Ratner’s Work on Flows and Its Impact Elon Lindenstrauss, Peter Sarnak, and Amie Wilkinson

Dani above. As the name suggests, these theorems assert that the closures, as well as related features, of the orbits of such flows are very restricted (rigid). As such they provide a fundamental and powerful tool for problems connected with these flows. The brilliant techniques that Ratner in- troduced and developed in establishing this rigidity have been the blueprint for similar rigidity theorems that have been proved more recently in other contexts. We begin by describing the setup for the of 푑×푑 matrices with real entries and equal to 1 — that is, SL(푑, ℝ). An element 푔 ∈ SL(푑, ℝ) is unipotent if 푔−1 is a nilpotent (we use 1 to denote the identity element in 퐺), and we will say a group 푈 < 퐺 is unipotent if every element of 푈 is unipotent. Connected unipotent subgroups of SL(푑, ℝ), in particular one-parameter unipo- Ratner presenting her rigidity theorems in a plenary tent subgroups, are basic objects in Ratner’s work. A unipo- address to the 1994 ICM, Zurich. tent group is said to be a one-parameter unipotent group if there is a surjective homomorphism defined by polyno- In this note we delve a bit more into Ratner’s rigidity theo- mials from the additive group of real numbers onto the rems for unipotent flows and highlight some of their strik- group; for instance ing applications, expanding on the outline presented by

Elon Lindenstrauss is Alice Kusiel and Kurt Vorreuter professor of mathemat- 1 푡 푡2/2 1 푡 ics at The Hebrew University of Jerusalem. His email address is elon@math 푢(푡) = ( ) and 푢(푡) = ⎜⎛ 1 푡 ⎟⎞ . .huji.ac.il 1 . ⎝ 1 ⎠ Peter Sarnak is Eugene Higgins professor of mathematics at Princeton Univer- sity and also a professor at the Institute for Advanced Study. His email address is [email protected]. In both cases it is easy to verify directly that these poly- Amie Wilkinson is a professor of mathematics at the University of Chicago. Her nomials do indeed define a homomorphism: i.e., for any email address is [email protected]. 푠, 푡 ∈ ℝ it holds that 푢(푡 + 푠) = 푢(푡) ⋅ 푢(푠). While For permission to reprint this article, please contact: there is essentially no loss of generality in discussing only [email protected]. the case of SL(푑, ℝ), a more natural context is that of lin- DOI: https://doi.org/10.1090/noti/1829 ear algebraic groups — subvarieties of SL(푑, ℝ) defined

MARCH 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 373 by polynomial equations that are closed under multiplica- of Numbers is how I have called this book, since I ar- tions and taking inverses (this notion actually makes sense rived at the methods, which deliver in it proofs of arith- for more general fields than the real numbers; if we wantto metic theorems, through spatial considerations. emphasize that we are working with the field of real num- Ratner’s work is a remarkable contribution in the general bers we will call such groups linear algebraic groups over theme of applying “analysis of the infinite” and “spatial ℝ). Connected unipotent subgroups of SL(푑, ℝ) are al- considerations” to number theory. ways linear algebraic groups. Another nice class of exam- So what did Ratner prove in these remarkable papers? ples are the orthogonal groups. Given a quadratic form Perhaps the easiest to explain is her Orbit Closure Classi- 푄(퐱) over ℝ (positive definite or not) in 푑 variables, one fication Theorem, confirming an important conjecture of can consider the group SO(푄) of all matrices in SL(푑, ℝ) M. S. Raghunathan: that preserve this form, i.e. 푑×푑-matrices 푀 so that 푄(푀퐱) = 푄(퐱) for all 푥 ∈ ℝ푑. This group will be compact if and Theorem 1 (Ratner’s Orbit Closure Theorem [M3]). Let only if 푄 is a positive definite or a negative definite form. 퐺 be a real linear algebraic group as above, Γ a lattice in 퐺 Ratner’s theorems on rigidity of unipotent group actions and 푈 < 퐺 a connected unipotent group. Then for any point deal with the action of a unipotent group 푈 on a quotient 푥 ∈ 퐺/Γ the closure of its 푈-orbit is a very nice object: a space of 퐺 by a discrete subgroup. An important exam- single orbit of some closed connected group 퐿 that is sandwiched ple of such a quotient space is when 퐺 = SL(푑, ℝ) and between 푈 and 퐺 (and may coincide with either). Moreover, Γ = SL(푑, ℤ), in which case 퐺/Γ can be identified with this single orbit of 퐿 has finite volume. 푑 the space lattices in ℝ that have unit covolume. A lattice Recall that the 푈-orbit of a point 푥 is simply the set 푑 in ℝ can be specified by giving 푑 linearly independent vec- {푢.푥 ∶ 푢 ∈ 푈}. Note that in particular this shows that any 푑 tors that generate it — i.e. vectors 푣1,..., 푣푑 ∈ ℝ (that 푈-orbit closure has a natural 푈-invariant probability mea- we prefer to think of as column vectors) so that Λ = ℤ푣1 + sure attached to it. We also remark that one can loosen the ⋯ + ℤ푣푑, and the condition that the lattice has unit covol- requirement that 푈 be unipotent to 푈 being generated by ume amounts to requiring that det(푣1, … , 푣푑) = 1, or one-parameter unipotent groups — the passage from The- in other words that the matrix 푔 = (푣1, … , 푣푑) obtained orem 1 to this more general statement is not very difficult. by joining together these 푑 vectors be in SL(푑, ℝ). The Unlike previous work towards Raghunathan’s Conjecture, generators of the lattice Λ are not uniquely determined: in particular Margulis’ proof in the mid 1980s of the (then) ′ ′ 푣1,..., 푣푑 generate the same lattice as 푣1,..., 푣푑 if and fifty year old Oppenheim Conjecture using a special case ′ ′ only if (푣1, … , 푣푑) = (푣1, … , 푣푑)훾 for 훾 ∈ SL(푑, ℤ), in of Raghunathan’s Conjecture, Ratner’s route to classifying other words, lattices of unit covolume in ℝ푑 are in one-to- orbit closures was not direct but by via a measure classifi- one correspondence with elements of SL(푑, ℝ)/ SL(푑, ℤ). cation result: Any matrix ℎ ∈ SL(푑, ℝ) acts on this space by left mul- Theorem 2 (Ratner’s Measure Classification Theorem tiplication; in terms of lattices this amounts to the map [M2, M1]). Let 퐺, Γ and 푈 be as in Theorem 1. Then the from the space of unit covolume lattices to itself taking a only (Borel) probability measures on 퐺/Γ that are invariant lattice Λ < ℝ푑 to the lattice {ℎ.푣 ∶ 푣 ∈ Λ}. and ergodic under 푈 are the natural measures on the orbit clo- This quotient space has the important property of hav- sures described in Theorem 1. ing finite volume, or more precisely an SL(푑, ℝ)-invariant probability measure. A subgroup Γ of a topological group This requires a bit of explanation: We equip 푋 = 퐺/Γ 퐺 which is discrete and such that 퐺/Γ has finite volume with the Borel 휎-algebra ℬ, and consider probability mea- is called a lattice (admittedly, this can be a bit confusing sures on the measurable space (푋, ℬ). Such a measure 휇 at first since our basic example of such 퐺/Γ is the space is 푈-invariant if the push forward of it under left multipli- of lattices in ℝ푑.. ., though this terminology is consistent). cation by every 푢 ∈ 푈 remains the same; 휇 is 푈-ergodic if Hermann Minkowski seems to have been the first to real- every 푈-invariant Borel subset of 푋 is either null or conull. ize the importance of such quotients, and in particular the Every 푈-invariant probability measure can be presented as space of lattices in ℝ푑, to number theory at the turn of the an average of ergodic ones, hence classifying the 푈-ergodic 19th century. In the introduction to his book Geometrie measures gives a description of all 푈-invariant probability der Zahlen, Minkowski writes1 measures on 푋. Dani conjectured this measure classifica- This book contains a new kind of applications of analy- tion result in the same paper where Raghunathan’s Conjec- sis of the infinite to the theory of numbers or, better, cre- ture first appeared. ates a new bond between these two areas. . . It is possible to reduce both Theorem 1 and Theorem 2 to the case where 푈 is a one-parameter unipotent group. 1Translated from the original German to English. The following theorem implies both of the theorems quoted

374 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 3 above in the one-parameter case, but is used by Ratner as a digression. It would have been interesting to have such bridge allowing her to pass from the measure classification rigidity results also for local fields of positive characteristic theorem (which, as we said in the outset, is the heart of her such as 픽푞((푡)) — the field of formal Laurent series with work on unipotent flows) to the obit closure theorem: coefficients in the finite field 픽푞 with 푞 elements — but there seem to be serious technical obstacles to doing so Theorem 3 (Ratner’s Distribution Rigidity Theorem and only partial results in this direction are known. [M3]). Let 퐺, Γ, 푈 be as above, and let 푥 ∈ 퐺/Γ. Then The rigidity theorems of Ratner have had numerous ap- there is a 푈-ergodic probability measure 푚푥 of the form given plications in many areas of mathematics. A highly non- above (i.e. the uniform measure on a finite volume orbit ofa trivial special case of her general measure classification re- connected group sandwiched between 푈 and 퐺) so that 푥 is in sult, namely the classification of measures on a reducible the support of 푚푥 and for any bounded continuous function 푓 product (SL(2, ℝ)/Γ ) × (SL(2, ℝ)/Γ ) invariant under on 퐺/Γ we have that the ergodic averages 1 1 a one-parameter unipotent group (the interesting case is 푇 1 classifying measures that project to the uniform measure ∫ 푓(푢(푡).푥) 푑푡 → ∫ 푓 푑푚푥 as 푇 → ∞. (0.1) 푇 0 on each (SL(2, ℝ)/Γ푖) factor, or in the ergodic theoretic The reader with some basic knowledge of ergodic theory terminology, joinings) was proved by Ratner already in the might be fooled to think that (0.1) is an application of the early 1980s. The original motivation of Ratner in study- Birkhoff Pointwise Ergodic Theorem. Not so! The Birkhoff ing these flows was to understand better (and give natural Pointwise Ergodic Theorem only gives information about examples for) a property of measure preserving systems almost every point (with respect to a given ergodic measure). called Loosely Bernoulli — we can view this somewhat The whole point of Ratner’s Distribution Rigidity Theorem anachronistically as an application of unipotent flows to is that it is true for each and every 푥 ∈ 퐺/Γ. Almost every- the abstract theory of dynamical systems. Since then her where results are almost always much easier to prove,2 but work has had several other applications to abstract ergodic in a mathematical manifestation of Murphy’s Law, such theory and descriptive set theory. There are very striking results might say something about virtually all points but applications of her work to mathematical physics, for in- if you are given a specific point and want to study its be- stance in the work of Marklof and Strömbergsson on the haviour under a given action they tell you absolutely noth- Lorentz gas, and to geometry. In this note we have chosen ing. To give a simple analogy, it is trivial to prove that for to highlight a couple of the many applications of her theo- a.e. 푥 ∈ [0, 1] the asymptotic density of occurrence of rems (as well as the extension to products of linear groups each of the digits 0,1,2,.. .,9 in the decimal expansion of 푥 over local fields as above) to number theory. is 1/10, but asking whether this holds for particular num- In making his famous conjecture, Raghunathan was mo- bers of interest such as 21/3 or 휋 seems at present to be a tivated by the connection to the Oppenheim Conjecture, hopelessly difficult question! a connection that allowed Margulis to resolve this long- As it turns out, for some of the most juicy applications standing open problem by establishing a special case of the of these rigidity results a more general setup is required. conjecture posed by Raghunathan [G]. Oppenheim conjec- To begin with, one may consider linear algebraic groups tured in the 1930s that for any indefinite quadratic form 푄 over other fields; and since the topological structure is very in 푑 ≥ 3 variables that is not proportional to a quadratic much in play here, the natural class of fields to look at are form with coefficients, the set of values attained by 푑 local fields, i.e. topological fields whose topology is locally 푄 at integer vectors, that is to say 푄(ℤ ), contains zero compact, such as ℝ or the 푝-adic numbers ℚ푝. Both Rat- as a non-isolated point. Using Ratner’s Measure Classifica- ner [M4] and independently Margulis and Tomanov [GAGM] tion Theorem, and relying upon prior work by Dani and extended the above results to this setting, and more gen- Margulis, Eskin, Margulis, and Mozes [AGS] were able not 푘 only to show that there are integer vectors 퐧 ∈ ℤ푑 for erally to quotients 퐺/Γ where 퐺 = ∏푖=1 퐺푖 with each 푄(퐧) 퐺푖 a linear algebraic group over a local field of character- which is close to a given value (say 0), but to count istic zero.3 We shall refer to such quotient spaces 퐺/Γ as the number of such vectors. More precisely, for indefinite 푆-arithmetic quotients, a terminology that probably needs quadratic forms as above, not of signature (1,2) or (2,2), some explanation which we omit to avoid too much of a Eskin, Margulis, and Mozes show that for any 푎 < 푏, the number of integer vectors 퐧 ∈ ℤ푑 inside a ball of radius 2This is a slight pun—“almost everywhere” is used in the above sentence in its precise math- 푅 for which 푎 < 푄(퐧) < 푏 is asymptotically given by the ematical sense, whereas “almost always” is used in the ordinary, non-mathematical sense of the phrase... volume of the corresponding shape cut by the two hyper- 3Note that our definitions of unipotent groups and one-parameter unipotent groups make surfaces 푄(퐱) = 푎 and 푄(퐱) = 푏 in this ball. Perhaps an sense over any field, and can be easily extended to the product case, e.g. asubgroup 푈 < 푘 illustration of the delicacy of the question is that this nat- ∏푖=1 퐺푖 (with each 퐺푖 defined over a different local field) is a one-parameter unipotent group if there is an 푖 so that 푈 is a one-parameter unipotent subgroup of 퐺푖. ural statement is false(!) for quadratic forms of signature

MARCH 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 375 (1,2) or (2,2), though in a follow-up paper Eskin, Mozes, argument, we note that even if a quadratic form 푄 is posi- and Margulis were able to prove this estimate for quadratic tive definite, hence its symmetry group over ℝ is compact, forms of signature (2,2) under a suitable Diophantine con- over the 푝-adic numbers in general for 푘 ≥ 3 variables it dition, a result which is of interest in the context of the would be a non-compact group with plenty of unipotents. study of the of energy levels of quantization of In truth, the relevant symmetry group for this case is not integrable dynamical systems. the symmetry group of 푄 but the subgroup of this symme- The reason unipotent dynamics is relevant to the Op- try group fixing a given quadratic form in ℓ variables, but penheim Conjecture is that the symmetry group of an in- this is precisely why in this problem one needs to employ definite (real) quadratic form with ≥ 3 variables contains 푝-adics. (indeed, is generated by) one-parameter unipotent groups. An even more surprising application of Ratner’s work to Surprisingly, there is a relatively recent application of the number theory was given by Vatsal and Cornut–Vatsal (e.g. S-arithmetic analogue of Ratner’s results to positive definite, [V]). We do not give details here, but in these works fam- integral forms. ilies of elliptic curve 퐿-functions, and in particular their Legendre’s Three Squares Theorem says that a positive central values (or derivatives thereof when their functional integer 푛 can be presented as a sum of three squares if and equation is odd rather than even), are considered. Us- only if it is not of the form 4푎(8푏 + 7) ,with 푎, 푏 inte- ing Ratner’s Orbit Closure Theorem as a basic ingredient gers. This is an example of a local-to-global principle: the Vatsal (and in the more general cases Cornut and Vatsal) quadratic form 푄(푥, 푦, 푧) = 푥2 + 푦2 + 푧2 represents an showed that all but finitely many of these values are not integer 푛 if and only if the congruences 푄(푥, 푦, 푧) ≡ 푛 zero. When combined with well-known results towards (mod 푝푎) are solvable for any prime 푝 and any 푎 ∈ ℕ the Birch and Swinnerton–Dyer Conjecture, this proves a (for a given 푝, consistency of this infinite set of congru- conjecture of Mazur: essentially all the points on an ellip- ences is equivalent to 푄(푥, 푦, 푧) = 푛 being solvable by tic curve whose coordinates lie in ring class fields with re- 푝-adic ). In this particular case, only the prime stricted ramification are generated by explicit special points 2 can be an obstacle though there is another restriction first constructed by Heegner. on 푛 implicit in the way that we set up the problem — The impact of Ratner’s work cannot be measured only that 푛 is positive — which can be said to come from the by direct application of her seminal results. Techniques “place at infinity,” in other words from the necessity that introduced by Ratner to study ergodic theoretic joinings 푄(푥, 푦, 푧) = 푛 be solvable over ℝ. in her early works on unipotent flows in the 1980s were Legendre’s Three Squares Theorem can be viewed as a a main inspiration in the work of the first named author special case of the following problem: Given a fixed posi- on diagonalizable flows and its applications to Arithmetic tive definite integral quadratic form 푄 in many (say 푘) vari- Quantum Unique Ergodicity and equidistribution. Benoist ables, which quadratic forms 푄′ in ℓ < 푘 variables can be and Quint were similarly inspired by Ratner’s work in their represented by 푄? That is to say, when can we find a 푘 × ℓ breakthrough work understanding stationary measures and integer matrix 푀 so that as quadratic forms 푄′ = 푄 ∘ 푀? orbit closures for actions of thin groups on homogeneous For ℓ = 1 and 푄 = 푥2 + 푦2 + 푧2 this reduces to the spaces, and Eskin and Mirzakhani transformed the study question addressed by Legendre: the form 푄′ = 푛푥2 can of moduli spaces of abelian and quadratic differentials on be represented by 푄 iff 푛 can be written as a sum of three Riemann surfaces by proving an analogue of Ratner’s work squares. Local solvability — the existence of such matrix in this setting. 푀 with entries in ℤ푝 for every 푝 — is an obvious necessary The prevalence of deep and suprising applications of condition that can be verified with a finite calculation. Ratner’s Rigidity Theorems on unipotent flows is remark- Hsia, Kitaoka, and Kneser in 1978 established the va- able, and shows the richness of the subject of homoge- lidity of such a local-to-global principle for representing neous dynamics and how interconnected it is with many any form 푄′ in ℓ variables with sufficiently large square other subjects. It is also a tribute to a wonderful mathe- free discriminant by a given form 푄 in 푘 variables once matician who has left a legacy to future mathematicians 푘 ≥ 2ℓ + 3 by using more traditional number theoretic for many years to come. methods. This remained the best result on this very classi- cal problem (essentially dating back to the work of Gauss) until Ellenberg and Venkatesh [JA] were able to use the 푆- References arithmetic extensions to Ratner’s Orbit Closure Theorem [AGS] Eskin A, Margulis G, Mozes S. Upper bounds and 푘 ℓ to very significantly reduce the restriction on and to asymptotics in a quantitative version of the Oppenheim be 푘 ≥ ℓ + 5. While we cannot get into the details of the conjecture, Ann. of Math. (2), no. 1 (147):93–141, 1998, DOI 10.2307/120984. MR1609447

376 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 3 [JA] Ellenberg J, Venkatesh A. Local-global principles for representations of quadratic forms, Invent. Math., no. 2 (171):257–279, 2008, DOI 10.1007/s00222-007-0077-7. MR2367020 [G] Margulis G. Discrete subgroups and ergodic theory, Num- ber theory, trace formulas and discrete groups (Oslo, 1987); 1989:377–398. MR993328 [GAGM] Margulis G A, Tomanov G M. Invariant measures for actions of unipotent groups over local fields on homoge- neous spaces, Invent. Math., no. 1-3 (116):347–392, 1994, DOI 10.1007/BF01231565. MR1253197 [M1] Ratner M. On measure rigidity of unipotent subgroups of semisimple groups, Acta Math., no. 3-4 (165):229–309, 1990, DOI 10.1007/BF02391906. MR1075042 [M2] Ratner M. On Raghunathan’s measure conjecture, Ann. of Math. (2), no. 3 (134):545–607, 1991. MR1135878 [M3] Ratner M. Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J., no. 1 (63):235–280, 1991, DOI 10.1215/S0012-7094-91-06311- 8. MR1106945 [M4] Ratner M. Raghunathan’s conjectures for Cartesian products of real and 푝-adic Lie groups, Duke Math. J., no. 2 (77):275–382, 1995, DOI 10.1215/S0012-7094-95-07710- 2. MR1321062 [V] Vatsal V. Uniform distribution of Heegner points, Invent. Math., no. 1 (148):1–46, 2002, DOI 10.1007/s002220100183. MR1892842

Credits Photo of Marina Ratner is courtesy of Anna Ratner.

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