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 ∈ 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) {Ua |a ∈ A} to a simplicial set replaces each Ua by a vertex of a simplex, and associates simplices ∆J with non–empty intersections ∩i∈J 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] := {0, 1, . . . , n}, and morphisms are nondecreasing maps f :[n] → [m] with f(0) = 0. Thus 0 ∈ [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;  ∂i σj−1 for i < j;  n−1 n−2 j i  σn−1∂n = id[n−1] for i ∈ {j, j + 1};  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 := {(x0, . . . , xn} ∈ R ∩ [0, 1] | xi = 1}. i=0

Starting with a simplicial set (Xn), we construct its geometric realisation |X|. It is the set, endowed with a topology,

∞ |X| := ∪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 ∈ A. Put for n ≥ 1, n Xn := {(α1, . . . , αn) | ∩i=1 Uαi 6= ∅},

X(f)(ϕ) := ϕ ◦ ∆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 |X| 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 )/{(X × ∗Y ) ∪ (∗X × Y )}.

• 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. Only after formal inversion of stable equivalences, requiring introduction of homotopy invariants, one can recon- struct commutative and associative smash product. • Γ–spaces. Γ–spaces were introduced by Graham Segal in [Se74]. Here we reproduce arguments of [Ly99] and mostly keep his notation. Let Γ (Γop in [Ly99]) be the category of based finite sets n+ := {0, 1, . . . , n} (former [n]), with base preserving maps as morphisms. So here we do not restrict morphisms by nondecreasing maps. A Γ–space E is a functor from Γ to based simplicial sets sending 0+ to point. Γ–spaces themselves are objects of the category denoted GS in [Ly99], morphisms in which are natural functors (we omit a precise description). Following Sec. 2 of [Ly99], we will call below based simplicial sets simply spaces.A space is called discrete, if its simplicial set is constant. -XV-

1.1. Theorem. One can define a functor ∧ : GS × GS → GS such that there is a canonical isomorphism of functors in three variables

GS(F ∧ F 0,F 00) =∼ GS(F, Hom (F 0,F 00)) where we denote by GS(F,F 0) the set of morphisms F → F 0 in GS. The category of Γ–spaces endowed with smash–product ∧ is a symmetric monoidal category. -XVI-

Sketch of Proof. Consider the category of Γ × Γ–spaces GGS. Its objects are pointed functors Γ × Γ → S. Define also the external smash product F ∧F 0 of two Γ–spaces F,F 0 as the functor that sends (m+, n+) to F (m+) ∧ F 0(n+). Then one can check that

GS(F, Hom (F 0,F 00)) =∼ GGS(F ∧F 0,RF 00), where R is a functor GS → GGS. After that one can prove that R has a left adjoint functor L : GGS → GS. Namely, LF 000 is the colimit of F 000(i+, j+) over all morphisms i+ ∧ j+ → n+. This is essentially the statement of Theorem 1.1. For many more details, see [Ly99]. -XVII-

• Definition. The sphere spectrum S is the unit object in the symmetric monoidal category of Γ–spaces. Here is a more detailed description of S. For any n+ we can define the Γ–space Γn by Γn(m+) := GS(n+, m+). From this, it follows that S is canonically isomorphic to Γ1, and is in fact the sphere sequential spectrum defined on p. XIII above. In particular, this construction suggests to consider homotopical enrichments of arithmetics passing through ring of integers Z =⇒ the sphere spectrum S -XVIII-

It also poses a challenge: discover a new information in number theory using the developed independently machinery of homotopy theory. In this talk, after a brief survey of relevant setups in homotopy theory given above and number theory below, I focus upon the ongoing research (joint with M. Marcolli) dedicated to the applications of spectra to the problems of distribution of ratio- nal/algebraic points on algebraic varieties. -XIX-

2. DIOPHANTINE EQUATIONS: DISTRIBUTION OF RATIONAL POINTS ON ALGEBRAIC VARIETIES

• Diophantine equations and heights. We will be studying here how fast the number of solutions of a system of equations can grow, if one first restricts the counting to solutions of height ≤ H, and then lets H grow. In order to define heights over general algebraic number fields, we need the following preparations.

Let K be a number field, ΩK = ΩK,f t ΩK,∞ the set of its places v represented as the union of finite and infinite ones. Kv denotes the respective completion of K.

For v ∈ ΩK,f , denote by Ov, resp. mv, the ring of integers of Kv, resp. its maximal ideal. The Haar measure dxv on Kv is normalised in such a way that the measure of Ov becomes 1. Moreover, for an archimedean v, the Haar measure will be the usual Lebesgue measure, if v is real, and for complex v it will be induced by Lebesgue measure on C, for which the unit square [0, 1] + [0, 1]i has volume 2. -XX-

∗ Let the map |.|v : Kv → R≥0 be defined by the condition d(λx)v = |λ|vdxv. ∗ Q Then for any λ ∈ K we have the following product formula: v |λ|v = 1. Let Pn be a projective space with a chosen system of homogeneous coordinates (x0 : x1 : ... : xn−1 : xn), n ≥ 1. Then we can define the exponential Weil height of a point n p = (x0(p): ··· : xn(p)) ∈ P (K) as Y h(p) := max{|x0(p)|v,..., |xn(p)|v}.

v∈ΩK

Because of the product formula, the height does not change, if we replace coordi- nates (x0 : ··· : xn) by (λx0 : ··· : λxn), λ ∈ K. -XXI-

• Height zeta functions. Let now (U, LU ) be a pair consisting of a projective variety U over K and an ample line bundle LU on it. Then we can define the height function hLv (p) on p ∈ U(K) using the same formula as above, but this time interpreting (xi) as a basis of sections in Γ(U, LU ). If h0 is another height, corresponding to a different choice of the basis of sections, LV then there exist two positive real constants C,C0 such that for all x,

Ch (x) ≤ h0 (x) ≤ C0h (x). LV LV LV

Now define the height zeta–function

X −s Z(V,LV , s) := hLV (x) . x∈V (K)

Let V ⊂ U be a locally closed subvariety of U, also defined over K. -XXII-

• Convergence boundary. Denote by σ(V,LV ) ∈ R the lower limit of positive reals σ for which Z(V,LV , s) absolutely converges if Re s ≥ σ.

We will call σ(V,LV ) the respective convergence boundary. Clearly, it is finite (because this is so for projective spaces), and non–negative whenever V (K) is infinite. Intuitively, we may say that V contains “considerably less” K–points than U, if

σ(V,LV ) < σ(U, LU ), and “aproximately the same” amount of K–points as U, if

σ(V,LV ) = σ(U, LU ). -XXIII-

• Example: accumulating subvarieties. Let V ⊂ U be a Zariski closed subvariety over K. If −1 card {x ∈ V (K) | hLV (x) ≤ H}· card {x ∈ (V \ U)(K)} | hLV (x) ≤ H} → 0 as H → ∞, then V is called an accumulating subvariety in U in the sense of [BaMa90], [FrMaTsch89].

Clearly, then σ(V,LV ) = σ(U, LU ). We will now describe a categorical environment appropriate for describing various versions of accumulation and connecting distributions with spectra. -XXIV-

3. RATIONAL POINTS, SIEVES, AND ASSEMBLERS

• Grothendieck topologies, sieves, and assemblers. (See [Za17a], [Za17b], [MaMar18]). Let C be a category with a unique initial object ∅. Two morphisms f1 : U1 → U and f2 : U2 → U are called disjoint, if U1 ×U U2 exists and is ∅.

Notice that if U1 ×U U2 exists, then U2 ×U U1 also exists, and these two relative products are canonically isomorphic, but in the case of disjoint pair we want this isomorphism to be unique. A sieve in C is its full subcategory C0 such that if f : V → U is a morphism in C, and U is an object of C0, then V is also an object of C0. Fixing an object U in C, can apply this notion to also to the category of morphisms f : V → U, or in other words to the “overcategory” C/U. Functors between the overcategories f ∗ : C/W → C/V induced by lifting morphisms f : W → V induce functors between sieves in the respective overcategories. -XXV-

This notion is convenient in order to define a Grohendieck topology on C: it is a collection of sieves J (U) in C/U, one for each object U of C, satisfying three axioms: (i) Any morphism f : W → V lifts to the map f ∗ : J (W ) → J (V ). (ii) The full overcategory C/U belongs to J (U) for any object of U of C. (iii) For any sieve S in J (U), any object f : V → U of this sieve, and any other sieve T in J (U), if we have f ∗(T ) ∈ J (V ), then T is in J (U). For any object U of a category with Grothendieck topology C (called also Grothen- dieck site) we can define the notion of covering family: it is a collection of morphisms {fi : Ui → U | i ∈ I} such that the full subcategory of C/U containing all morphisms in C factoring through fi belongs to the initial collection of sieves. -XXVI-

• Assemblers. (See [Za17a], [Za17b], [MaMar18]). An assembler is a small category C with a Grothendieck topology and initial object ∅. All morphisms in C must be monomor- phisms. Empty family is a covering of ∅.

Two morphisms f : V → U and g : W → U are called disjoint, if V ×U W = ∅. We also require that any two pairwise disjoint finite covering families of U admit a common refinement which is also a finite disjoint covering family. Assemblers themselves form a category, in which a morphism is a functor continuous wrt respective Grothenieck topologies, sending initial object to initial object, and disjoint morphisms to disjoint morphisms. Later, we will study assemblers related to the distribution of rational points, and to this end we will have to define formally certain sieves via point distribution. But now we will deal with the spectra associated to assemblers. -XXVII-

Let C is a Grothendieck site. Denote by C◦ its full subcategory of noninitial objects.

If we have a family of assemblers {Cx} numbered by elements x of a set X, we will W F ◦ denote by x∈X Cx the category whose non–initial objects are x∈X Ob Cx and to which one initial object is formally added. • From assemblers to Γ–spaces and spectra. (See [Za17a]; [MaMar18], Sec. 4.4). Starting with an assembler C, we can construct the following category W(C): (a) One object of W(C) is a map I → Ob (C) where I is a finite set, and the map lands in non–initial objects. We may write it as {Ai}i∈I ,Ai ∈ Ob (C).

(b) One morphism f : {Ai}i∈I → {Bj}j∈J consists of a map of finite sets f : I → J and −1 a family of morphisms fi : Ai → Bf(i) such that {fi : Ai → Bj : i ∈ f (j)} for each j ∈ J is a. disjoint covering family. -XXVIII-

• Proposition. (See [Za17a], Prop. 2.11.) (i) All morphisms in W(C) are monomorphims. (ii) If C has all pullbacks (i.e. is closed), then WC is closed as well.

(iii) Given a family of assemblers {Cx | x ∈ X} indexed by elements of a set X, denote by Q ⊕ W(Cx) the full subcategory of W(Cx) whose objects are families of objects of Cx for which all but finitely many of them are indexed by ∅. Consider the functor

_ Y P : W( Cx) → W(Cx) x∈X x∈X induced by the morphisms of assemblers Fy : ∨x∈X Cx → Cy that map each Cx to the initial object if x 6= y and identically to Cy if x = y. Then P induces an equivalence of categories

_ M W( Cx) → W(Cx). x∈X x∈X -XXIX-

Sketch of Proof. The main part of our argument is a detailed study of pullback squares.

Start with a morphism f : {Ai}i∈I } → {Bj}j∈J , and two more morphisms

g, h : {Ck}k∈K → {Ai}i∈I .

In order to show that f is a monomorphism we must check that if fg = fh, then g = h. Choose any k ∈ K and look at the respective commutative square with vertices Ck,Ag(k),Ah(k),Bfg(k). We must have g(k) = h(k), because otherwise fg(k) , fh(k) are disjoint. Remind that the nerve N of a category A is the simplicial set NA whose vertices are indexed by objects A of A; 1–simplices by morphisms A1 → A2 of A; and generally, n–simplices by sequences of morphisms A1 → · · · → An+1. Faces and degenerations (see 0.3 above) are determined via composition of morphisms in a pretty obvious way. The remaining statements easily follow from these remarks.  -XXX-

• Move to Γ–spaces, spectra, and K–theory. (Cf. Sec. 2.2 of [Za17a]). To move from here to Γ–spaces, start with the category Γ0 whose objects are finite sets n := {0, 1, 2, . . . , n} with base point 0, and morphisms are all maps sending 0 to 0 (as above, but with slightly changed notation). If X is a pointed set, and C an assembler, we can construct the assembler X ∧ C := W ◦ x∈X◦ Cx, where X := X \ {∗} . After one step further, we get the “operadic” smash product ∧ (see [SpWh53] and Sec. 1 above) enabling a passage from assemblers C to n n spectra Xn := NW(S ∧C), with structure maps induced by X∧N W(S ∧C) → N W(X∧C): _ M X ∧ N W(C) =∼ NW(C) → N ( W(C)) ≡ N W(X ∧ C). X◦ X◦ -XXXI-

I. Zakharevich defines the symmetric sequential spectrum K(C) as spectrum of simplicial sets in which the k–th space is given by the diagonal of the bisimplicial set k [n] → N W((S )n ∧ C) and writes

Ki(C) := πiK(C).

The following result (Theorem 2.13 in [Za17a]) furnishes the first justification of the intuition encoded in the word “assembler”.

• Theorem. Let C be an assembler. Then K0(C) is canonically isomorphic to the abelian group generated by (isomorphism classes of) objects [A] of C modulo the family of relations indexed by finite disjoint covering families {Ai → A | i ∈ I}: each such family produces the relation X [A] = [Ai]. i∈I -XXXII-

Sketch of Proof. The calculation of generators and relations for K0(C) can be reduced to making explicit the composition of functors

P −1 : W(C) ⊕ W(C) → W(C ∧ C) and µ : W(C ∨ C) → W(C).

−1 Here P sends a pair of objects ({Ai | i ∈ I}, {Bj | j ∈ J}) to the object {Ck | k ∈ I ∪J}, where for k ∈ I, resp. k ∈ J, we put Ck = Ak, resp. Ck = Bj. The map µ−1 is induced by the folding map of assemblers.

Finally, it remains to remark that relations in π0 are generated by morphisms {Ai → A | i ∈ I} → {A}.  -XXXIII-

We will now move to our main preoccupation: construction of assemblers related to the distribution of rational points on algebraic varieties. We will first of all define formally certain sieves via point distribution.

• Categories C(U, LU ). Let U be a projective variety over K and LU an ample rank 1 vector bundle on U over K.

By definition, objects of C(U, LU ) are locally closed subvarieties V ⊂ U also defined over K, and morphisms are the structure embeddings iV,U , or simply iV : V → U. Here ∗ we did not mention L explicitly, but it is natural to endow each V by LV := iV (LU ). Structure embeddings are compatible with these additional data so that we have in fact structure functors C(V,LV ) → C(U, LU ) which make of each C(V,LV ) a full sub- category of C(U, LU ) closed under precomposition, that is, a sieve .

We will call such categories C(U, LU ) geometrical sieves, and now introduce the arith- ar metical sieves C (U, LU ) in the following way. -XXXIV-

• Lemma. The family of those morphisms iV,U as above, together with their sources and targets, for which 0 < σ(V,LV ) < σ(U, LU ), ar forms a sieve in C(U, LU ) denoted C (U, LU ). Proof. If we have a two–step ladder of locally closed embeddings W ⊂ V ⊂ U such that 0 < σ(V,LV ) < σ(U, LU ) and 0 < σ(W, LW ) < σ(V,LV ), then of course 0 < σ(W, LW ) < ar σ(U, LU ), so that the composition of these embeddings is also a morphism in C (U, LU ).

Notice, that if V (K) is a finite set, then σ(V,LV ) = 0, but the converse is not true: σ(V,LV ) = 0 for any abelian variety V/K and for many other classes of V . A complete geometric description of this class of varieties seemingly is not known. ar • Arithmetic assemblers. Using sieves C (U, LU ), we can easily introduce the respective arithmetic assemblers CU : the relevant Grothendieck topology is simply the Zariski topology over K, and ∅ is the empty scheme. -XXXV-

The remaining two sections sketch two diverging paths leading from distribution of ar K–rational points U(K) in U to various further versions of the arithmetic sieve C (U, LU ). In the Sec. 4 we look at the more narrow class of varieties, Fano varieties, for which more precise data about behaviour of some heights are known, or at least, conjectured. In the Sec. 5 (restricted to K = Q), we consider rational points as a subset of adelic points, and try to go beyond heights by using new tools for studying the distribution of adelic points U(AQ) themselves. -XXXVI-

4. ANTICANONICAL HEIGHTS AND POINTS COUNT

• Anticanonical heights: dimension one. Let (U, LU ) be as above a pair consisting of a variety and ample line bundle defined over K, [K : Q] < ∞. Choose a exponential height function hL, and set for B ∈ R+

N(U, LU ,B) := card{x ∈ U(K) | hL(x) ≤ B}.

On page XIV, we based the definition of an arithmetical sieve upon an intuitive idea that iV : V → U belongs to this sieve, if the number of K–points on U is “considerably less” that such number on V . To make this idea precise, we used convergence boundaries. Below, we will use considerably more precise count of points in order to define subtler sieves on a more narrow class of varieties U, using counting functions themselves N(U, LU ,B) in place of convergence boundaries. Start with one–dimensional U. -XXXVII-

If U is a smooth irreducible curve of genus g, with nonempty set U(K), we have the following basic alternatives:

1 g = 0 : U = P ,LU = −KV ,N(U, LU ,B) ∼ cB.

r−1 g = 1 : U = an elliptic curve, with rank of P icard groupe r, N(U, LU ,B) ∼ c(log B) .

g > 0 : N(U, LU ,B) = const, if B is big enough. A survey of expected typical behaviours of multidimensional analogs can be found in the Introduction to [FrMaTsch89]. Below, our attention will be focussed upon Fano varieties, that is, varieties with −1 ample anticanonical bundle ωV , as a wide generalisation of the one–dimensional case g = 0. -XXXVIII-

• Anticanonical heights for Fano varieties. The most precise conjectural asymptotic formula for Fano varieties (or Zariski open subsets of them) with dense U(K) has the form −1 t N(U, ωU ,B) ∼ cB(log B) , t := rk P ic U − 1. It certainly is wrong for many subclasses of Fano varieties. On the other hand, it is (i) stable under the direct products; (ii) compatible with predictions of Hardy–Littlewood for complete intersections; (iii) true for quotients of semisimple algebraic groups modulo parabolic subgroups. We reproduce below some of the arguments [FrMaTsch89], Sections 1–2, proving these statements. -XXXIX-

• Direct products. Let (U, LU ) and (V,LV ) be data as above. We must study the ∗ ∗ behaviour of N(U × V,LU×V ) where LU×V := prU (LU ) ⊗ prV (LV ). In this context, we may restrict ourselves by consideration of logarithmic heights satisfying the exact equality

hLU×V (x, y) = hLU (x)hLV (y).

This will be applied to the case of anticanonical heights. We will now change notation without referring anymore to the specical properties of heights on Fano varieties etc, as in [FrMaTsch89]. Consider two infinite families of nondecreasing real numbers indexed by 1, 2,... : {λi} and {µj}. We allow each number to be repeated several times, so that they can have finite mulltiplicities. We can then form a new family: λµ := {λiµj} again ordered nondecreasingly. Put Nλ(B) := card {i | λi < B}, and similarly for Nµ,Nλµ. -XL-

Now “stability under the direct product” from 4.2 is a consequence of the following • Lemma. If s s−1 Nλ(B) = cλBlog B + O(Blog B), r r−1 Nµ(B) = cµBlog B + O(Blog B), then r+s+1 r+s Nλµ(B) = C cλcµBlog B + O(Blog B), where the constant C = C(r, s) is the Euler beta–function. -XLI-

Proof. Directly from definitions and assumptions, we get

Nλ(B/µi) Nλ(B/µi) X X r Nλµ(B) = Nµ(B/λi) = cµ B/λi log (B/λi) i=1 i=1

Nλ(B/µi) X r−1 +O( B/λi log (B/λi)) (4.1) i=1 Since the error term has the same structure as the main one, with r replaced by r − 1, we can apply this formula inductively, and get the following expression for the main term N X r cλcµB a(j)B/j log (B/j) (4.2) j=1 -XLII-

where a(j) := card {i | λ1 + j ≤ λi < λ1 + j + 1},N := [B/µ1 − λ1] + 1. So the main term of (4.1) can be rewritten as

N Z j+1 X r −2 s cλcµB j log j x log (B/x)dx. j=1 j

After approximating the sum by the integral, we get the expected result.  -XLIII-

• Hardy–Littlewood method and complete intersections. Below we will sketch, following Igusa, methodology and results of applications of Hardy–Littlewood method in the setup of Fano complete intersections in projective spaces, explained in [FrMaTsch89]. Besides showing the “typical” asymptotic behaviour for some of them, they can serve as an example of study of distribution of rational points in adelic spaces. Later we will survey some recent developments in this direction, cf. Section 5 below. -XLIV-

n Let P be a projective space with a fixed system of homogeneous coordinates (x0 : x1 : ··· : xn) as in 2.1 above, K a base field of finite degree over Q. Consider m ≥ 1 forms fi ∈ K[x0, . . . , xn] and put di := deg fi. Denote by V the projective variety over K which is the nonsingular complete intersection of hypersurfaces fi = 0. Then its anticanonical line bundle is

−1 ωV = OV (n + 1 − d0 − · · · − dm).

Therefore V is a Fano variety, iff n + 1 > d0 + ··· + dm. Finally, how summation over points may be approximated by integration over adelic spaces, is briefly explained in Proposition 4 of [FrMaTsch89] and Sec. 5 below. -XLV-

• Generalised flag manifolds. The last class of Fano varieties we considered in [Fr- MaTsch89] consists of generalised flag manifolds V = P \G, where G is a semisimple linear algebraic group, and P is a parabolic subgroup, both defined over K. For con- venience, we assumed moreover that P contains a fixed minimal parabolic subgroup P0. Denote by π : G → V the canonical projection. Let X ∗(P ) be the group of characters of P defined over K. Each character χ ∈ X ∗(P ) defines a line bundle on V that we will denote Lχ. Its local sections come from those local functions on G upon which the left multiplication by p ∈ P lifts to the multiplication by χ(p).

Anticanonical line bundle is located among Lχ. It is denoted L−2ρ in Sec. 2 of [FrMaTsch89]. -XLVI-

Now comes the main part of the construction: explicit description of the anticanon- ical height. Let AK be the adele ring of K. Q There exists a maximal compact subgroup K = v Kv ⊂ AK such that G(AK ) = P0(AK )K, where P0 is the fixed minimal parabolic subgroup. Each bundle Lχ ⊗ Fv on V ⊗Kv is endowed by a canonical Kv–invariant v–adic norm: if s is a local section of Lχ which is a lift of local function f, we put |s|v = |f(k)|v and define the height function by Y −1 hχ = hLχ := |s|v . v Using this description, we can identify the anticanonical height zeta–function of V with one of the Eisenstein series. We omit further details. -XLVII-

• Heights with respect to more general line bundles LU on Fano varieties. In [BaMa90] it was suggested that only a slight generalisation of formula on page XXXVIII should be “typical” (although valid in a much more restricted set of cases):

β t N(U, LU ,B) ∼ cB (log B) , t := rk P ic U − 1.

As was argued in [BaMa90], β should be defined by the relative positions of LU and −KU in the cone of pseudo–effective divisors of U: see precise conjectures there. Sh. Tanimoto in [Ta19] provided arguments proving various inequalities for these numbers related to the conjectures in [BaMa90]. Finally, the subtlest information about such asymptotics is given by several conjec- tures and proofs regarding exact value of the constant c. -XLVIII-

• From asymptotic formulas to sieves. If we restrict ourselves by those V for which we can define a Grothendieck topology, objects of which satisfy strong asymptotic formulas dis- cussed above, or their weaker versions, then we can try to define sieves in it by some inequalities weakening the inequalities between convergence boundaries, such as

N(V,LV ,B)/N(U, LU ,B) = o(1) or even, for β(U, LU ) = β(V,LV ),

t(V,LV ) < t(U, LU ). where t(V,LV ) and β(V,LV ) refer to formula on previous page. -XLIX-

5. SIEVES “BEYOND HEIGHTS” ?

• Thin sets and Tamagawa measures. Below we survey recent attempts to define geome- try of subsets of rational points of V (K) containing “considerably less” points than V . From our viewpoint, these definitions should also be tested on compatibility with our philosophy of sieves and assemblers. Below we adopt the framework of [Pe18] which was further studied in [Sa20] (with additional restriction K = Q). The paper [Sa20] starts with Conjecture 1.1 called “Modern formulation of Manin’s conjecture”, and involves several shifts from the setup in our previous Section 4: (i) Summation over points x of height ≤ B is replaced by the averaging of the measures δx of the same points embedded into the adelic space V (AQ). This averaging is done using the Tamagawa measure. -L-

(ii) The definition of the respective Tamagawa measure τ assumes that V is geo- metrically integral smooth projective Fano variety with rank r of Picard group. Let

V be its proper integral model over Z, and L(s, P ic VQ) and the respective local zetas Lv are zetas of lattices.

Namely, for any place w∈ / S, denote by qw the cardinality of the residual field of w, and by (w, L/K) the linear operator upon P ic (V ), lift of the Frobenius (we now write

V in place of VQ ). Then

−1 −s Lv(s, P ic V ) := det(1 − qw (w, L/K) | P ic (V )), and r Y −1 τ := (lims→1(s − 1) L(s, P ic VQ)) Lv(s, P ic VQ) ωv, v where ωv is defined by the natural measure on local non–archimedean points of V or archimedean volume form. -LI-

(iii) Finally, define the numbers α(V ) and β(V ) by

eff ∨ α(V ) := r vol {y ∈ ((Pic(V ) ⊗ R) ) | KV · y ≤ 1}, and 1 β(V ) := card H (Gal(Q/Q), Pic VQ). Then the modern formulation of the conjecture on the number of rational points of bounded height on a Fano variety, according to [Pe95], [BaTsch98], [Pe03], [Sa20] can be stated as follows. Let f : U → V be a morphism of geometrically integral smooth projective varieties. Call it a thin morphism, if the induced map U → f(U) is generically finite of degree 6= 1. -LII-

• Conjecture. There exists the complement W to the union of a finite family of thin morphisms such that we have an exact formula for weak limit of the form

1 X lim δ = α(V )β(V )τ Br, B→∞ B(logB)r−1 x x∈W (Q) H(x)

Br where τ is the restriction of Tamagawa measure on the subset of V (AQ) on which the Brauer– Manin obstruction vanishes. Regarding Brauer–Manin obstruction, see the monograph [CThSk19] and many references therein. For a class of varieties for which this conjecture is valid, we obtain interesting new possibilities for defining sieves. This remains to be studied. -LIII-

References

[BaMa90] V. Batyrev, Yu. Manin. Sur le nombre des points rationnels de hauteur born´ee des vari´et´esalg´ebriques. Math. Ann. 286 (1990), pp. 27–43. [BaTsch98] V. Batyrev, Yu. Tschinkel. Tamagawa numbers of polarized algebraic vari- eties. Ast´erisque251 (1998), pp. 299–340. [BlBrDeGa20] V. Blomer, J. Br¨udern,U. Derenthal, G. Gagliardi. The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one. arXiv: 2004.09357, 76 pp. [CThSk19] J.-L. Colliot-Th´elene,A. Skorobogatov. The Brauer–Grothendieck group. imperial.ac.uk 2019, 360 pp. [DePi19] U. Derenthal, M. Pieropan. The split torsor method for Manin conjecture. arXiv:1907.09431, 36 pp. [FMa20] F. Manin. Rational homotopy type and computability. arXiv:200710632, 18 pp. [FrMaTsch89] J. Franke, Yu. Manin, Yu. Tschinkel. Rational points of bounded height on Fano varieties. Invent. Math., 95, no. 2 (1989), pp. 421–435. -LIV-

[GeMa03] S. Gelfand, Yu. Manin. Methods of homological algebra. Second Ed., Springer Monograph in Math., (2003), xvii + 372 pp. [LeSeTa18] B. Lehman, A. Sengupta, S. Tanimoto. Geometric consistency of Manin’s conjecture. arXiv:1805.10580 [Ly99] M. Lydakis. Smash products and Γ–spaces. Math. Proc. Cambridge Phil. Soc., 126 (1999), pp. 311–128. [MaMar18] Yu. Manin, M. Marcolli. Homotopy types and geometries below Spec Z. In: Dynamics: Topology and Numbers. Conference to the Memory of Sergiy Kolyada. Contemp. Math., AMS 744 (2020). arXiv:math.CT/1806.10801. [Pe95] E. Peyre. Hauteurs et mesures de Tamagawa sur les vari´et´esde Fano. Duke Math. J. 79 (1995), pp. 101–218. [Pe03] E. Peyre. Points de hauteur born´ee,topologie ad´elique et mesures de Tamagawa. J. Th´eor.Nombres Bordeaux 15 (2003), pp. 319–349. [Pe17] E. Peyre. Libert´eet accumulation. Documenta Math., 22 (2017), pp. 1616– 1660. [Pe18] E. Peyre. Beyond heights: slopes and distribution of rational points. arXiv:1806.11437, 53 pp. [Pe19] E. Peyre. Les points rationnels. SMF Gazette, 159 -LV-

[Sal98] P. Salberger. Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Ast´erisque251 (1998), pp. 91–258. [Sa20] W. Sawin. Freeness alone is insufficient for Manin–Peyre. arXiv:math.NT/2001.06078. 7 pp. [Sc01] S. Schwede. S–modules and symmetric spectra. Math. Ann. 319 (2001), pp. 517–532. [Se74] G. Segal. Categories and cohomology theories. Topology, vol. 13 (1974), pp. 293–312 . [Ta19] Sh. Tanimoto. On upper boundaries of Manin type. arXiv:1812.03423v2, 29 pp. [SpWh53] E. H. Spanier, J. H. C. Whitehead. A first approximation to homotopy theory. Proc. Nat. Ac. Sci. USA, 39 (1953), pp. 656–660. [Za17a] I. Zakharevich. The K–theory of assemblers. Adv. Math. 304 (2017), pp. 1176–1218 .

[Za17b] I. Zakharevich. On K1 of an assembler. J. Pure Appl. Algebra 221, no.7 (2017), pp. 1867–1898 .

Yuri I. Manin, Max-Planck-Institut f¨urMathematik, Vivatsgasse 7, 53111 Bonn, Germany. e-mail: [email protected] -LVI-

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