, 14N20, 52C30.Manuscript received 30 December 2024, revised 11 March 2025, accepted 14 March 2025.1. IntroductionLine arrangements, i.e. finite unions of lines in the projective plane, are studied in variousdomains: algebra, topology, combinatorics... A double point (respectively triple point) on aline arrangement is a point where exactly 2 (respectively 3) lines of the arrangement meet. Thecelebrated Sylvester–Gallai Theorem asserts that any line arrangement over the real field has atleast one double point, or is the trivial example: the union of lines passing through the samepoint.Hirzebruch [2] then proved that over the complex field, a line arrangement is either the trivialexample or possesses double or triple points. To date, there exists a unique non-trivial exampleof a complex line arrangement with only triple points. Even before that result, the constructionof line arrangements with many or exclusively triple points had attracted attention, see e.g. [3] fora historical account.In this paper, we present a new construction of line arrangements over finite fields thatexhibit only triple points. Line arrangements with only triple points are also interesting froma combinatorial perspective, as they are linked to Steiner triple systems. A Steiner triple systemis a set T of 3-element subsets of a finite set, such that any 2-element subset is contained in aunique 3-element subset t ∈ T .Before presenting our results, let us introduce some notations and recall some properties ofline arrangements. If L = ℓ1 +·· ·+ℓn is a line arrangement labeled by {1, . . . ,n}, the data M(L )ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/594 Xavier Roulleauof the triples {i , j ,k} ⊂ {1, . . . ,n} such that the lines ℓi , ℓ j and ℓk meet at a common point is calledthe matroid associated with L ; it encodes the combinatorics of L . If L has only triple points,then M(L ) is a Steiner triple system.If γ is a projective transformation of the plane, one may define γL = γℓ1 + ·· · + γℓn . Theline arrangements L , γL have the same combinatorics, meaning that M(L ) = M(γL ). Given amatroid M , a labelled line arrangement L such that M(L ) = M is said to be a realization of M .Given a field K , there exists a scheme parametrizing the realizations over K , and the quotientR(M)/K of these realizations modulo the action of PGL3(K ) is called the moduli space of M . Aline arrangement over a field K is said to be rigid if R(M)/K is zero-dimensional.In the present paper, we describe a new infinite family of line arrangements in the plane withonly triple points, and study some of their associated moduli spaces:Theorem 1. For q = 3n , n ≥ 1, there exists a line arrangement Lq of q lines over Fq with q(q−1)6triple points and no other singularities.Let Mq = M(Lq ) be the matroid associated to Lq . For n ≥ 3, the matroid Mq has no realizationsover any field of characteristic ̸= 3.The moduli space R(M27)/F3 is 3-dimensional with a 2-dimensional immersed component ofmultiplicity 2.One obtains a model of the reduced scheme of R(M27)/F3 as an open sub-scheme of a quadricin P4.Our first example of a line arrangement Lq was obtained as the dual of the q points definedover Fq on the smooth part of the cuspidal cubic y2z = x3. However the generic realization of M27is such that the 27 points of its dual are not contained in any plane cubic.The previously known line arrangements with only triple points are:(a) The trivial example: three lines meeting at one point.(b) Over a field containing a primitive third root of unity (in particular over any algebraicallyclosed field of characteristic ̸= 3): the Ceva(3) line arrangement with 9 lines, given by(x3 − y3)(x3 − z3)(z3 − y3) = 0.It has 12 triple points and it is the dual of the Hesse configuration, i.e. of the 9 flex pointsof a smooth cubic. In characteristic 3, there is also a line arrangement of 9 lines havingthe same associated matroid as Ceva(3). This is the 9 lines in P2(F3) not containing agiven point p in P2(F3). These line arrangements are rigid.(c) The Fano plane: the 7 lines in P2(F2). It is also a rigid line arrangement.(d) The sequence of line arrangements C ′2n−1: for n ≥ 4, the line arrangement C ′2n−1 consistsof 2n −1 lines defined over F2. It is constructed in [3] and obtained by taking the dual of ageneric projection to P2 of the 2n −1 points Pn−1(F2) contained in Pn−1.(e) A rigid example over F11: a rigid line arrangement with 19 lines and 57 triple points,constructed in [3].It is an open question [5] whether or not (a) and (b) are the unique examples of complex linearrangements with only triple points.For any n ≥ 2, we also obtain line arrangements C2n−1 over F2 with 2n −1 lines and only triplepoints. These are constructed by using the dual of the non-zero points in A1(F2n ) in the smoothpart of the cuspidal cubic curve (which is isomorphic toA1).For n ≥ 4, we obtain that our line arrangement C2n−1 and the line arrangement C ′2n−1 ofexample (d) define isomorphic matroids. Thus C2n−1 is another construction of C ′2n−1.Our construction of Cn generalizes a bit the one of C ′n since it holds for n = 2 and 3. The linearrangement C3 is the trivial example, and C7 is the Fano plane (example (c)). Regarding theline arrangement Ln : for n = 3, it is the trivial example, and L9 corresponds to example (b). WeXavier Roulleau 595checked that the 19 points in the dual of example (e) are not contained in a cubic curve. Thisexample is therefore the only one known that does not appear to be related to cubic curves.2. Line arrangements using the cuspidal cubic2.1. The cuspidal cubicLet C ,→P2 be the cuspidal cubic over a field K , with a flex point O defined over K . Let C∗ be thecomplement of the singular point. As for a smooth cubic curve, one can define a compositionlaw (denoted by +) on C∗ by chords and tangents, so that O is the neutral element.Theorem 2 ([4, Chapter III, Proposition 2.5]). The composition law + makes C∗(K ) an abeliangroup isomorphic to (K ,+). Three points a,b,c ∈ C∗(K ) (identified with elements of K ) are on aline if and only ifa +b + c = 0 in K .In particular the tangent to a point a in C∗ cuts C∗ at a residual point b such that 2a+b = 0. Asa concrete example, one may consider the curve C∗ :{x3 − z y2 = 0} ,→P2 with flex point (0 : 1 : 0)and the parametrization t → (t : 1 : t 3).Now, suppose that K is a finite field of order #K = q = pn , where p is the characteristic of K .From Theorem 2, the lines containing three distinct points of K correspond bijectively to the setsof the formSa,b ={a,b,−(a +b)}that have cardinality 3. One has |Sa,b | = 3 if and only ifa ̸= b and a ̸= −(a +b) and b ̸= −(a +b)which is equivalent to conditiona ̸= b and b ̸= −2a and a ̸= −2b. (1)2.2. Characteristic 32.2.1. General constructionSuppose that Char(K ) = 3. Let a,b ∈ K and Sa,b ={a,b,−(a + b)}. Since −2 = 1 in K , theconditions in (1) for #Sa,b = 3 reduce to a ̸= b. Therefore the number of lines containing 3points is 13(q2) = q(q−1)6 , and through each point in K , there are 12 (q − 1) such lines. This leadsto a configuration (q 12 (q−1),(q(q−1)6)3)of points and lines. Let Lq denote the dual of the q points of K .Theorem 3. The line arrangement Lq of q lines hasq(q−1)6 triple points and no other singularities.Proof. The dual of the lines containing three points of K are triple points of the line arrangementLq . Thus Lq hasq(q−1)6 triple points. A line arrangement with q lines can have at most13q(q−1)2triple points, and if that bound is reached there are no other singularities. □We note that every point a on C∗ is a flex point: for any a one has 3a = 0. Geometrically, theintersection of the tangent line at a with the cubic curve is three times the point a.596 Xavier RoulleauRemark 4. As a group, or Fp -vector space, Fq is isomorphic to (Fp )n , where q = pn . Sincethe relation a +b + c = 0 is preserved under such an isomorphism, the group GLn(Fp ) inducessymmetries on the matroid Mq associated to Lq . For p = 3, since for any element t ∈ Fq , we have(a + t )+ (b + t )+ (c + t ) = a +b + c, the relation a +b + c = 0 is preserved by translation. Thus, theautomorphism group of M3n contains the general affine group of An/F3 , which is the semi-directproduct of (Z/3Z)n with GLn(F3).2.2.2. Examples in characteristic 3• The line arrangement L3 consists of 3 lines passing through the same point.• The line arrangement L9 consists of 9 lines with 12 double points. The matroid associatedto L9 is the same as the matroid associated to the Ceva(3) line arrangement, after appropriatelylabeling the lines.• Let M27 be the matroid associated with the line arrangement L27. Using [1] and OSCARsoftware, one computes that the moduli space R(M27) over F3 of M27 is 3-dimensional witha non-reduced immersed component Z of dimension 2 and multiplicity 2. A model for thismoduli space is an open sub-scheme of a rank 3 quadric inA4. The equations for this model, aswell as the associated 27 normal vectors of the line arrangements, are provided in the ancillaryfile of the arXiv version of this paper.Using the generic point of R(M27), one can check using MAGMA software that the 27 pointsof the dual of the generic realization of M27 are not contained on a cuspidal cubic. However,for the generic element of Z , (which is such that Zred is a sub-scheme of R(M27)), the 27 pointsdual to a realization of M27 lie on a cuspidal cubic curve.We now state the following result:Proposition 5. For any q = 3n with n ≥ 3, the matroid Mq has no realizations over any field ofcharacteristic ̸= 3.Proof. Using [1] and OSCAR software, one computes that the matroid M27 has no realizationsover any field of characteristic ̸= 3. The matroid Mq , q = 3n , can also be described as triples{a,b,c} in (F3)n such that a +b + c = 0. For n ≥ 3, the group (F3)3 is a sub-group of (F3)n , and arealization of Mq gives a realization of M27 by taking the labels indexed by that sub-group. Thisforces the realization to be over a field of characteristic 3. □2.3. Characteristic 22.3.1. General constructionSuppose that Char(K ) = 2 and K = Fq , where q = 2n for n > 1. Conditions in (1) for Sa,b ={a,b,−(a +b)} to be of order 3 are then a ̸= b and a ̸= 0, b ̸= 0. Geometrically, since for any a ∈ K ,one has 2a = 0, the tangent line trough a contains 0, and the point a with multiplicity 2, so itcontains no other points of the cuspidal cubic. This implies that for any pair a ̸= b in K ∗, the linepassing through a and b intersects the cubic at a third point c =−(a+b) ̸∈ {0, a,b}. Through eachpoint a ̸= 0 of K there are 12 (q−2) lines containing a and another point of K ∗. There are(q−1)(q−2)6such lines, and each line contains 3 points of K ∗. This results in a((q −1) 12 (q−2),((q−1)(q−2)6)3)configuration of points and lines. Let Cq−1 be the dual of the q −1 points in K ∗. We obtain:Theorem 6. The line arrangement Cq−1 of q − 1 lines has (q−1)(q−2)6 triple points and no othersingularities.Xavier Roulleau 5972.3.2. Examples in characteristic 2• The line arrangement C22−1 is 3 lines passing through the same point.• The line arrangement C23−1 is the Fano plane.• Let N2n−1 be the matroid associated to C2n−1; this is the set of triples {a,b, a +b} ⊂ F2n witha ̸= b, a ̸= 0, b ̸= 0. Using [1], one computes that the moduli space R(N15)/F2 is an opensubscheme in P3/F2, in particular it is 3-dimensional. We checked that the generic element C inR(N15) has the property that there is a cuspidal cubic containing the 15 points that are dual tothe 15 lines of C .For n > 3, in [3], are constructed some line arrangements C ′2n−1 (over the field F23n−1 ) withonly triple points and with the same number q −1 = 2n −1 of lines as C2n−1. The 2n −1 pointsdual to C ′2n−1 are the image of the 2n−1 points ofPn−1(F2) inPn−1 by a generic projection toP2.In fact, one can check that the line arrangements Cq−1 and C ′q−1 define isomorphic ma-troids as follows. As a group, the field K = Fq (for q = 2n) is the group (F2)n and the set (F2)n \{0}is contained inPn−1(F2). Two distinct points a, b of (F2)n \{0} generate a line inPn−1 which con-tains the points a, b, c = a +b (so that a +b +c = 0), and no other points of Pn−1(F2). As shownin [3], a generic projection f : Pn 99K P2 induces a bijection between the set Pn−1(F2) and itsimage, and f preserves the alignments. Specifically, for a,b,c ∈Pn−1(F2), one has a+b+c = 0 in(F2)n if and only if f (a), f (b) and f (c) are on a line inP2. Thus a bijective morphism η of groupsbetween K and (F2)n defines an isomorphism between the matroids M(Cq ) and M(C ′q ), i.e.sends the triples {a,b,c} of N = M(Cq ) bijectively to the triples of N ′ = M(C ′q ). If C ′q = (ℓa)a∈Fn2is a realization of N ′, then (ℓη(a))a∈K is a realization of N .The construction of C ′15 in [3] explains why the moduli space R(N15) is an open sub-schemeof P3/F2, since the line arrangement C ′15 is obtained by the projection to P2 from a generic pointin P3 of the 15 points of P3(F2) in P3.Remark 7. For n ≥ 3, the projections Pn 99K P2 are parametrized by the Grassmannian G(n −2,n + 1). For n > 4, it would be of interest to determine whether, as in the case n = 3, an opensub-scheme of the Grassmannian also corresponds to the moduli spaces of the matroid N2n−1.For n = 4, we have verified that the moduli space R(N31) is 6-dimensional and rational, as G(2,5).Remark 8. The line arrangements C2n−1 are defined over F2n . This addresses the question posedbefore Theorem 3.4 in [3] regarding the identification of a smaller field over which C ′q can bedefined.2.4. Characteristic ̸= 2,3Suppose that Char(K ) = p > 3. Then for b =−2a and b =− 12 a, Sa,b has order 2. Thus, the numberof lines containing 3 points and passing through a ∈C∗ is 12 (q −3). Furthermore, the number oflines containing 3 distinct points of K is16 q(q −3)and the number of lines containing exactly two points is q . This leads to a(q 12 (q−3),(q(q−3)6)3)configuration of points and lines. The dual is a line arrangement of q lines and q(q−3)6 triplepoints. It hasq(q−1)2 −3q(q−3)6 = qdouble points (corresponding to the q 2-rich lines in the dual space).598 Xavier RoulleauDeclaration of interestsThe author does not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and has declared no affiliations other than their researchorganizations.References[1] D. Corey, L. Kühne and B. Schröter, “Matroids in OSCAR”. To appear as a chapter in the bookThe Computer Algebra System OSCAR.[2] F. E. P. Hirzebruch, “Arrangements of lines and algebraic surfaces”, in Arithmetic and geom-etry, Vol. II , Progress in Mathematics, Birkhäuser, 1983, pp. 113–140.[3] L. Kühne, T. Szemberg and H. Tutaj-Gasińska, “Line arrangements with many triple points”,Rend. Circ. Mat. Palermo (2) 73 (2024), no. 7, pp. 2501–2512.[4] J. H. Silverman, The arithmetic of elliptic curves, Second edition, Graduate Texts in Mathe-matics, Springer, 2009, pp. xx+513.[5] G. Urzúa, “Some open questions about line arrangements in the projective plane”, Proyec-ciones 41 (2022), no. 2, pp. 517–536.

The cuspidal cubic and line arrangements with only triple points

Xinxin Chen

Yong Han

Yanqi Qiu

Zipeng Wang

The Mandelbrot–Kahane problem of Benoît Mandelbrot model of turbulence

Dario Corona

Alessandro Della Corte

Comptes RendusMathématiqueDario Corona and Alessandro Della CorteCorrigendum to “The critical exponent functions”Volume 363 (2025), p. 791-792Online since: 4 July 2025https://doi.org/10.5802/crmath.667This 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. 791-792https://doi.org/10.5802/crmath.667Erratum / ErratumDynamical systems / Systèmes dynamiquesCorrigendum to “The critical exponentfunctions”Corrigendum de « Les fonctions de l’exposant critique »Dario Corona a and Alessandro Della Corte aa School of Science and Technology, University of Camerino, ItalyE-mails: dario.corona@unicam.it, alessandro.dellacorte@unicam.itAbstract. We give a corrigendum to our paper [1] entitled “The critical exponent functions”.Résumé. Nous donnons un corrigendum à notre article [1] intitulé « Les fonctions de l’exposant critique ».Manuscript received 2 July 2024, accepted 9 July 2024.In [1], a family of interval maps based on the word-combinatorial concept of critical exponentwas introduced, hence referred to as “critical exponent functions”. A purely combinatorial result,namely [1, Theorem 7], was presented and employed in that paper to prove some analytical anddynamical properties of such interval maps. However, [1, Theorem 7] does not hold. Denotingby {0,1}∗ and {0,1}ω the sets of finite and infinite binary words, respectively, and using E(w)to represent the critical exponent of a word, [1, Theorem 7] was formulated as follows: “Letw ∈ {0,1}∗. For every α> max{2,E(w)}, there exists y ∈ {0,1}ω such that E(w y) =α”. This claim isfalse, as shown for instance by the counterexample w = 00100100. Indeed, in this case we haveE(w) = 8/3, while both w0 and w1 are 3-powers, so that E(w y) ≥ 3 > 8/3 for every y ∈ {0,1}ω.However, most of the properties obtained through that combinatorial result remain valid withminor adjustments. Indeed, after reviewing all the properties of κ2, we find that all of them hold,except for the following ones:• The property of being left- or right-Darboux at every point, as stated in [1, Proposi-tion 19]; it remains unclear if this property holds or not.• The characterization of the almost fixed points of κ2, specifically in [1, Corollary 24], andthe subsequent uncountability of them (cf. [1, Corollary 25]).• The existence of the fixed point can be granted only on any left-neighbourhood of xτ, notin a neighbourhood of any almost fixed point (cf. [1, Corollary 26]); however, this sufficesto prove the existence of countably many fixed points, so that [1, Corollary 27] holds.Analogously, most of the properties of κn hold. The main results that fail are:(1) being left- or right-Darboux at every point (cf. [1, Corollary 42]);(2) the characterization of the almost-fixed points.All the properties presented in [1] for κ hold.The interested reader can find more details in [2].ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/792 Dario Corona and Alessandro Della CorteAcknowledgmentsWe are grateful to Saúl Rodríguez Martín for pointing out the error.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] D. Corona and A. Della Corte, “The critical exponent functions”, C. R. Math. 360 (2022),pp. 315–332.[2] D. Corona, A. Della Corte and M. Farotti, “Dynamical properties of critical exponent func-tions”, 2024. Online at https://arxiv.org/abs/2406.14487.

Corrigendum to “The critical exponent functions”

Florent Noisette

Uniqueness for the Camassa–Holm equation with non-homogeneous boundary conditions

Antoine Mouzard

El Maati Ouhabaz

A simple construction of the Anderson operator via its quadratic form in dimensions two and three

Roberta Bianchini

Vincent Laheurte

Frank Sueur

A new look at the controllability cost of linear evolution systems with a long gaze at localized data

Claire Burrin

Comptes RendusMathématiqueClaire BurrinEquidistribution of continuous low-lying pairs of horocycles via Ratner’s theoremVolume 363 (2025), p. 1-5Online since: 6 March 2025https://doi.org/10.5802/crmath.693This 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. 1-5https://doi.org/10.5802/crmath.693Research article / Article de rechercheDynamical systems / Systèmes dynamiquesEquidistribution of continuous low-lying pairsof horocycles via Ratner’s theoremÉquidistribution de paires d’horocycles longs via lethéorème de RatnerClaire Burrin aa Institute of Mathematics, University of Zurich, SwitzerlandE-mail: claire.burrin@math.uzh.chAbstract. We record an alternative proof of a recent joint equidistribution result of Blomer and Michel, basedon Ratner’s topological rigidity theorem. This approach has the advantage of extending to non-uniformlattices.Résumé. Nous présentons une preuve alternative d’un résultat récent d’équidistribution jointe de Blomer etMichel, basée sur le théorème de rigidité topologique de Ratner. Cette approche a l’avantage de s’étendre auxréseaux non uniformes.Keywords. Joint equidistribution, measure classification, commensurator, Hecke operators, matrix coeffi-cients.Mots-clés. Équidistribution jointe, classification des mesures, commensurateur, opérateurs de Hecke, coef-ficients matriciels.2020 Mathematics Subject Classification. 37A10, 37A44, 22F30.Funding. The author acknowledges the support of the Swiss National Science Foundation (Grant No. 201557).Manuscript received 1 August 2024, revised 23 September 2024 and 28 September 2024, accepted 28 Septem-ber 2024.Expanding closed horocycles are known to become equidistributed on any non-compactfinite-area hyperbolic orbifold, such as, e.g., the low-lying horocycle{SL2(Z)(x+ iT)∣∣x ∈ [0,1]} onthe modular curve X = SL2(Z)\H as T →∞. Equidistribution also holds for various arithmeticallyinteresting growing discrete subsets of points along expanding closed horocycles (see, e.g.,[2,3,5,9,12]). Most recently Blomer and Michel [1] investigated the joint distribution properties ofsets of the shape {(SL2(Z)(a + iq),SL2(Z)(ab + iq))∣∣∣∣a (mod q)}⊂ X ×X (1)as q → ∞ while b ∈ (Z/qZ)∗ varies with q in a prescribed manner. In parallel, they consider a“continuous version” of this setup, which establishes the joint equidistribution of two low-lyinghorocycles of different speeds under a weak diophantine condition; see Theorem 1 below. ByDirichlet’s theorem, given any real numbers y and Q > 0, there is a rational number aq so that|y − aq | < 1qQ and q ≤Q. In the following statement, y is irrational and Q is picked to depend on agrowing parameter T so that as T →∞ we can find a sequence q →∞ for which y has a rationalapproximation of that order of magnitude.ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/2 Claire BurrinTheorem 1 (Blomer–Michel). Let X = SL2(Z)\H, T > 1, y ∈ [1,2] and write y = a/q +O(1/(qQ))for positive coprime integers a, q with q ≤Q := T 0.99. Let I ⊆R be a fixed non-empty interval. Then{(SL2(Z)(x + iT),SL2(Z)(x y + iT))∣∣∣∣x ∈ I}⊆ X ×Xequidistributes as T →∞ for pairs (y,T ) with q →∞.1The argument of proof is based on the estimation of Weyl sums via a shifted convolutionproblem, Sato–Tate and a sieving argument, and crucially the measure classification theoremof Einsiedler–Lindenstrauss [4, Theorem 1.4]. The purpose of this note is to record the followingversion of their result, using exclusively homogeneous dynamics. It would be very interesting tohave such an argument for the discrete equidistribution analogue of Theorem 1 concerning setsof the shape (1), which is the main result of [1] (see [1, Theorem 1.1]).Writeat =(pt 00pt −1)for t ∈R>0 and ux =(1 x0 1)for x ∈R.Theorem 2. Let G = SL2(R), let Γ be a non-uniform lattice in G, set L =G×G,Λ= Γ× Γ. Let T > 1,and let y > 0 be irrational if Γ is arithmetic or y ̸= 1 if Γ is non-arithmetic, and let I ⊆ R be a fixednon-empty interval. Then {(Γux a−1T , Γux y a−1T): x ∈ I}⊆Λ\Lequidistributes as T →∞.Proof. Let π be the composition of the smooth immersion ιy : G → L, ιy (g ) = (g , ay g a−1y ) and thecanonical projection L →Λ\L so that for any test function ϕ ∈Cb(Λ\L), we have∫ 10ϕ(π(ux a1/T ))dx =∫ 10ϕ(Γux a1/T , Γux y a1/T)dx,where we use the standard fact that the expansion (or contraction) of closed horocycles by thegeodesic flow is algebraically expressed by ay ux a−1y = ux y . A result of Shah [13, Theorem 1.4]asserts that if π(G) =Λ\L thenlimT→∞∫ 10ϕ(Γux a1/T , Γux y a1/T)dx =∫Λ\Lϕ.We may introduce a conjugation so that L = G ×G , Λ = Γ× a−1y Γay and π corresponds tothe diagonal embedding. Let ∆(G) denote the diagonal embedding of G in L. By Ratner’stopological rigidity theorem [11], we know that π(G) =π(H) for some closed connected subgroupH < L containing ∆(G), where H ∩Λ is a lattice in H . Since G is simple, ∆(G) is a maximalconnected subgroup in L and so the only options are that H = L (in which case π(G) =Λ\L andequidistribution follows) or that H = ∆(G). In the latter case, the requirement that Λ∩ H be alattice in H is equivalent to asking that Γy := Γ∩a−1y Γay be a lattice in G . This holds if and only ifay ∈ Comm(Γ), where Comm(Γ) is the commensurator of Γ in G , i.e.,Comm(Γ) = {α ∈ GL+2 (R) ∣∣ Γ∩α−1Γα has finite index in both Γ and α−1Γα} .A non-uniform lattice Γ < G is arithmetic if and only if it is commensurable to SL2(Z). SinceComm(SL2(Z)) = R× ·GL+2 (Q), we have ay ∈ Comm(Γ) if and only if y ∈Q. If Γ is non-arithmetic,we claim that ay ∈ Comm(Γ) if and only if y = 1.By Margulis’ arithmeticity criterium [6, Theorem 1.16, Chapter IX] Comm(Γ) contains Γ asa finite-index subgroup. In particular, Comm(Γ) is itself a lattice in G and by discreteness, itsmaximal unipotent subgroup is infinite cyclic. Let ut be its generator, i.e., ut ′ ̸∈ Comm(Γ) for all1In the statement of [1, Theorem 1.3], x is replaced by x/T . The argument of proof does however refer to the low-lyinghorocycles stated above; see [1, Theorem 4.2 and p. 55].Claire Burrin 3|t ′| < t . Let ay ∈ Comm(Γ); up to replacing ay by its inverse, we may assume that y > 1. Then thefact that a−1y ut ay = ut/y ∈ Comm(Γ) contradicts the minimality of t . □Remark 3. The statement extends to finitely many factors as follows. If y1, . . . , yn ∈R are chosensuch that ayi /y j ̸∈ Comm(Γ) for i ̸= j , then{(Γux y1 a−1T , . . . , Γux yn a−1T)∣∣x ∈ I}⊂ Γn\Gnequidistributes as T →∞. This can be compared to [7, Theorem 10.2] and [14, Theorem 4]. See [8]for a higher rank version, which also relies on commensurability.In the rest of this note, we come back to the case of X = SL2(Z)\H. By spectral expansion andWeyl’s equidistribution criterion, equidistribution in Theorem 1 holds if and only if∫ 10ϕ(x + iT, x y + iT)dx = oϕ(1)holds for all ϕ = f1 ⊗ f2, where f1 and f2 are non-constant Hecke eigenforms. We will show thatwhen y ∈Q (and some additional constraints on y) this fails, with a limit that can be expressed asa product of Hecke eigenvalues and matrix coefficients. To proceed we first recall some notationconnected to the construction of Hecke operators (see, e.g., [10]). For each elementα ∈ Comm(Γ)we haveΓαΓ=M⊔m=1Γαhmwhere the hm ’s are coset representatives Γ = ⊔Mm=1 Γαhm , with Γα = Γ∩α−1Γα. For everyfunction f on Γ\G we set ( f | ΓαΓ)(g ) = ∑Mm=1 f (αhm g ). Further, if α ∈ GL+2 (Q) has integerentries, there is a uniquely determined pair of positive integers l ,m such that l |m and ΓαΓ =Γ(l 00 m)Γ. The Hecke operator arising from the double coset ΓαΓ is then given byTn f = n−1/2∑lm=nl |m(f∣∣∣∣ Γ( l 00 m)Γ).We say that f is a Hecke eigenform if f is an eigenfunction of the Hecke operators Tn , n ≥ 1. Wedenote the corresponding Hecke eigenvalues λ f (n).By abuse of notation, we view functions on X as functions on G that are left SL2(Z)-invariantand right SO(2)-invariant. We denote by σ1 the usual sum-of-divisors function σ1(n) =∑d |n d .Proposition 4. Let Γ = SL2(Z). Let y = pq with p, q distinct, coprime and squarefree, and letϕ= f1 ⊗ f2 ∈Cb(X ×X ), where f1, f2 are non-constant Hecke eigenforms. ThenlimT→∞∫ 10ϕ(x + iT,xpq+ iT)dx = λ f2 (pq)ppqσ1(pq)〈f1, ap/q . f2〉= λ f1 (pq)ppqσ1(pq)〈aq/p . f1, f2〉,where〈f1, ap/q . f2〉= ∫Γ\G f1(g ) f2(g a−1p/q )dµ(g ) and µ is normalized so that µ(Γ\G) = 1.Proof. Set Γ = SL2(Z). Since ap/q ∈ Comm(Γ), the group Γp/q = a−1p/q Γap/q ∩ Γ is a finiteindex subgroup of Γ; in particular Γp/q < G is a lattice. Then ϕ ◦ ιp/q ∈ Cb(Γp/q \G), and theequidistribution of pieces of expanding closed horocycles on Γp/q \G yields [14]limT→∞∫Iϕ(ux a−1T ,uxp/q a−1T)dx = 1µ(Γp/q \G)∫Γp/q \Gϕ(g , ap/q g a−1p/q)dµ(g ),where the Haar measure µ is normalized so that µ(Γ\G) = 1.For p, q distinct, coprime and squarefree, we find Γap/q Γ= δ−1ppq Γ(1 00 pq)Γ, where δx =( xx),and µ(Γp/q \G) = M = σ1(pq) [10, Lemma 4.5.6]. Let F be a fundamental domain for Γ\G ; then4 Claire Burrin⊔Mm=1 hmF , where the hm ’s are coset representatives of Γp/q in Γ, is a fundamental domain forΓp/q \G . We can now compute1µ(Γp/q \G)∫Γp/q \Gf1(g ) f2(ap/q g a−1p/q )dµ(g ) =1MM∑m=1∫hmFf1(g ) f2(ap/q g a−1p/q )dµ(g )= 1M∫Ff1(g )M∑m=1f2(ap/q hm g a−1p/q )dµ(g )=ppqσ1(pq)∫Γ\Gf1(g )Tpq f2(g a−1p/q )dµ(g ). □AcknowledgmentsThanks are due to Alex Gorodnik for organizing the workshop Distribution of orbits: Arithmeticsand Dynamics in the Swiss Alps, where Valentin Blomer’s nice talk motivated this note. Follow-ing his presentation, Amir Mohammadi and Andreas Strömbergsson both independently alsopointed out the Ratner-based argument presented in this note. The computation that followsbenefitted from separate discussions with Andreas Strömbergsson, and I want to thank JensMarklof for references and comments, and the referees for their careful reading and helpful sug-gestions to improve the exposition of this note.Declaration of interestsThe author does not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and has declared no affiliations other than their researchorganizations.References[1] V. Blomer and P. G. Michel, “The unipotent mixing conjecture”, J. Anal. Math. 151 (2023),pp. 25–57.[2] C. Burrin, U. Shapira and S. Yu, “Translates of rational points along expanding closedhorocycles on the modular surface”, Math. Ann. 382 (2022), no. 1, pp. 655–717.[3] L. Clozel, H. Oh and E. Ullmo, “Hecke operators and equidistribution of Hecke points”,Invent. Math. 144 (2001), no. 2, pp. 327–380.[4] M. L. Einsiedler and E. Lindenstrauss, “Joinings of higher rank torus actions on homoge-neous spaces”, Publ. Math., Inst. Hautes Étud. Sci. 129 (2019), pp. 83–127.[5] D. A. Hejhal, “On value distribution properties of automorphic functions along closedhorocycles”, in XVIth Rolf Nevanlinna Colloquium (Joensuu, 1995), Walter de Gruyter, 1996,pp. 39–52.[6] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematikund ihrer Grenzgebiete. 3. Folge, Springer, 1991, pp. x+388.[7] J. Marklof, “Spectral theta series of operators with periodic bicharacteristic flow”, Ann. Inst.Fourier 57 (2007), pp. 2401–2427.[8] J. Marklof, “Horospheres and Farey fractions”, in Dynamical numbers—Interplay betweendynamical systems and number theory, Contemporary Mathematics, American Mathemat-ical Society, 2010, pp. 97–106.[9] J. Marklof and A. Strömbergsson, “Equidistribution of Kronecker sequences along closedhorocycles”, Geom. Funct. Anal. 13 (2003), no. 6, pp. 1239–1280.Claire Burrin 5[10] T. Miyake, Modular forms, Springer, 1989, pp. x+335.[11] M. Ratner, “Raghunathan’s topological conjecture and distributions of unipotent flows”,Duke Math. J. 63 (1991), pp. 235–281.[12] P. C. Sarnak and A. Ubis, “The horocycle flow at prime times”, J. Math. Pures Appl. 103(2015), no. 2, pp. 575–618.[13] N. A. Shah, “Limit distribution of expanding translates of certain orbits on homogeneousspaces”, Proc. Indian Acad. Sci., Math. Sci. 106 (1996), pp. 105–125.[14] A. Strömbergsson, “On the uniform equidistribution of long closed horocycles”, DukeMath. J. 123 (2004), pp. 507–547.

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