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HOMOTOPY SPECTRA and DIOPHANTINE EQUATIONS Yuri I. Manin

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HOMOTOPY SPECTRA and DIOPHANTINE EQUATIONS Yuri I. Manin HOMOTOPY SPECTRA AND DIOPHANTINE EQUATIONS Yuri I. Manin -I- CONTENTS 0. Introduction and summary .................................................................... II 1. Homotopy spectra: a brief presentation ................................................ XIII 2. Diophantine equations: distribution of rational points on algebraic varieties ................................... XIX 3. Rational points, sieves, and assemblers ................................................ XXIV 4. Anticanonical heights and points count ............................................. XXXVI 5. Sieves \beyond heights" ? ..................................................................... XLIX References ..................................................................................................... LIII -II- 0. INTRODUCTION AND SUMMARY • For a long stretch of time in the history of mathematics, Number Theory and Topology formed vast, but disjoint domains of mathematical knowledge. Emmanuel Peyre reminds us in [Pe19] that the Babylonian clay tablet Plimpton 322 (about 1800 BC) contained a list of integer solutions of the \Diophantine" equation a2+b2 = c2: archetypal theme of number theory, named after Diophantus of Alexandria (about 250 BC). Babilonian Clay Tablet Plimpton 322 -III- • \Topology" was born much later, but arguably, its cousin { modern measure theory, { goes back to Archimedes, author of Psammites (\Sand Reckoner"), who was approximately a contemporary of Diophantus. In modern language, Archimedes measures the volume of observable universe by counting the number of small grains of sand necessary to fill this volume. Of course, many qualitative geometric models and quantitative estimates of the relevant dis- tances precede his calculations. Moreover, since the estimated numbers of grains of sands are quite large (about 1064), Archimedes had to invent and describe a system of notation for large numbers going far outside the possibilities of any of the standard ancient systems. The construction of the first bridge between number theory and topology was ac- complished only about fifty years ago: it is the theory of spectra in stable homotopy theory -IV- • Integers and finite sets. Below we will appeal to the intuition of listeners who are somewhat accustomed to categorical reasoning. Intuitively, the set of non{negative integers N can be imagined as embodiment of \sizes" (formally, cardinalities) of all finite sets (including empty set). The latter form a category F Set, and already to interprete the inequality m ≤ n in N one needs to look at monomorphisms in F Set. A categorical interpretation of multiplication of integers uses direct products of finite sets. Associativity of multiplication is a reflection of the monoidal structure on F Set. Addition in N already requires more sophisticated constructions in F Set: if two sets X; Y are disjoint, then cardinality of X [ Y is the sum of cardinalities of X and Y , but far from all pairs (X; Y ) are disjoint. Finally, the passage from N to Z requires one step higher: constructions in category of categories. -V- • Integers and topological spaces. Much less popular is the background vision of a natural number n 2 N as the dimension of a manifold, say sphere Sn. Here already the definition of equality of numbers requires quite sophisticated tech- nicalities, if we want to lift it: it involves functorial definitions of classes of morphisms that we will declare invertible ones (e.g. homotopy equivalences) after passing to the cate- gory of categories. But as soon as we accomplish this, addition and multiplication in N become liftable, and, after some more work, passage from N to Z as well. • From finite sets to topological spaces, and back. The critically important bridging of finite sets and topological spaces is furnished by the machinery of simplicial sets. The simplest and most intuitive constructions refer to simplicial sets associated to coverings. -VI- Roughly speaking, this passage replaces a finite set X by the simplex σ(X) in which X is only subset of vertices, and a set theoretical map X ! Y by the respective glueing together (parts of) boundaries of σ(X) and σ(Y ). In reverse direction, the passage from a topological space with its covering (say, by a family of open subsets) fUa ja 2 Ag to a simplicial set replaces each Ua by a vertex of a simplex, and associates simplices ∆J with non{empty intersections \i2J Uai . For a detailed treatment, see [GeMa03]. -VII- Here we will only recall the combinatorial setup, which embodies SF Sets, the minimal category of based finite sets with the following properties. Its objects are totally ordered sets [n] := f0; 1; : : : ; ng, and morphisms are nondecreasing maps f :[n] ! [m] with f(0) = 0: Thus 0 2 [n] is the based point, and [0] is the initial object. i The set of all morphisms is generated by two classes of maps: \i{th face" @n and i \i{th degeneration" σn. The i{th face is the increasing injection [n−1] ! [n] not taking i the value i, and σn is the nondecreasing surjection taking the value i twice. All the relations between faces and degenerations are generated by the relations j i i j−1 @n+1@n = @n+1@n for i < j; j i i j+1 σn−1σn+1 = σnσn+1 for i ≤ j; 8 @i σj−1 for i < j; > n−1 n−2 j i < σn−1@n = id[n−1] for i 2 fj; j + 1g; :> i−1 j @n−1σn−2 for i > j + 1: -VIII- A simplicilal set X• (or simply X) can be then defined as a functor from SF Sets to some category Set of sets. Thus, one simplicial set is the structure consisting of a family of sets (Xn), n = 0; 1; 2;::: and a family of maps X(f): Xn ! Xm, corresponding to each nondecreasing map f :[m] ! [n], such that X(id[n]) = idXn , and X(g ◦ f) = X(f) ◦ X(g). i i Restricting ourselves by consideration of only X(@n) and X(σn), and the respective relations, we get the convenient and widely used description of simplicial sets. In order to pass from combinatorics to geometry, we must start with geometric simplices. -IX- In order to pass from combinatorics to geometry, we must start with geometric simplices. n{dimensional simplex ∆n is the topological space embedded into real affine n + 1{ dimensional space endowed with coordinate system: n n+1 n+1 X ∆n := f(x0; : : : ; xng 2 R \ [0; 1] j xi = 1g: i=0 Starting with a simplicial set (Xn), we construct its geometric realisation jXj. It is the set, endowed with a topology, 1 jXj := [n=0(∆n × Xn)=R; where R is the weakest equivalence relation identifying boundary subsimplices in ∆n ×Xn and ∆m ×Xm corresponding to nondecreasing maps f :[m] ! [n]: see [GeMa03], pp. 6{7. In particular, the boundary of ∆n+1 for n ≥ 2 is the geometric realisation of the union of n + 1 boundary (n − 1){dimensional simplices: this is a simplicial model of Sn−1. -X- One popular path in opposite direction produces a simplicial set from a topological space Y endowed with a covering by open (or closed) sets (Uα), α 2 A. Put for n ≥ 1, n Xn := f(α1; : : : ; αn) j \i=1 Uαi 6= ;g; X(f)(') := ' ◦ ∆f ; where f :[m] ! [n]; ∆f : ∆m ! ∆n: The role of homotopy appearing at this point finds a beautiful expression in the following fact: the geometric realisation jXj is homotopically equivalent to Y, if U is a locally n finite covering, and all nonempty finite intersections \i=1Uαi are contractible. These constructions form the background for the much more sophisticated machin- ery of homotopical spectra sketched below in Sec. 1. -XI- • Plan of exposition. Section 1 is dedicated to a brief, but sufficiently formal presenta- tion of homotopy spectra. In particular, we would like to draw attention of a listener to the category of Γ{spaces that has several different applications to the construction and study of new geometries: for example, geometries \in characteristic 1", or more generally \under Spec Z". Section 2 introduces an approach to distributions of rational/algebraic points on algebraic varieties based upon counting of points of bounded sizes. Here the central role is played by the notion of heights: ways to measure sizes of rational/algebraic points on algebraic varieties taking in account their positions in a projective space to which the variety is/can be embedded. -XII- The pure categorical Section 3 is dedicated to the basic machinery of assemblers that we later use to bridge diophantine geometry and homotopical algebra. Its central intuitive notion is similar to one that lies in the background of the definition of functor K0 of an abelian category: we replace each object of a category by the formal sum of its \irreducible" pieces, neglecting ways in which these pieces are assembled together. In the Section 4 we introduce arithmetical/geometric environments that are more restricted, but better describable by homotopy means. Finally, Section 5 considers possibilities to avoid point count on algebraic varieties, replacing it by a machinery of measurable sets in adelic spaces. This allows us to include consideration of Brauer{Manin obstructions and more intuitive notions of equidistribution of rational points. -XIII- 1. HOMOTOPY SPECTRA: A BRIEF PRESENTATION The notion of spectra in homotopy theory evolved through several stages. Below I will briefly describe two of them: sequential spectra, and Γ{spaces. We have to start with the definition of the smash product ^ in the category of finite pointed sets: (X; ∗X ) ^ (Y; ∗Y ) := (X × Y )=f(X × ∗Y ) [ (∗X × Y )g: • Sequential spectra (see [SpWh53]). A sequential spectrum E is a sequence of based simplicial sets En, n = 0; 1; 2;::: , and the the structure maps 1 σn :ΣEn := S ^ En ! En+1: n 1 1 The sphere (sequential) spectrum S consists of sets S := S ^ :::S and identical σn's. -XIV- The smash product can be extended to the category of sequential spectra them- selves, however there it loses commutativity property.
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