The motivic Poisson formula and motivic height zeta functions

The motivic Poisson formula and motivic height zeta functions Antoine Chambert-Loir Université Paris 7 Denis Diderot SIMONS SYMPOSIA Geometry over non-closed fields, April 17 23, 2016. p. 1

Contents 1 Introduction Heights in number theory Geometric height zeta functions Grothendieck groups of varieties Statement of the theorem 2 The motivic Poisson formula of Hrushovski Kazhdan Introduction Grothendieck rings of varieties with exponentials Local Schwartz Bruhat functions Global Schwartz Bruhat functions The formula of Hrushovski Kazhdan 3 Proof of the theorem The arithmetic analogue The motivic case Further developments and open questions Introduction. p. 2

The goal of the talk is to discuss the application of the Poisson formula to the study of height zeta functions of some algebraic varieties. Two frameworks: arithmetic case (number fields) estimating the behaviour of the number of rational points of bounded height. The Poisson formula is the generalization to topological groups of the classical formula f(m) = f(n), f S(R). m Z n Z geometric case (function fields over an algebraically closed field) estimating the behaviour of some moduli spaces of curves when the degree varies. The Poisson formula is a sophisticated rewriting of the Riemann Roch theorem for curves in the context of motivic integration, due to Hrushovski and Kazhdan. This is joint work with François Loeser. Introduction. p. 3

Heights Classically, the Height of a rational number is defined as H(x) = max( a, b ), x = a/b, a, b Z, gcd(a, b) = 1. It is some measure of its arithmetic complexity. More generally, one defines the Height of a point x P n (Q) by H(x) = max( x 0,..., x n ) where x = [x 0 : : x n ] for coprime integers x 0,..., x n. Let F be a number field and M F a complete set of normalized absolute values on F. One defines the Height of a point x = [x 0 : : x n ] P n (F) by H F (x) = max( x 0 v,..., x n v ). v M F By the product formula, this is independent from the choice of homogeneous coordinates. Introduction. Heights in number theory p. 4

Heights over number fields Northcott s finiteness theorem. Given F, for every B 0, there are only finitely many x P n (F) with Height B. (More generally, there are only a finite number of points of bounded Height and bounded degree.) Corollary (Kronecker). α F is a root of unity iff H F (α) = 1. Schanuel s theorem. When B, one has N(P n (F), B) := Card x P n (F) ; H F (x) B C F,n B n+1 for some explicit constant C F,n, e.g. C Q,n = 1 ζ(n+1) 2n. Height zeta function. The asymptotic behaviour of the counting function N(P n (F), B) is encoded by the generating series Z(s) = H F (x) s. x P n (F) Introduction. Heights in number theory p. 5

Heights Manin s question Generalizations: One can replace P n by an arbitrary projective variety; One can vary the embedding and consider polarized varieties (X, L) L is an ample line bundle on the projective variety X; One could also replace F by a function field in one variable over a finite field. Manin s question: What can be said of the corresponding counting function N X (B)? Does it have an asymptotic expansion: N X (B) cb s log(b) t. What are the analytic properties of the associated generating series Z X (s)? Abscissa of convergence? meromorphic continuation? poles? Introduction. Heights in number theory p. 6

Manin s proposal Manin proposed a geometric interpretation which involves the geometry of the effective cone of the Néron-Severi group of X, and of the classes of L and K 1. For example, when L = K 1, he suggest that X X one has s = 1 and t = rank(ns(x)) 1. Peyre refined this interpretation by interpreting the constant c as the volume of the adelic space X(A F ) for some Tamagawa measure. Necessary precautions: One generally assumes that X(F) is Zariski dense; It may be necessary to replace the ground field by a finite extension; It may be necessary to restrict the counting problem to a dense open subset. Introduction. Heights in number theory p. 7

Theorems... Manin Peyre s expectation has been verified in a number of situations: Hypersurfaces of small degree compared to the dimension (circle method); Flag varieties and their generalizations (Franke Manin Tschinkel, using Langlands s theorems on Eisenstein series); Equivariant compactifications of some algebraic groups: tori (Batyrev Tschinkel), vector spaces (Chambert-Loir Tschinkel), simply connected semi-simple groups (Shalika Takloo-Bighash Tschinkel, Gorodnik Maucourant Oh), Heisenberg groups (Shalika Tschinkel)... Del Pezzo surfaces (de la Bretèche, Browning, Derenthal, Le Boudec...). The list is still growing. Introduction. Heights in number theory p. 8

... and counterexamples There are also counterexamples which show that it is sometimes necessary to avoid a thin subset. Total space of the universal family of diagonal cubic surfaces (Batyrev Tschinkel); The Hilbert space of two points on P 2 or P 1 P 1 (Schmidt, Le Rudulier). These counterexamples affect the finer aspects of the conjectural asymptotic expansion N(B) cb s log(b) t, namely the parameter t and the constant c, but not (not yet?) the parameter s. Introduction. Heights in number theory p. 9

Geometric heights Let C be a projective curve over a field k and let F = k(c). Then there is a correspondence: arithmetic over F projective space P n F point x P n (F) height h(x) = log(h(x)) geometry over C (trivial) fibration P n k C C morphism σ x : C P n k degree deg(σx O(1)) polarized variety (X, L)/F polarized fibration (X C, L) point x X(F) section σ x : C X height h(x) degree deg(σx L) In the geometric context, Northcott s theorem becomes the statement that sections σ of given degree d form a algebraic variety M d over k. It is then natural to follow Peyre s suggestion (around 2000) to investigate how these varieties M d vary with d. Introduction. Geometric height zeta functions p. 10

Geometric height zeta functions Let C be a projective curve over a field k and let F = k(c). Let X C be a proper flat morphism, let L be a line bundle on X. One lets X = X F, L = L X and assumes that L is big. One adds a natural condition on points x X(F)/sections σ x such as: x belongs to an appropriate subset U of X, which insures that for every d Z, sections σ : C X such that deg(σ L) = d form a constructible set M d, empty for d 0. Geometric height zeta function: Z(T) = [M d ]T d. d Z This is a formal Laurent series with coefficients in the Grothendieck ring of varieties KVar k. Introduction. Geometric height zeta functions p. 11

Grothendieck groups of varieties As a group, the Grothendieck ring KVar k is defined by its generators: isomorphism classes [X] of algebraic varieties X over k subject to the scissor relations: [X] = [Y] + [X Y], for any closed subscheme Y of X. Its ring structure is given by: [X] [Y] = [X k Y] This is a huge, complicated ring (non-noetherian, non-reduced,...) whose structure is not yet well understood. Unit element: class of the point, 1 = [Spec(k)] Lefschetz element: class of the affine line, L = [A 1 k ] Example: [P n k ] = 1 + L + + Ln. Introduction. Grothendieck groups of varieties p. 12

Motivic measures One grasps properties of the Grothendieck ring KVar k thanks to motivic measures which are functions µ from Var k to a ring A such that µ(x) = µ(y) if X and Y are isomorphic; µ(x) = µ(y) + µ(x Y) if Y is a closed subset of X; µ(x k Y) = µ(x)µ(y). Indeed these are exactly the ring morphisms from KVar k to A. One knows many interesting motivic measures: If k is finite, counting measure X #X(k); If k = C, Hodge-Deligne polynomial X E X (u, v); Cohomological realizations into various other Grothendieck groups (Hodge structures, l-adic representations, crystals...), and many other... Introduction. Grothendieck groups of varieties p. 13

Example: The geometric Schanuel theorem Let us consider the case of X = P n C, polarized with L = O(1). For simplicity, we assume that C has a k-rational point a. Let M d be the space of morphisms σ : C P n such that k deg(σ O(1)) = d. The height zeta function for P n is the generating series Z(T) = [M d ]T d. d 0 Using results of Kapranov, Peyre has shown the motivic analogue of Schanuel s theorem: Z(T) is a rational power series; The power series (1 L n+1 T)Z(T) converges at T = L 1 in a suitable completion M of the localized Grothendieck ring KVar k [L 1 ]; When d, dim(m d ) (n + 1)d has a finite limit, and log(κ(m d ))/ log(d) converges to 0. Here, κ(m d ) is the number of irreducible components of M d of maximal dimension. Introduction. Grothendieck groups of varieties p. 14

Objects: the context The situation is as follows: Let C be an irreducible smooth projective curve over an algebraically closed field k of characteristic zero, let F = k(c). Let C 0 be a non-empty open subset of C. Let X be a proper smooth variety endowed with a non-constant morphism to C, let L be a line bundle on X, and let U be a Zariski open subset of X. Let X = X F, U = U F and L = L X be the corresponding objects on the generic fiber. One makes the following geometric assumptions: U is isomorphic to the affine space G = A n, viewed as an F algebraic group over F; X is a smooth equivariant compactification of G whose divisor at infinity D = X G has strict normal crossings; L = (K X + D). Introduction. Statement of the theorem p. 15

Objects: the moduli spaces For every integer d Z, we consider the moduli space M d of sections σ : C X satisfying the following properties: deg(σ L) = d sections of geometric height d; σ(c 0 ) U. The second condition means that we consider a variant for integral points of the geometric Manin problem. For this reason, the relevant line bundle is the log-anticanonical line bundle. By general results on equivariant compactifications of affine spaces The space M d exists as a constructible set, and even as an algebraic variety over k if L is effective. In particular, it has a class [M d ] KVar k. M d is empty for d 0. Introduction. Statement of the theorem p. 16

Objects: the height zeta function Let M be the localization of KVar k by the multiplicative subset of KVar k generated by L and by the classes L a 1, for a > 0. We consider the generating Laurent series Z(T) = [M d ]T d M[[T]][T 1 ] d Z with coefficients in this localization. We introduce subrings M[T, T 1 ] M{T} M{T} M[[T]][T 1 ], where M{T} is generated over M[T, T 1 ] by the inverses of the polynomials 1 L a T b, for b a 0 (non both 0), and M{T} is generated by the inverses of the polynomials 1 L a T b, for b > a 0. An element P of M{T} has a value P(L 1 ) at T = L 1. Introduction. Statement of the theorem p. 17

Statement of the theorem (with F. Loeser) 1 There exists an integer a 1, an element P(T) M{T}, and an integer t 1, such that (1 L a T a ) t Z(T) = P(T) and such that P(L 1 ) is effective and non-zero. 2 In particular Z(T) belongs to M{T}. 3 For every integer p {0,..., a 1}, one of the following cases occurs when d tends to infinity in the congruence class of p modulo a: 1 Either dim(m d ) = o(d), 2 Or dim(m d ) d has a finite limit and log(κ(m d ))/ log(d) converges to some integer in {0,..., t 1}. Moreover, the second case happens at least for one integer p. Recall that κ(m d ) is the number of irreducible components of M d of maximal dimension. Introduction. Statement of the theorem p. 18

The classical Poisson summation formulas Poisson summation formula: For every smooth rapidly decreasing function f on R, with Fourier transform f, one has f(a) = f(b). a Z b Z More general formula: Let G be a locally compact abelian group, G its Pontryagin dual, Γ a discrete cocompact subgroup of G. For matching Haar measures dg on G and dχ on G, f(γ) = vol(g/γ) γ Γ χ (G/Γ) f(χ), for every suitable function f on G. Here, ˆf(χ) = G f(g)χ(g) dg is the Fourier transform of f. The motivic Poisson formula of Hrushovski Kazhdan. Introduction p. 20

The Poisson summation formula: Example Let F be a number field. Then F is a discrete cocompact subgroup of the adele group A F. Let C be a connected projective smooth curve over a finite field k, let F = k(c). F is a discrete cocompact subgroup of the adele group A F. Similar results hold for other algebraic groups, for example tori. Used by Batyrev Tschinkel/Bourqui to solve Manin s problem on toric varieties on number fields/function fields. What about a function field over an algebraically closed field? The motivic Poisson formula of Hrushovski Kazhdan. Introduction p. 21

Hrushovski-Kazhdan s motivic Poisson formula Let C be a connected projective smooth curve over an algebraically closed field k, let g be its genus and F = k(c). The motivic Poisson formula of Hruhovski-Kazhdan states formally ϕ(x) = L (1 g)n F ϕ(y), x F n y F n in which ϕ S, the space of motivic Schwartz Bruhat functions on A n F, built from suitable relative Grothendieck ring of varieties; F ϕ S is the Fourier transform of ϕ; For ϕ S, the sum x Fn ϕ(x) is an element of a Grothendieck ring of varieties. The motivic Poisson formula of Hrushovski Kazhdan. Introduction p. 22

Varieties with exponentials Let k be a field. As a group, the Grothendieck ring of k-varieties with exponentials KExpVar k is defined by its generators: isomorphism classes pairs [X, f], where X is a k-scheme of finite type and f : X A 1 is a morphism, subject to the scissor relations: [X, f] = [Y, f Y ] + [X Y, f X Y ], for any closed subscheme Y of X and to the additional relation: [X A 1, pr 2 ] = 0, where pr 2 is the second projection. Ring structure: [X, f] [Y, g] = [X k Y, pr 1 f + pr 2 g]. Unit element: class 1 = [Spec(k), 0]. Lefschetz element: class L of [A 1, 0]. Localizations: M k, ExpM k. The canonical ring morphism M k ExpM k, [X] [X, 0] is injective. The motivic Poisson formula of Hrushovski Kazhdan. Grothendieck rings of varieties with exponentials p. 23

Relative Grothendieck rings with exponentials Let S be a noetherian k-scheme. One defines analogously a Grothendieck ring KExpVar S of S-varieties with exponentials, generated by pairs [X, f] S where X is an S-scheme of finite type and f : X A 1 is a morphism. A morphism u: S T gives rise to: A ring morphism u : KExpVar T KExpVar S, [X, f] T [X T S, f pr 1 ] S ; A group morphism u! : KExpVar S KExpVar T, [X, f] S [X, f] T. Functional interpretation: An element ϕ KExpVar S is thought of as a motivic function on S, x ϕ(x) = x ϕ KExpVar k(x). The morphism u corresponds to the composition of functions. The morphism u! corresponds to summation over the fibers. The motivic Poisson formula of Hrushovski Kazhdan. Grothendieck rings of varieties with exponentials p. 24

Local Schwartz Bruhat functions of given level Let F = k((t)) and let F = k[[t]]. For every integers N M, t M F /t N F is identified with the k-rational points of A N M by the formula k x = x i t i (mod t N ) (x M,..., x N 1 ). Define S(F, (M, N)) = ExpM A N M, the ring of Schwartz Bruhat k functions of level (M, N) on F. For ϕ S(F, (M, N)), one defines ϕ = L N π! ϕ ExpM k, F where π : A N M k Spec(k) is the canonical morphism. The motivic Poisson formula of Hrushovski Kazhdan. Local Schwartz Bruhat functions p. 25

Compatibilities The closed immersion ι : A N M A N (M 1), k k (x M,..., x N 1 ) (0, x M,..., x N 1 ) gives rise to two maps: restriction: ι : S(F, (M 1, N)) S(F, (M, N)); extension by zero: ι! : S(F, (N, M)) S(F, (M 1, N)). One has ι ι! = Id. The projection π : A (N+1) M k gives rise to two maps: π : S(F, (M, N)) S(F, (M, N + 1)); A N M, (x k M,..., x N ) (x M,..., x N 1 ) convolution: π! : S(F, (M, N + 1)) S(F, (M, N)). One has π! π (ϕ) = Lϕ. The motivic Poisson formula of Hrushovski Kazhdan. Local Schwartz Bruhat functions p. 26

Local Schwartz Bruhat functions Writing F = lim lim t M F /t N F, M N M one defines the ring of smooth functions: D(F) = lim lim S(F, (M, N)), M,ι N,π and its ideal of Schwartz Bruhat functions: S(F) = lim lim S(F, (M, N)). M,ι! N,π Unit : 1 F S(F) is induced by the unit element of S(F, (0, N)). Every ϕ S(F) has an integral F ϕ ExpM k. For example, F 1 F = 1. The motivic Poisson formula of Hrushovski Kazhdan. Local Schwartz Bruhat functions p. 27

Fourier transform of local Schwartz Bruhat functions Fix ω Ω 1 F, non-zero. Fourier kernel: e(xy) = res(xyω). It is defined as an element of D(F 2 ) defined by the class [A N M A N M, u], where u is the k k composition (for suitable N, M ) A N M k A N M k A N M k A 1 k, the first map being induced by the product at finite levels, and the second by the residue. Fourier transform of ϕ S(F): F ϕ(y) = ϕ(x)e(xy) dx. It is an element of S(F). Fourier inversion: Let ν be the order of the pole of ω. Then F F ϕ(x) = L ν ϕ( x). Self-duality: If ν = 0, then F 1 F = 1 F. The motivic Poisson formula of Hrushovski Kazhdan. Local Schwartz Bruhat functions p. 28 F

Global Schwartz Bruhat functions Let C be a connected projective smooth curve over an algebraically closed field k. Let F = k(c); fix a non-zero form ω Ω 1 F/k. For v C(k), let F v k((t)) be the completion of F at v. The preceding constructions generalize naturally to finite products of fields isomorphic to k((t)). In particular: every finite set S of C(k), one has a space S( v S F v ), a Fourier transform, a Fourier inversion formula, etc. For S S, multiplications by units 1 F v for v S S gives rise to maps S( v S F v ) S( v S F v). Rng of global Schwartz Bruhat functions on A F : S(A F ) = lim S( F v ). S C(k) v S It admits a Fourier transform. Fourier inversion: For ϕ S(A F ), F F ϕ(x) = L 2g 2 ϕ( x). The motivic Poisson formula of Hrushovski Kazhdan. Global Schwartz Bruhat functions p. 29

Summation over rational points Let ϕ S(A F ). How to define x F ϕ(x) ExpM k? Assume that ϕ is represented by an element in S( v S A N v M v ). For an effective divisor D on C, let L(D) be the corresponding Riemann-Roch space L(D). Observation: F is the inductive limit of the spaces L(D). We view L(D) as a k-scheme and let π D be the canonical projection to Spec(k). For finite sets S C(k), the natural map L(D) v S F v gives rise to a morphisms of schemes u D,S : L(D) v S A N v M v. k For large enough D, one can then set ϕ(x) = (π D )! (u D,S ) ϕ ExpM k. x F The motivic Poisson formula of Hrushovski Kazhdan. Global Schwartz Bruhat functions p. 30

The motivic Poisson formula We now can state Hrushovski Kazhdan s motivic Poisson formula: For every ϕ S(A F ), one has F ϕ(y). x F ϕ(x) = L 1 g y F Once one unwields all definitions, this formula boils down to a combination of the Riemann-Roch formula and the Serre duality theorem on the curve C in its adelic formulation using repartitions (Groupes algébriques et corps de classes). The motivic Poisson formula of Hrushovski Kazhdan. The formula of Hrushovski Kazhdan p. 31

Sketch of the proof (arithmetic case) Our theorem is a geometric analogue of a theorem over number fields by Chambert-Loir and Tschinkel. Let us explain the strategy of the proof, following the one already set up by Batyrev Tschinkel for toric varieties. So one has a polarized variety (X, L), X is an equivariant compactification of G = G n a X. We want to count points of G(F) of bounded height. 1 One extends the height, G(F) R, to a function on the adelic space G(A F ) R. Proof of the theorem. The arithmetic analogue p. 33

Sketch of the proof (arithmetic case) Our theorem is a geometric analogue of a theorem over number fields by Chambert-Loir and Tschinkel. Let us explain the strategy of the proof, following the one already set up by Batyrev Tschinkel for toric varieties. So one has a polarized variety (X, L), X is an equivariant compactification of G = G n a X. We want to count points of G(F) of bounded height. 1 One extends the height, G(F) R, to a function on the adelic space G(A F ) R. 2 Recall that G(F) is a cocompact discrete subgroup of G(A F ). Fixing a non-trivial additive character ψ of A F, then (G(A F )/G(F)) is identified with G(F) by the pairing x, y = ψ(xy). Proof of the theorem. The arithmetic analogue p. 33

Sketch of the proof (arithmetic case) (...) So one has a polarized variety (X, L), X is an equivariant compactification of G = G n a X. We want to count points of G(F) of bounded height. 1 One extends the height, G(F) R, to a function on the adelic space G(A F ) R. 2 Recall that G(F) is a cocompact discrete subgroup of G(A F ). Fixing a non-trivial additive character ψ of A F, then (G(A F )/G(F)) is identified with G(F) by the pairing x, y = ψ(xy). 3 For Re(s) large enough, the function x H(x) s on G(A F ) can be applied the Poisson summation formula and we get a formula Z(s) = H(x) s = F (H( ) s )(y). x G(F) y G(F) Proof of the theorem. The arithmetic analogue p. 33

Sketch of the proof (arithmetic case) (...) 1 One extends the height, G(F) R, to a function on the adelic space G(A F ) R. 2 Recall that G(F) is a cocompact discrete subgroup of G(A F ). Fixing a non-trivial additive character ψ of A F, then (G(A F )/G(F)) is identified with G(F) by the pairing x, y = ψ(xy). 3 For Re(s) large enough, the function x H(x) s on G(A F ) can be applied the Poisson summation formula and we get a formula Z(s) = H(x) s = F (H( ) s )(y). x G(F) y G(F) 4 In this formula, the main term is for y = 0. The Fourier transform is a product of variants of local Igusa zeta functions; on can establish its meromorphic continuation and locate its main pole. Proof of the theorem. The arithmetic analogue p. 33

Sketch of the proof (arithmetic case) (...) 3 For Re(s) large enough, the function x H(x) s on G(A F ) can be applied the Poisson summation formula and we get a formula Z(s) = H(x) s = F (H( ) s )(y). x G(F) y G(F) 4 In this formula, the main term is for y = 0. The Fourier transform is a product of variants of local Igusa zeta functions; on can establish its meromorphic continuation and locate its main pole. 5 The other terms vanish for y outside of a lattice in G(F); they also have a meromorphic continuation, with less poles, and one can show that the same hold for the sum over all y 0. Proof of the theorem. The arithmetic analogue p. 33