Height zeta functions - PDF Free Download

Geometry Mathematisches Institut July 19, 2006

Geometric background Let X P n be a smooth variety over C. Its main invariants are: Picard group Pic(X ) and Néron-Severi group NS(X ) Λ eff (X ), Λ ample (X ) Pic(X ) R - (the closure of) the effective and the ample cone Example K X - the canonical class Let X = X f P n be a hypersurface, n 4. Then Pic(X ) = Z, generated by the hyperplane class L, K X = (n + 1 deg(x ))L and Λ eff (X ) = Λ ample (X ) = R 0L.

Exercise Let X P 3 be a smooth cubic surface, over Q. Then Pic(X ) = Z r with r {1,..., 7}. Implement (in Magma) an algorithm to determine r.

Exercise Let X P 3 be a smooth cubic surface, over Q. Then Pic(X ) = Z r with r {1,..., 7}. Implement (in Magma) an algorithm to determine r. Compute Λ eff (X ).

Exercise Let X P 3 be a smooth cubic surface, over Q. Then Pic(X ) = Z r with r {1,..., 7}. Implement (in Magma) an algorithm to determine r. Compute Λ eff (X ). Allow X to be singular (rational double points). Compute r and Λ eff ( X ) for the minimal desingularization X of X.

Let X P 3 be a smooth surface of degree 4, over Q. Then Pic(X ) = Z r with r {1,..., 20}. Example (Shioda) If X is given by xy 3 + yz 3 + zx 3 + w 4 = 0 then Pic(X ) = Z 20 (van Luijk 2005, 2006) Examples of X over Q with geometric Picard number 1 (and 3).

Classification According to the location of K X with respect to Λ ample (X ): K X ample: X Fano K X ample: X of general type otherwise, X of intermediate type Remark More advanced classification schemes, based on the Minimal Model Program or Campana s program, will be explained by D. Abramovich.

Examples dim Fano Intermediate type General type 1 P 1 elliptic curves C, g(c) 2 2 P 2, P 1 P 1, K3 surfaces : X 4 P 3,...... X 2,2 P 4, X 3 P 3,... abelian surfaces,... 3 120 families Calabi Yau varieties...

Zariski density Let X be a smooth projective variety over a number field F. Then X of general type X (F ) not Zariski dense; X Fano there exists a finite extension E/F such that X (E) is Zariski dense; X intermediate type??? More precise versions by Campana (2001). He formulates (conjectural) necessary and sufficient conditions on X to insure potential density of rational points. For our purposes, varieties with many rational points are close to being Fano. Most interesting cases: Del Pezzo surfaces.

Cones in geometry If X is Fano, i.e., K X Λ ample (X ), then Λ ample(x ) is finitely generated. For dim(x ) 3 Batyrev (1992) proved the following general: Conjecture Assume that X is Fano. Then Λ eff (X ) is a finitely generated rational cone. Let X be smooth projective, with a polarization L. Assume that Λ eff (X ) is finitely generated. Put a(l) := inf{a al + K X Λ eff (X )}, b(l) := min codim of the face of Λ eff (X ) containing a(l)l + K X. We have a( K X ) = 1 and b( K X ) = r := rk Pic(X ).

Cones in geometry Geometry Example (Hassett-T. 2002) Let X = M0,6 be the moduli space of 6 points in P 1. Then Pic(X ) = Z 16. The dual to the effective cone has 3905 generators.

Cones in geometry Example (Hassett-T. 2002) Let X = M0,6 be the moduli space of 6 points in P 1. Then Pic(X ) = Z 16. The dual to the effective cone has 3905 generators. Question Let X be Fano. Is there an effective algorithm to determine rk Pic(X )? Λ eff (X )?

Cones in analysis (A, Λ) = (lattice, convex cone in A R ); (Ǎ, ˇΛ) = (dual lattice, dual cone); α(λ, s) := ˇΛ e s,ǎ dǎ, R(s) Λ. This goes back to Köcher, Rothaus, Vinberg, Farauth. Example If Λ = r j=1 R 0e j then, for s = r j=1 s je j, α(λ, s) = 1 s 1... s r, R(s) > 0.

Cones in analysis II Geometry If Λ is finitely generated, then α(λ, s) is rational function of degree r (triangulate the dual cone ˇΛ). For κ, λ Λ consider α(λ, sλ + κ). If Λ is finitely generated, we can define a(λ), b(λ) as above. Then α(λ, sλ + κ) is holomorphic for R(s) > a(λ), and has an isolated pole at s = a(λ) of order b(λ).

Heights Data: F /Q number field X = X F projective algebraic variety over F X (F ) its F -rational points L adelically metrized very ample line bundle H L : X (F ) R>0 associated height, depends on the embedding and the metrization (choice of norms) H L is not invariant with respect to field extensions H L+L = H L H L (height formalism)

Main problem Geometry Counting function: N(X (F ), L, B) := #{x X (F ) H L (x) B} where X (F ) X (F ). Problem: Describe the asymptotic behaviour of N(X (F ), L, B), as B.

Height zeta function Z(X (F ), L, s) := x X (F ) It is holomorphic for R(s) 0 (X projective). Conjecture [Batyrev-Manin (1990)] H L (x) s Let X be Fano. Then, after allowing finite extensions of the groundfield and after passing to some Zariski open X, one has: 1 Z(X, L, s) is holomorphic for R(s) > a(l); 2 Z(X, L, s) admits a meromorphic extension to R(s) > a(l) δ (for some δ > 0), with an isolated pole at s = a(l) of order b(l). In other words, the analytic properties of Z(X, L, s) mirror the properties of α(λ, sλ + κ) defined above.

Tauberian theory Geometry Let (a n ) n be a sequence of nonnegative real numbers and put Z(s) := n a n n s. Assume that there exist a > 0, b N, δ > 0 such that Z(s) = c (s a) b + g(s) (s a) b 1 with g(s) holomorphic for R(s) > a δ and c > 0. Then as B. N(B) := n B a n = c aγ(b) Ba log(b) b 1 (1 + o(1))

Expected asymptotic Geometry Apply this to the height zeta function: N(X (F ), L, B) c a(l)γ(b(l)) Ba(L) log(b) b(l) 1, Later we will discuss the constant c.

Fano cubic bundles [Batyrev-T. (1996)] Let X P 3 P 3 be given by 3 j=0 x j y 3 j = 0. It is Fano, with Pic(X ) = Z Z; Λ ample (X ) = Λ eff (X ) = R 0L x R 0 L y ; K X = (3, 1).

Fano cubic bundles [Batyrev-T. (1996)] Let X P 3 P 3 be given by 3 j=0 x j y 3 j = 0. It is Fano, with Pic(X ) = Z Z; Λ ample (X ) = Λ eff (X ) = R 0L x R 0 L y ; K X = (3, 1). We have b( K X ) = 2, but N(X (F ), K X, B) B log(b) 6, for every Zariski open X, provided F contains ζ 3. So far, no counterexamples for a(l).

Algebraic dynamics Data: G/F linear algebraic group (Heisenberg group, G d a - additive group, G d m - algebraic torus) ρ : G PGl n+1 algebraic representation PGl n+1 acts on P n fix x P n (F ), consider the flow ρ(g) x H : P n (F ) R>0 - height {γ G(F ) H(ρ(γ) x) B} Problem: Count F -rational points on G/H, (where H is the stabilizer of x).

Translation Let X be the Zariski closure of ρ(g) x P n. Then X is an equivariant compactification of G/H with a G-linearized very ample line bundle L which is equipped with an adelic metrization. New problem: #{γ G/H(F ) H L (γ) B} We get a supply of algebraic varieties with many rational points.

Example 1: unipotent varieties X U - equivariant compactification of a unipotent group Pic(X ) = i ZD i K X = i κ id i, with κ i 2 [Hassett-T., (1999)] Λ eff (X ) = i R 0D i

Example 2: toric varieties X T - equivariant compactification of an algebraic torus 0 X (T) i ZD i π Pic(X ) 0 K X = π( i D i) Λ eff (X ) = π( i R 0D i ) any finitely generated cone can occur

Results Let X be one of the following varieties: Then (Franke 89) G/P, (Strauch 01) twisted products of G/P (Batyrev-T. 95) X T (Strauch-T. 99) X G/U (Chambert-Loir-T. 02) X G n a (Shalika-T. 02) X U (bi-equivariant) (Shalika-Takloo-Bighash-T. 05) X G De Concini-Procesi varieties Z(X, L, s) = c(l) (s a(l)) b(l) + g(s) (s a(l)) b(l) 1, with c(l) > 0 and g holomorphic for R(s) > a(l) δ, for some δ > 0.

c(l) c(k X ) = α(x )β(x )τ(k X ) α(x ) := α(λ eff (X ), K X ) β(x ) = Br(X )/Br(F ) = H 1 (Gal, Pic( X )) τ(k X ) = X (F ) ω K X, Tamagawa measure [Peyre (1995)]

c(l) α(x ) := α(λ eff (X ), K X ) c(k X ) = α(x )β(x )τ(k X ) β(x ) = Br(X )/Br(F ) = H 1 (Gal, Pic( X )) τ(k X ) = X (F ) ω K X, Tamagawa measure [Peyre (1995)] In general, c(l) = y Y (F ) c(k y ), where X Y is a fibration arising in Fujita s version of the Minimal Model Program.

Example For Del Pezzo surfaces of degree 4 β(x ) {1, 2, 4}. For cubic surfaces, β(x ) {1, 2, 4, 3, 9}.

Example For Del Pezzo surfaces of degree 4 β(x ) {1, 2, 4}. For cubic surfaces, β(x ) {1, 2, 4, 3, 9}. Example α(x ) = 7/18, where X : x0 3 + x 1 3 + x 2 3 + x 3 3 = 0. For split Del Pezzo surfaces X we have (Derenthal 2006): deg(x ) 7 6 5 4 3 2 1 α(x ) 1/24 1/72 1/144 1/180 1/120 1/30 1