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Rational points of algebraic varieties: a homotopical approach Yu. I. Manin Max-Planck-Institut für Mathematik, Bonn, Germany
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Izvestiya: Mathematics, 2023, Volume 87, Issue 3, Pages DOI: https://doi.org/10.4213/im9315e 
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     IntroductionIn the earlier works (cf. [1] and a brief summary [2]), it was found that $k$-rational points on projective varieties $V/k$ (with $[k:\mathbf{Q}] < \infty$) are generally not “equidistributed” : they tend to accumulate on some subvarieties which are locally closed in the Zariski topology (cf. also [3] and [4]). Precise statements, both of definitions and theorems, and also conjectures, vary. They depend on the choice of a class of varieties, ground fields, and the questions asked. Especially important is the choice of definition of “accumulation”. Here we consider various versions of accumulation based upon counting points of height ${\leqslant}\,H$ with big $H$. In this respect, we follow generally the survey [5], but the main new material is related to various structures of homotopical and homological algebra. In the paper [6] the interested reader will find a very detailed description of the tools of homological and homotopical algebra, applicable in this environment. We mainly focus on the class of projective varieties of dimension ${\geqslant}\,1$ whose set of $k$-points is dense in $V(k)$. In the last section, we consider also varieties with finite sets $V(k)$ and discuss obstructions, in particular the applicability of the tools of [6] in this setting. § 1. Sieves and assemblersWe start with a survey of Grothendieck topologies on various categories, in full detail explained in Chapters 16 and 17 of [7]. Let $\mathcal{C}$ be a category. 1.1.Definition. A sieve over an object $U$ of $\mathcal{C}$ is a subset of morphisms $\mathcal{S}$, $V\to U$, closed with respect to precompositions with morphisms in $\mathcal{C}$: if $V\to U$ belongs to $\mathcal{S}$, then for any $W\to V$, the composition $W\to V \to U$ also belongs to $\mathcal{S}$. Denote the family of sieves over $U$ by $\mathcal{S}_U$. As explained in [7], there is a natural bijective correspondence between sieves in $\mathcal{S}_U$ and subobjects of the category of contravariant functors $\mathcal{C}^{\mathrm{op}} \to S$. We always work with small categories: see explanations in [7; Section 1.4 and further Definition 16.1.2 on p. 390]. 1.2.Definition. A Grothendieck topology on $\mathcal{C}$ is a family of sieves $\mathcal{S} \mathrm{Cov}_U$, one for every object $U$ of $\mathcal{C}$, called coverings of $U$, that satisfies the following restrictions. (i) Any isomorphism $U' \to U$ belongs to this family. (ii) Let $\mathcal{S}_1, \mathcal{S}_2$ be sieves such that $\mathcal{S}_2$ is a sieve over $U$ and $\mathcal{S}_1$ is a union of sieves over all objects of $\mathcal{S}_2$. Then $\mathcal{S}_2$ also belongs to $\mathcal{S}\mathrm{Cov}_V$. (iii) Let $U\to V$ be a morphism in $\mathcal{C}$. Then the family of lifts to $V$ of all sieves in $\mathcal{S}\mathrm{Cov}_U$ belongs to $\mathcal{S} \mathrm{Cov}_V$. Here a lift of a morphism $S\to U$ is defined as a natural map $S\times_V U\to U$. (iv) Consider two sieves over $U$: $S$ and $S'$. Assume that $S' \in \mathcal{S}\mathrm{Cov}_U$, and that for any morphism $V\to U$ in $S'$, the lifts of these coverings to $V$ also belong to $S'$. Then $S$ is a covering of $U$. 1.3. Examples: Zariski, étale, and flat topologiesThe definition and functorial properties of assemblers, axiomatized in [8], are motivated by examples of several Grothendieck topologies in algebraic geometry. As illustrative examples, we choose three: Zariski topology, étale topology and flat topology. In each case the central notion is that of covering. A (finite) covering of $U$ is a surjective morphism $\bigsqcup_i U_i \to U$, defined up to renumbering the components $U_i$, and such that restrictions $U_i \to U$ are open embeddings of Zariski subvarieties, or their étale, resp. flat, covers. The main property of such coverings is that their system is stable with respect to iterations: if, for each $i$, we are given a family of coverings $\bigsqcup_j U_{ji} \to U_i$, then $\bigsqcup_{i,j} U_{ji} \to U$ is a covering of the same type. If we do not make any preliminary assumptions about nature of objects of our basic category $\mathcal{C}$, but wish to imitate the properties of coverings discussed above, then we may start with a symmetric monoidal category $(\mathcal{C}, \sqcup)$, endowed with the identity object $\varnothing$ (as in [8; Section 2.1], we will also assume that it is unique). A minimal requirement, imposed in [8; Definition 2.2] on the family of all morphisms $U_i \to U$ used in the coverings above, is that it forms a sieve. 1.4. AssemblersA category $\mathcal{C}$ with Grothendieck topology will be called a Grothendieck site. 1.4.1.Definition. An assembler $\mathcal{C}$ is a small Grohendieck site, in which all morphisms are monomorphisms, and which is endowed with the following additional structures and conditions. (i) $\mathcal{C}$ has an initial object denoted by $\varnothing$, and the empty family of objects is a covering family of $\varnothing$. (ii) Any two disjoint covering families of an object in $\mathcal{C}$ have a common refinement, which is a disjoint covering family. Here a family of morphisms $\{ f_i\colon A_i \to A\}$ is said to be disjoint if for $i\neq i'$ the lift of $f_{i'}$ exists and is empty. 1.5. Example: the assembler of all subsetsConsider the category whose objects are subsets of a fixed set and morphisms are their natural embeddings. A covering family of a subset is a family whose union is that subset. Obviously, it is an assembler. 1.6. Example: the assembler of half-open intervalsConsider the category $\mathcal{C}$ whose objects are finite unions of pairwise disjoint half-open intervals
$$
\begin{equation}
(a,b],\quad \text{with}\quad a,b\in \mathbf{Q},\ \ 0< a< b
\end{equation}
\tag{1.1}
$$
and morphisms are natural embeddings. Finite union of such objects considered as a bi-functor defines a monoidal structure on this category. If the intersection $U:= U_1\cap U_2$ is non-empty, then it is also an object of $\mathcal{C}$. We formally add the empty “half-open interval” $\varnothing$ to this category, so that the intersection is now defined for any finite family of objects. With respect to union, $\varnothing$ becomes the double-sided unit. Again, we get an assembler. 1.7. The category of assemblers(See [8; Section 2].) Given two assemblers $\mathcal{C}$ and $\mathcal{D}$, a morphism $F\colon \mathcal{C} \to \mathcal{D}$ is, by definition, a functor continuous with respect to their Grothendieck topologies, sending the initial object of $\mathcal{C}$ to the one of $\mathcal{D}$, and disjoint morphisms in $\mathcal{C}$ to disjoint morphisms in $\mathcal{D}$. Assemblers and their morphisms form the category $\mathbf{Asm}$, with arbitrary products and coproducts ([8; Lemma 2.8]). One can also consider the category $c\mathbf{Asm}$ of closed assemblers. Objects of this category are assemblers having all pullbacks. Morphisms are functors preserving pullbacks. It turns out that $c\mathbf{Asm}$ also has arbitrary products and coproducts. § 2. Height assemblers2.1. Heights of rational points(Cf. [6; Section 2.1].) For any field $k$ with $[k:\mathbf{Q}\,] < \infty$, its set $\Omega_k$ of places $v$ is the disjoint union of finite and infinite ones: $\Omega_k = \Omega_{k,f} \sqcup \Omega_{k,\infty}$. Let $k_v$ be the respective completion of $k$. The completions at infinite $v$ have natural Haar measures. We will normalize them as follows: for $k_v = \mathbf{R}$, $dx_v$ denotes the usual Lebesgue measure with $\int_{[0,1]} dx_v= 1$, and for $k_v = \mathbf{C}$ the normalization will be $\int_{[0,1]+i[0,1]} dx_v = 1$. Haar measures of the completions at finite $v$ also have natural normalizations, for which the subrings of $v$-integers $\mathcal{O}_v$ have measure $1$. For any $v$, we will introduce the map $|\,{\cdot}\,|_v\colon k_v \to \mathbf{R}^*_{\geqslant 0}$ such that $d(\lambda x)_v = |\lambda |_v \, dx_v$. Then we can define the (exponential) Weil height for a projective space $\mathbf{P}^n$ with a chosen and fixed system of homogeneous coordinates $(x_0 : x_1 : \dots : x_n)$: the point $(x_0(p) : \dots : x_n(p))$ has height
$$
\begin{equation*}
h(p) := \prod_{v\in \Omega_k} \mathrm{max} \{ |x_0(p)|_v, \dots, |x_n(p)|_v \}.
\end{equation*}
\notag
$$
If we replace $(x_0: \dots : x_n)$ by $(\lambda x_0 \dots \lambda x_n)$, the height will not change. More generally, if we choose a different system of homogeneous coordinates and produce the respective height $h'$, we can always find two positive real constants $C$, $C'$ such that for all $p$ we will have 2.2. Height zetas and convergence boundariesAny non-empty subset $U\subset \mathbf{P}^n(k)$ determines its height zeta-function Define its convergence boundary as
$$
\begin{equation}
\sigma (U) := \inf \{ \sigma \in \mathbf{R}_{\geqslant 0}\mid Z(U,s)\text{ absolutely converges for }\operatorname{Re}s \geqslant \sigma \}.
\end{equation}
\tag{2.2}
$$
For example, $\sigma (\mathbf{P}^n(k)) = n+1$ ([1; Section 1.3]) and $\sigma (U) = 0$ if $U$ is empty or if $U$ is the set of all $k$-points of an abelian subvariety in $\mathbf{P}^n$. From (2.1) one easily sees that convergence boundaries do not depend on the choice of homogeneous coordinates on $\mathbf{P}^n$. In many arithmetic applications, it is convenient to include the following additional restrictions and information into the notation for height zetas. Write $V$ for a locally closed (in Zariski topology) subvariety of a projective variety $U$ (over $k$), and $L_U$ for an ample line bundle on it. We can then define the height $h_{L_V} (p)$ for points $p\in V(k)$, depending on the choice of a basis of sections of $L_U$, restricted to $V$. Changing the basis of sections, we generally get a different height, but related to the previous one as in (2.1). The respective height zeta will be denoted by $Z(V, L_V, s)$, and its convergence boundary will be denoted by $\sigma (V, L_V)$. After having established the connections between the asymptotic bounds as (2.2) and the asymptotic behaviour of the number of $k$-points of height ${\leqslant}\,H$, we can now introduce arithmetic assemblers as in [6; Proposition 3.6.2]. 2.2.1.Proposition. (i) Let $\mathcal{C}_U$ be the category whose objects are locally closed subvarieties of $U$ and whose morphisms can be described as follows. Open embeddings are Zariski open subsets $W\,{\subset}\,V$ with $0\,{<}\,\sigma (W,L_W)\,{<}\,\sigma (V,L_V)$. Closed embeddings are complements to open ones. Finite compositions of open and closed embeddings are all morphisms. The Grothendieck topology is generated by the disjoint covering families of the form $\{W\subset U,\, U \setminus W \subset U\}$. (ii) Then the category $\mathcal{C}_U$ is an assembler. 2.3. Height morphisms of assemblersGenerally, a morphism $F\colon \mathcal{C}_1 \to \mathcal{C}_2$ of assemblers is a functor which is continuous with respect to their Grothendieck topologies. Moreover, it must send the initial object of $\mathcal{C}_1$ to the initial object of $\mathcal{C}_2$, and disjoint morphisms in $\mathcal{C}_1$ to disjoint morphisms in $\mathcal{C}_2$. Consider now the map $H\colon \mathcal{C}^{\mathrm{ar}} (k,n) \to \mathcal{C}^{\mathrm{rat}}$ sending each subset $U\subset \mathbf{P}^n(k)$ to $(0, \sigma (U)]$ (cf. (2.2) above) regarded as an object of $\mathcal{C}^{\mathrm{rat}}$. 2.3.1.Proposition. The map $H$ naturally extends to a morphism of assemblers. Proof. This map sends any finite subset of $\mathbf{P}^n(k)$, in particular, its initial object $\varnothing$, to the “empty half-open interval $(0, 0\,]$”.
Generally, we extend it to objects of $\mathcal{C}^{\mathrm{ar}}(k,n)$ by sending a subset $U\subset \mathbf{P}^n(k)$ to the half-open interval $(0, \sigma (U)]$. The proposition is proved. § 3. Homotopy formalism and assemblers3.1. Sequential spectra and $\Gamma$-spaces(See [9]–[11].) Let $\Gamma$ be the category of based finite sets $n^+ := \{0,1, \dots,n\}$, with morphisms sending $0$ to $0$. A $\Gamma$-space is a functor from $\Gamma$ to based simplicial sets, sending $0^+$ to the point. The category of $\Gamma$-spaces, with natural functors as morphisms, will be denoted by $\mathcal{GS}$. 3.1.1.Theorem. (i) The category of $\Gamma$-spaces is a symmetric monoidal category with respect to a natural functor $\wedge \colon \mathcal{GS} \times \mathcal{GS} \to \mathcal{GS}$. (ii) Using $\wedge$, one can define a canonical isomorphism of functors in three variables
$$
\begin{equation*}
\operatorname{Hom}_{\mathcal{GS}} (F\wedge F', F'') \simeq \operatorname{Hom}_{\mathcal{GS}} \bigl(F, \operatorname{Hom}_{\mathcal{GS}} (F', F'')\bigr).
\end{equation*}
\notag
$$
In all contexts above, the wedge product $\wedge$ is defined as a natural extension of the wedge product of two pointed sets: $(X,*_X)\wedge (Y, *_Y)$ is the result of contraction of the subset $(X\times *_Y)\cup (*_X\times Y)$ in $X\times Y$. 3.1.2.Proposition. The monoidal category of $\Gamma$-spaces has the unit object called the sphere spectrum $\mathbf{S}$. 3.2. Categories and their nervesLet $\mathcal{A}$ be a category. Its nerve $\mathcal{NA}$ is the simplicial set whose vertices (resp. $n$-simplices) are bijectively indexed by the objects $A$ of $\mathcal{A}$ (resp. the sequences of morphisms $A_1 \to \dots \to A_{n+1}$ for $n\geqslant 1$). The face maps $\partial^j_n$ and the degeneration maps $\sigma^i_n$ are defined in a pretty obvious way. We now apply this construction to the simplest assembler among the ones introduced above, in § 1.5: the category $P(X)$ of all pointed subsets of a finite pointed set $(X, x_0)$, with natural embeddings as morphisms. Consider in addition a category $\mathcal{C}$, endowed with a categorical direct sum $\oplus$ and zero object. Starting with $(P(X), \mathcal{C})$, define the notion of a summing functor: it is a functor $\Phi \colon P(X) \to \mathcal{C}$ such that if $A$, $A'$ are subsets in $X$ with intersection $\{\varnothing \}$, then $\Phi (A\oplus A') = \Phi (A) \oplus \Phi (A')$. In particular, $\Phi (\{x_0\})= 0$. The summing functors themselves form objects of a category $\Sigma_{\mathcal{C}} (X)$, whose morphisms are invertible natural transformations. Finally, define the $\Gamma$-space $F_{\mathcal{C}}$ by assigning the pointed simplicial set nerve $\mathcal{N}(\Sigma_{\mathcal{C}(X)})$ to every finite pointed set $(X,c_0)$. § 4. Obstructions4.1. Stratifications and sievesAccording to [12; Remark 3.6], a stratification of a variety $V$, indexed by a finite partially ordered set $(I, \leqslant )$, is a family of locally closed subsets $\{U_i \subseteq V,\, i\in I \}$ such that $U_i\cap U_j = \varnothing$ for $i\ne j$ and the closure of any $S_i$ in $V$ equals $\bigcup_{j\leqslant i} S_j$. We can omit the restriction of finiteness of $I$ in this definition. Let $\mathcal{C}_V$ be the category of locally closed subvarieties of $V$, with natural embeddings as morphisms. Since it is closed with respect to finite unions, intersections and complements, any finite subset of its objects is contained in a stratification of the union of all objects of this subset. Thus, $\mathcal{C}_V$ is a sieve, which in a very definite sense is a limit of stratifications. 4.2. Generalized obstructionsWe will work with the categories $\operatorname{Var}_k$ of algebraic $k$-varieties or $\operatorname{Sch}_k$ of $k$-schemes. Here $k$ is a field of algebraic numbers, an extension of $\mathbf{Q}$ of finite degree, as in § 2.1, or a more general global field. Let $\mathbf{A}_k$ be the ring of adèles of $k$. Below, we will reproduce a functorial description of obstructions developed in [12; Section 2.1]. Consider a covariant functor $\omega\colon \operatorname{Var}_k \to \mathrm{Sets}$, which is a subfunctor of the functor $\operatorname{Var}_k \to \mathrm{Sets}$ sending $V$ to $V(\mathbf{A}_k)$. Following [12], we will write $V(\mathbf{A})^{\omega}$ in place of $\omega(V)$. Such a functor $\omega$ is called a generalized obstruction to the local-global principle if we have $V(k) \subseteq V(\mathbf{A}_k)^{\omega}$ for any object $V$ of $\operatorname{Var}_k$. We need also take into account the generalizations $\mathbf{A}_{k,S}$ of the ring of adèles, corresponding to subsets $S$ of the set of places of $k$. We denote the projection of $V(\mathbf{A}_k)^{\omega}$ to $V(\mathbf{A}_{k,S})$ by $V(\mathbf{A}_{k,S})^{\omega}$. Such a triple $(\omega, S, k)$ is called an obstruction datum. 4.3. VSA: Very Strong Approximation principleWe say that $V$ satisfies VSA for the obstruction datum $(\omega, S,k)$ if $V(k)=V(\mathbf{A}_{k,S})^{\omega}$. 4.4. Examples of obstruction data: descent obstructions(See [12; Section 2.3].) Given a contravariant functor $F\colon \operatorname{Sch}_k \to \mathrm{Sets}$, consider the following commutative diagram for every $X$ in $\operatorname{Sch}_k$: where the vertical arrows are pullbacks.Write $X(\mathbf{A}_k)^A$ for the subset of $X(\mathbf{A}_k)$ whose image in $\prod_vF(k_v)$ is contained in the image of $\mathrm{loc}$. Finally, denote the obstruction set $\bigcap_{A\in F(X)} X(\mathbf{A}_k)^A$ by $X(\mathbf{A}_k)$. For a family $\mathcal{F}$ of such functors put
$$
\begin{equation}
X(\mathbf{A}_k)^{\mathcal{F}} := \bigcup_{F\in \mathcal{F}} X(\mathbf{A}_k)^F.
\end{equation}
\tag{4.2}
$$
It is easy to check that the functor $X \mapsto X(\mathbf{A}_k)^{\mathcal{F}}$ is a generalized obstruction. We will focus on the case when $\mathcal{F}$ in (4.2) is the family of all finite smooth abelian group schemes over $k$ and denote the left-hand side of (4.2) by $X(\mathbf{A}_k)^{f-ab}$. This is a family of data called generalized descent obstructions in [12]. We now state one of the main (unconditional) new results of [12]. 4.5.Theorem ([12; Theorem 3.3]). Let $k$ be an imaginary quadratic or totally real number field. Then the obstruction datum $(f-ab, f, k)$ satisfies the very strong approximation principle (VSA). An interested reader can find several variations on this subject in [12], including conditional proofs of VSA which are based upon Grothendieck’s section conjecture. § 5. Tamagawa numbers and stratifications5.1. A class of Fano varietiesHere we briefly sketch some constructions, used by E. Peyre ([13]) and further developed by W. Sawin ([14]), in which stratifications from [12] appear through the so-called Tamagawa measures. In this section, $V$ is assumed to be a geometrically integral smooth projective Fano variety, with Zariski dense set $V(\mathbf{Q})$, satisfying the following additional restrictions ([13; Hypotheses 3.27]). (a) A certain multiple of the anticanonical class $-K_V$ is represented by the sum of an ample divisor and a divisor with normal crossings. (b) $H^1(V,O_V) = H^2(V,O_V) = \{0\}$. (c) The Brauer group of $\overline{V}:= V\otimes\overline{\mathbf{Q}} $ is trivial. The Picard group $\operatorname{Pic}\overline{V}$ has no torsion. (d) The closure of the cone of effective divisors $C^1_{\mathrm{eff}} (\overline{V})$ is generated by the classes of a finite set of effective divisors. 5.2. Local factors of the $L$-function of $V$There is a finite extension $k$ of $\mathbf{Q}$, over which the Picard group of $V$ becomes the same as $\operatorname{Pic}\overline{V}$. Write $S$ for a finite set of places of $\mathbf{Q}$ containing $\infty$ and all places of bad reduction of $V$. For each prime $p$ outside $S$, we can lift the respective Frobenius element to an element $(p, k/\mathbf{Q})\in \operatorname{Gal} (\overline{\mathbf{Q}}/\mathbf{Q})$, defined uniquely up to conjugation. The respective (inverted) local factors of the $L$-function are then
$$
\begin{equation*}
L_p(s)^{-1} := \det \bigl(\mathrm{id}-p^{-s} (p, k/\mathbf{Q}) \mid \operatorname{Pic}(\overline{V})\bigr).
\end{equation*}
\notag
$$
For complex $s$ with $\operatorname{Re} s > 1$, the product
$$
\begin{equation*}
L_S\bigl(s, \operatorname{Pic} (\overline{V})\bigr) := \prod_{p \notin S} L_p(s)
\end{equation*}
\notag
$$
converges. We also put $\lambda_p := L_p(1)^{-1}$ if $p \notin S$, and $\lambda_w := 1$ for all other places $w$ (including the Archimedean place). 5.3. From convergence boundaries to Tamagawa measuresOur constructions of assemblers based upon comparison of convergence boundaries (see (2.2)) will now be replaced by constructions of stratifications involving several shifts of setups. (a) Assemblers themselves are replaced by stratifications, which can be considered as finite approximations to assemblers; see § 4.1. (b) The height zetas of varieties, involving summation over all rational points $V(k)$, are replaced by averaging of delta-measures over $V(k)$ of these points in the space of adèles $V(\mathbf{A}_k)$ (we recall that $k=\mathbf{Q}$ in this section). (c) Numbers $r=r(V)$, $\tau(V)$, $\alpha(V)$, $\beta(V)$. Let $r=r(V)$ be the rank of the Picard group of $V$. Given a place $v$ of $k= \mathbf{Q}$, we write $\omega_v$ for the integral of the measure that was called $dx_v$ in § 2.1 and put
$$
\begin{equation*}
\tau (V) := \bigl(\lim_{s\to 1} (s-1)^r L\bigl(s, \operatorname{Pic}(\overline{V})\bigr)\bigr) \prod_v \lambda_v\omega_v.
\end{equation*}
\notag
$$
Finally, put
$$
\begin{equation*}
\begin{aligned} \, \alpha(V) &:= r\operatorname{vol} \{ y\in ((\operatorname{Pic} (V)\otimes\mathbf{R})^{\mathrm{eff}})^{\wedge}\mid {-K_V}\cdot y \leqslant 1\}, \\ \beta (V) &:= \operatorname{card} H^1\bigl(\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}), \operatorname{Pic} (\overline{V})\bigr). \end{aligned}
\end{equation*}
\notag
$$
We now write $\tau^{\mathrm{Br}}$ for the restriction of the Tamagawa measure to the subset of $V(\mathbf{A}_{\mathbf{Q}})$ where the Brauer–Manin obstruction vanishes. 5.3.1.Conjecture. There exists a finite family of maps $f_i\colon Y_i \to X$ with the following properties. (i) Each $f_i$ is generically finite of degree $\neq 1$. (ii) Let $W$ be the complement of $\bigcup_i f_i(Y_i(\mathbf{Q}))$. Then
$$
\begin{equation*}
\lim B^{-1}(\log B)^{1-r} \sum_{x\in W(\mathbf{Q}),\, H(x) < B} \delta_x = \alpha(V)\beta(V) \tau^{\mathrm{Br}}.
\end{equation*}
\notag
$$
Various particular cases, studied/proved in [12]–[14] and other references, show to which degree they provide us with subtler information than just convergence boundaries of height zetas.
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