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Orbital Counting Via Mixing and Unipotent Flows Clay Mathematics Proceedings Volume 8, 2008 Orbital counting via Mixing and Unipotent flows Hee Oh Contents 1. Introduction: motivation 1 2. Adeles: de¯nition and basic properties 5 3. General strategy on orbital counting 8 4. Equidistribution of semisimple periods via unipotent flows 11 5. Well-rounded sequence and Counting rational points 14 6. Mixing and Hecke points 20 7. Bounds toward the Ramanujan conjecture on the automorphic spectrum 22 8. Counting via mixing and the wavefront property 25 9. A problem of Linnik: Representations of integers by an invariant polynomial II 29 References 32 This note is an expanded version of my lectures given at the Clay summer school in 2007. 1. Introduction: motivation Let X be a projective algebraic variety de¯ned over Q, that is, X is the set of (equivalence classes of) zeros of homogeneous polynomials with coe±cients in Q. The set X(Q) of rational points in X consists of rational zeros of the polynomials. The following is a classical question in number theory: \understand the set X(Q) of rational points": More detailed questions can be formulated as follows: (1) Is X(Q) non-empty? (2) If non-empty, is X(Q) in¯nite? (3) If in¯nite, is X(Q) Zariski dense in X? (4) If Zariski dense, setting XT := fx 2 X(Q) : \ size"(x) < T g; what is the asymptotic growth rate of #XT as T ! 1? °c 2008 Clay Mathematics Institute 1 2 HEE OH (5) Interpret the asymptotic growth rate of #XT in terms of geometric invariants of X (6) Describe the asymptotic distribution of XT as T ! 1? A basic principle in studying these questions is that (1.1) \the geometry of X governs the arithmetic of X": A good example demonstrating this philosophy is the Mordell conjecture proved by Faltings [30]: Theorem 1.2. If X is a curve of genus at least 2, then X(Q) is ¯nite. Note in the above theorem that a purely geometric property of X imposes a very strong restriction on the set X(Q). Smooth projective varieties are classi¯ed roughly into three categories according to the ampleness of its canonical class KX. For varieties of general type (KX ample), the following conjecture by Bombieri, Lang and Vojta is a higher dimensional analogue of the Mordell conjecture [65]: Conjecture 1.3. If X is a smooth projective variety of general type, then X(K) is not Zariski dense for any number ¯eld K. At the other extreme are Fano varieties (¡ KX ample). It is expected that a Fano variety should have a Zariski dense subset of rational points, at least after passing to a ¯nite ¯eld extension of Q. Moreover Manin formulated conjectures in 1987 on the questions (4) and (5) for some special \size functions", which were soon generalized by Batyrev and Manin for more general size functions [1], and Peyre made a conjecture on the question (6) [50]. In these lecture notes, we will focus on the question (4). First, we discuss the notion of the size of a rational point in X(Q). One way is simply to look at the Euclidean norm of the point. But this usually gives us in¯nitely many points of size less than T , and does not encode enough arithmetic information of the point. Height function|{ A suitable notion of the size of a rational point is given by a height function. For a general variety X over Q and to every line bundle L of X over Q, one can associate a height function HL on X(Q), unique up to multiplication by a bounded function, n via Weil's height machine (cf. [38]). On the projective space P , the height H = HOPn(1) associated to the line bundle OPn(1) of a hyperplane is de¯ned by: q 2 2 H(x) := x0 + ¢ ¢ ¢ + xn n where (x0; ¢ ¢ ¢ ; xn) is a primitive integral vector representing x 2 P (Q), which is unique up to sign. It is clear that there are only ¯nitely many rational points in Pn(Q) of height less than T , as there are only ¯nitely many (primitive) integral vectors of Euclidean norm at most T . Schanuel [57] showed that as T ! 1, n n¡1 #fx 2 P (Q):HOPn(1) (x) < T g » c ¢ T for an explicit constant c > 0, and this is the simplest case of Manin's conjecture. For a general variety X over Q, a very ample line bundle L of X over Q de¯nes a Q- n embedding ÃL : X ! P . Then a height function HL is the pull back of the height function HOPn(1) to X(Q) via ÃL. For an ample line bundle L, we set 1=k HL = HLk COUNTING RATIONAL POINTS 3 for k 2 N such that Lk is ample. Note that for any ample line bundle L, #fx 2 X(Q):HL(x) < T g < 1: For a subset U of X and T > 0, we de¯ne NU (L; T ) := #fx 2 X(Q) \ U :HL(x) < T g: Manin's conjecture|{ Manin's conjecture (or more generally the Batyrev-Manin conjec- ture) says the following (cf. [1]): Conjecture 1.4. Let X be a smooth Fano variety de¯ned over Q. For any ample line bundle L of X over Q, there exists a Zariski open subset U of X such that (possibly after passing to a ¯nite ¯eld extension), as T ! 1, aL bL¡1 NU (L; T ) » c ¢ T ¢ (log T ) where aL 2 Q>0 and bL 2 Z¸1 depend only on the geometric invariants of L and c = c(HL) > 0. More precisely, the constants aL and bL are given by: aL := inffa : a[L] + [KX] 2 ¤e®(X)g; bL := the codimension of the face of ¤e®(X) containing aL[L] + [KX] in its interior where Pic(X) denotes the Picard group of X and ¤e®(X) ½ Pic(X) ­ R is the cone of e®ective divisors. Remark 1.5. The restrictionP to a Zariski open subset is necessary in Conjecture 1.4: for the cubic surface X: 3 x3 = 0, the above conjecture predicts the T (log T )3 order of q i=0 i P3 2 rational points of height i=0 xi < T , but the curve Y given by the equations x0 = ¡x1 2 and x2 = ¡x3 contains the T order of rational points of height bounded by T . Note that the asymptotic growth of the number of rational points of bounded height, which is arithmetic information on X, is controlled only by the geometric invariants of X; so this conjecture as well embodies the basic principle (1.1) for Fano varieties. Conjecture 1.4 has been proved for smooth complete intersections of small degree [6], flag varieties [31], smooth toric varieties [2], smooth equivariant compacti¯cations of horo- spherical varieties [60], smooth equivariant compacti¯cations of vector groups [11], smooth bi-equivariant compacti¯cations of unipotent groups [59] and wonderful compacti¯cations of semisimple algebraic groups ([58] and [33]). We refer to survey articles ([62], [63], [10]) for more backgrounds. In the ¯rst part of these notes, we will discuss a recent work of Gorodnik and the author [35] which solves new cases of Conjecture 1.4 for certain compacti¯cations of homogeneous varieties. In contrast to most of the previous works which were based on the harmonic analysis on the corresponding adelic spaces in order to establish analytic properties of the associated height zeta function, our approach is to use the dynamics of flows on the homo- geneous spaces of adele groups. Approach|{We will be interested in the projective variety X which is the compacti¯cation of an a±ne homogeneous variety U, and try to understand the asymptotic of the number of rational points of U of height less than T (note that U is a Zariski open subset of X and hence it su±ces to count rational points lying in U.). More precisely, let U be an orbit u0G 4 HEE OH n n where G ½ PGLn+1 is an algebraic group de¯ned over Q and u0 2 P (Q). And let X ½ P be the Zariski closure of U, and consider the height function H on X(Q) obtained by the pull pack of HOPn (1). We attempt to forget about the ambient geometric space X for the time being and to focus on the rational points U(Q) of the a±ne homogeneous variety U. We would like to prove that a b¡1 (1.6) NT := #fx 2 U(Q) : H(x) < T g » c ¢ T ¢ (log T ) for some a; c > 0 and b ¸ 1. How does one prove such a result? Or where should the growth rate T a(log T )b¡1 come from? Consider for a moment how to count integral vectors of Euclidean norm less than T in the plane. How does one know that the asymptotic of the number NT of such integral vectors 2 is of the form ¼T , as T ! 1? It is because that one can show that NT is asymptotic to the area of disc of radius T and compute that the area is ¼T 2, using calculus. It turns out that one can follow the same basic strategy for counting rational points. We will ¯rst understand the set U(Q) of rational points as a discretely imbedded subset in certain ambient locally compact space and show that NT is asymptotic to the volume of a suitably de¯ned height ball in this ambient space.
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