Distribution of Rational Points on Toric Varieties: A Multi-Height Approach

Arda H. Demirhan

University of Illinois at Chicago

March 19, 2021

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Height Machine associates counting devices to divisor classes. ( ) ( ) geometric facts arithmetic facts ⇔ given by divisor relations given by height relations

Height Machine is useful for other fields too.

How to Count Rational Points?

X: an algebraic variety in Pn over a number field

Main Idea: We use height functions to count rational points.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Height Machine is useful for other fields too.

How to Count Rational Points?

X: an algebraic variety in Pn over a number field

Main Idea: We use height functions to count rational points.

Height Machine associates counting devices to divisor classes. ( ) ( ) geometric facts arithmetic facts ⇔ given by divisor relations given by height relations

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach How to Count Rational Points?

X: an algebraic variety in Pn over a number field

Main Idea: We use height functions to count rational points.

Height Machine associates counting devices to divisor classes. ( ) ( ) geometric facts arithmetic facts ⇔ given by divisor relations given by height relations

Height Machine is useful for other fields too.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Counting Function:

N(U(K), B) = card{x ∈ U(K): HK (x) ≤ B}

Let U ⊆ X be a Zariski open with some rational points.

Geometry shapes Arithmetic (for curves) 2 g = 0 N(B) ≈ cB r/2 g = 1 N(B) ≈ c(log(B) ) g > 1 N(B) ≈ c

Another important geometric invariant is the canonical class.

Counting functions

Slogan: Geometry Arithmetic

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Another important geometric invariant is the canonical class.

Counting functions

Slogan: Geometry Arithmetic

Counting Function:

N(U(K), B) = card{x ∈ U(K): HK (x) ≤ B}

Let U ⊆ X be a Zariski open with some rational points.

Geometry shapes Arithmetic (for curves) 2 g = 0 N(B) ≈ cB r/2 g = 1 N(B) ≈ c(log(B) ) g > 1 N(B) ≈ c

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Counting functions

Slogan: Geometry Arithmetic

Counting Function:

N(U(K), B) = card{x ∈ U(K): HK (x) ≤ B}

Let U ⊆ X be a Zariski open with some rational points.

Geometry shapes Arithmetic (for curves) 2 g = 0 N(B) ≈ cB r/2 g = 1 N(B) ≈ c(log(B) ) g > 1 N(B) ≈ c

Another important geometric invariant is the canonical class.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach This is unfortunately false in general:

Manin’s Conjecture

X: Fano variety i.e. −KX is ample. Then Conjecture (Batryev-Manin) X: smooth projective variety. There is a finite extension F of k with X (F ) Zariski dense in X . Moreover, for a small enough open set U

N(U(F ), B) = CB(log(B))t−1(1 + o(1))

as B → ∞.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Manin’s Conjecture

X: Fano variety i.e. −KX is ample. Then Conjecture (Batryev-Manin) X: smooth projective variety. There is a finite extension F of k with X (F ) Zariski dense in X . Moreover, for a small enough open set U

N(U(F ), B) = CB(log(B))t−1(1 + o(1))

as B → ∞. This is unfortunately false in general:

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 1 Xn+2: Smooth Fano variety n+2 2 A ,→ Xn+2 as a Zariski open set 2 3 Pic(X ) =∼ Z √ n+2 4 Q( −3) ⊆ k0 a number field

3 N(U(k0), B) ≥ cB(log(B))

A Counter-Example

Example n 3 (Batyrev-Tschinkel) Xn+2: hypersurface in P × P given by

3 3 3 3 `0(x)y0 + `1(x)y1 + `2(x)y2 + `3(x)y3 = 0

where n > 2

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach n+2 2 A ,→ Xn+2 as a Zariski open set 2 3 Pic(X ) =∼ Z √ n+2 4 Q( −3) ⊆ k0 a number field

3 N(U(k0), B) ≥ cB(log(B))

A Counter-Example

Example n 3 (Batyrev-Tschinkel) Xn+2: hypersurface in P × P given by

3 3 3 3 `0(x)y0 + `1(x)y1 + `2(x)y2 + `3(x)y3 = 0

where n > 2

1 Xn+2: Smooth Fano variety

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 2 3 Pic(X ) =∼ Z √ n+2 4 Q( −3) ⊆ k0 a number field

3 N(U(k0), B) ≥ cB(log(B))

A Counter-Example

Example n 3 (Batyrev-Tschinkel) Xn+2: hypersurface in P × P given by

3 3 3 3 `0(x)y0 + `1(x)y1 + `2(x)y2 + `3(x)y3 = 0

where n > 2

1 Xn+2: Smooth Fano variety n+2 2 A ,→ Xn+2 as a Zariski open set

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach √ 4 Q( −3) ⊆ k0 a number field

3 N(U(k0), B) ≥ cB(log(B))

A Counter-Example

Example n 3 (Batyrev-Tschinkel) Xn+2: hypersurface in P × P given by

3 3 3 3 `0(x)y0 + `1(x)y1 + `2(x)y2 + `3(x)y3 = 0

where n > 2

1 Xn+2: Smooth Fano variety n+2 2 A ,→ Xn+2 as a Zariski open set ∼ 2 3 Pic(Xn+2) = Z

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach A Counter-Example

Example n 3 (Batyrev-Tschinkel) Xn+2: hypersurface in P × P given by

3 3 3 3 `0(x)y0 + `1(x)y1 + `2(x)y2 + `3(x)y3 = 0

where n > 2

1 Xn+2: Smooth Fano variety n+2 2 A ,→ Xn+2 as a Zariski open set 2 3 Pic(X ) =∼ Z √ n+2 4 Q( −3) ⊆ k0 a number field

3 N(U(k0), B) ≥ cB(log(B))

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Toric Varieties Some equivariant compactifications Flag varieties Smooth complete intersections of small degree

Some Good News

Some Examples MC is true for some classes of varieties including:

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Some Good News

Some Examples MC is true for some classes of varieties including: Toric Varieties Some equivariant compactifications Flag varieties Smooth complete intersections of small degree

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Abscissa of Convergence: β = βU (L) := inf {σ : ZU (L, s) converges for σ} The Connection with the growth function:

log N(L, U, B) β = lim sup B→∞ log B Accumulating subvarieties: X ⊂ U is accumulating when

βU (L) = βX (L) > βU\X (L)

Accumulating Subvarieties

Let U be a countable set, hL : U → R+ a function such that its growth is finite for all bounds. Zeta function: X −s ZU (L, s) = hL(x) x∈U

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach The Connection with the growth function:

log N(L, U, B) β = lim sup B→∞ log B Accumulating subvarieties: X ⊂ U is accumulating when

βU (L) = βX (L) > βU\X (L)

Accumulating Subvarieties

Let U be a countable set, hL : U → R+ a function such that its growth is finite for all bounds. Zeta function: X −s ZU (L, s) = hL(x) x∈U Abscissa of Convergence: β = βU (L) := inf {σ : ZU (L, s) converges for σ}

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Accumulating Subvarieties

Let U be a countable set, hL : U → R+ a function such that its growth is finite for all bounds. Zeta function: X −s ZU (L, s) = hL(x) x∈U Abscissa of Convergence: β = βU (L) := inf {σ : ZU (L, s) converges for σ} The Connection with the growth function:

log N(L, U, B) β = lim sup B→∞ log B Accumulating subvarieties: X ⊂ U is accumulating when

βU (L) = βX (L) > βU\X (L)

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach xm+1 = ··· = xn = 0

n V : blow up of P along L0 n n−m−1 Projections: π1 : V → P and π2 : V → P

The Exceptional divisor E and the hyperplane section L of π1 satisfy:

1 π1 is an isomorphism away from E. 2 The Picard group of V , Pic(V ) = Z[L] ⊕ Z[E].

3 The canonical class, KV = −(n + 1)L + (n − m − 1)E. 4 Ample divisors D are of the form D = aL − bE for a > b > 0.

Blown up Projective Space

Blowing up Pn along Pm n 0 ≤ m ≤ n − 2 and L0: linear subspace of P defined by

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach n n−m−1 Projections: π1 : V → P and π2 : V → P

The Exceptional divisor E and the hyperplane section L of π1 satisfy:

1 π1 is an isomorphism away from E. 2 The Picard group of V , Pic(V ) = Z[L] ⊕ Z[E].

3 The canonical class, KV = −(n + 1)L + (n − m − 1)E. 4 Ample divisors D are of the form D = aL − bE for a > b > 0.

Blown up Projective Space

Blowing up Pn along Pm n 0 ≤ m ≤ n − 2 and L0: linear subspace of P defined by

xm+1 = ··· = xn = 0

n V : blow up of P along L0

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach The Exceptional divisor E and the hyperplane section L of π1 satisfy:

1 π1 is an isomorphism away from E. 2 The Picard group of V , Pic(V ) = Z[L] ⊕ Z[E].

3 The canonical class, KV = −(n + 1)L + (n − m − 1)E. 4 Ample divisors D are of the form D = aL − bE for a > b > 0.

Blown up Projective Space

Blowing up Pn along Pm n 0 ≤ m ≤ n − 2 and L0: linear subspace of P defined by

xm+1 = ··· = xn = 0

n V : blow up of P along L0 n n−m−1 Projections: π1 : V → P and π2 : V → P

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 1 π1 is an isomorphism away from E. 2 The Picard group of V , Pic(V ) = Z[L] ⊕ Z[E].

3 The canonical class, KV = −(n + 1)L + (n − m − 1)E. 4 Ample divisors D are of the form D = aL − bE for a > b > 0.

Blown up Projective Space

Blowing up Pn along Pm n 0 ≤ m ≤ n − 2 and L0: linear subspace of P defined by

xm+1 = ··· = xn = 0

n V : blow up of P along L0 n n−m−1 Projections: π1 : V → P and π2 : V → P

The Exceptional divisor E and the hyperplane section L of π1 satisfy:

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 2 The Picard group of V , Pic(V ) = Z[L] ⊕ Z[E].

3 The canonical class, KV = −(n + 1)L + (n − m − 1)E. 4 Ample divisors D are of the form D = aL − bE for a > b > 0.

Blown up Projective Space

Blowing up Pn along Pm n 0 ≤ m ≤ n − 2 and L0: linear subspace of P defined by

xm+1 = ··· = xn = 0

n V : blow up of P along L0 n n−m−1 Projections: π1 : V → P and π2 : V → P

The Exceptional divisor E and the hyperplane section L of π1 satisfy:

1 π1 is an isomorphism away from E.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 3 The canonical class, KV = −(n + 1)L + (n − m − 1)E. 4 Ample divisors D are of the form D = aL − bE for a > b > 0.

Blown up Projective Space

Blowing up Pn along Pm n 0 ≤ m ≤ n − 2 and L0: linear subspace of P defined by

xm+1 = ··· = xn = 0

n V : blow up of P along L0 n n−m−1 Projections: π1 : V → P and π2 : V → P

The Exceptional divisor E and the hyperplane section L of π1 satisfy:

1 π1 is an isomorphism away from E. 2 The Picard group of V , Pic(V ) = Z[L] ⊕ Z[E].

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 4 Ample divisors D are of the form D = aL − bE for a > b > 0.

Blown up Projective Space

Blowing up Pn along Pm n 0 ≤ m ≤ n − 2 and L0: linear subspace of P defined by

xm+1 = ··· = xn = 0

n V : blow up of P along L0 n n−m−1 Projections: π1 : V → P and π2 : V → P

The Exceptional divisor E and the hyperplane section L of π1 satisfy:

1 π1 is an isomorphism away from E. 2 The Picard group of V , Pic(V ) = Z[L] ⊕ Z[E].

3 The canonical class, KV = −(n + 1)L + (n − m − 1)E.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Blown up Projective Space

Blowing up Pn along Pm n 0 ≤ m ≤ n − 2 and L0: linear subspace of P defined by

xm+1 = ··· = xn = 0

n V : blow up of P along L0 n n−m−1 Projections: π1 : V → P and π2 : V → P

The Exceptional divisor E and the hyperplane section L of π1 satisfy:

1 π1 is an isomorphism away from E. 2 The Picard group of V , Pic(V ) = Z[L] ⊕ Z[E].

3 The canonical class, KV = −(n + 1)L + (n − m − 1)E. 4 Ample divisors D are of the form D = aL − bE for a > b > 0.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach ∴ E is an accumulating subvariety for the anticanonical divisor. Corollary: The exceptional curve of the blowing up of P2 at a point is accumulating for the anticanonical divisor (3, 1).

Distribution of Rational Points on the Blown up Projective Space

Blowing up Pn along Pm

 B(m+1)/(a−b) if m+1 > n−m  a−b b  (m+1)/(a−b) m+1 n−m N(E(Q), aL−bE, B) = B log B if a−b = b  (n−m)/b m+1 n−m  B if a−b < b  B(m+2)/(a−b) if n−m−1 < b  n+1 a  (n+1)/a n−m−1 b N(U(Q), aL − bE, B) = B log B if n+1 = a  (n+1)/a n−m−1 b  B if n+1 > a

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Corollary: The exceptional curve of the blowing up of P2 at a point is accumulating for the anticanonical divisor (3, 1).

Distribution of Rational Points on the Blown up Projective Space

Blowing up Pn along Pm

 B(m+1)/(a−b) if m+1 > n−m  a−b b  (m+1)/(a−b) m+1 n−m N(E(Q), aL−bE, B) = B log B if a−b = b  (n−m)/b m+1 n−m  B if a−b < b  B(m+2)/(a−b) if n−m−1 < b  n+1 a  (n+1)/a n−m−1 b N(U(Q), aL − bE, B) = B log B if n+1 = a  (n+1)/a n−m−1 b  B if n+1 > a

∴ E is an accumulating subvariety for the anticanonical divisor.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Distribution of Rational Points on the Blown up Projective Space

Blowing up Pn along Pm

 B(m+1)/(a−b) if m+1 > n−m  a−b b  (m+1)/(a−b) m+1 n−m N(E(Q), aL−bE, B) = B log B if a−b = b  (n−m)/b m+1 n−m  B if a−b < b  B(m+2)/(a−b) if n−m−1 < b  n+1 a  (n+1)/a n−m−1 b N(U(Q), aL − bE, B) = B log B if n+1 = a  (n+1)/a n−m−1 b  B if n+1 > a

∴ E is an accumulating subvariety for the anticanonical divisor. Corollary: The exceptional curve of the blowing up of P2 at a point is accumulating for the anticanonical divisor (3, 1).

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach We will show that the multi-height approach eliminates the role played by the accumulating subvarieties for toric varieties.

There are several approaches to the study of the distribution of rational points on toric varieties: 1 Harmonic analysis by Batyrev and Tschinkel, and also by Strauch and Tschinkel, 2 Universal torsors by Salberger, de la Bret`eche,et al.

Distribution of Rational Points on the Blown up Projective Space: A Few Remarks

The blow up of Pn along Pm is an example of a toric variety.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach There are several approaches to the study of the distribution of rational points on toric varieties: 1 Harmonic analysis by Batyrev and Tschinkel, and also by Strauch and Tschinkel, 2 Universal torsors by Salberger, de la Bret`eche,et al.

Distribution of Rational Points on the Blown up Projective Space: A Few Remarks

The blow up of Pn along Pm is an example of a toric variety.

We will show that the multi-height approach eliminates the role played by the accumulating subvarieties for toric varieties.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 1 Harmonic analysis by Batyrev and Tschinkel, and also by Strauch and Tschinkel, 2 Universal torsors by Salberger, de la Bret`eche,et al.

Distribution of Rational Points on the Blown up Projective Space: A Few Remarks

The blow up of Pn along Pm is an example of a toric variety.

We will show that the multi-height approach eliminates the role played by the accumulating subvarieties for toric varieties.

There are several approaches to the study of the distribution of rational points on toric varieties:

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 2 Universal torsors by Salberger, de la Bret`eche,et al.

Distribution of Rational Points on the Blown up Projective Space: A Few Remarks

The blow up of Pn along Pm is an example of a toric variety.

We will show that the multi-height approach eliminates the role played by the accumulating subvarieties for toric varieties.

There are several approaches to the study of the distribution of rational points on toric varieties: 1 Harmonic analysis by Batyrev and Tschinkel, and also by Strauch and Tschinkel,

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Distribution of Rational Points on the Blown up Projective Space: A Few Remarks

The blow up of Pn along Pm is an example of a toric variety.

We will show that the multi-height approach eliminates the role played by the accumulating subvarieties for toric varieties.

There are several approaches to the study of the distribution of rational points on toric varieties: 1 Harmonic analysis by Batyrev and Tschinkel, and also by Strauch and Tschinkel, 2 Universal torsors by Salberger, de la Bret`eche,et al.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 1 Define a height pairing 2 Define the Height Zeta function 3 Poisson formula 4 Compute Fourier Transforms 5 Get a meromorphic continuation 6 Tauberian Theorem

A Brief Roadmap to use Harmonic Analysis

Our method is based on harmonic analysis. The Idea:

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 2 Define the Height Zeta function 3 Poisson formula 4 Compute Fourier Transforms 5 Get a meromorphic continuation 6 Tauberian Theorem

A Brief Roadmap to use Harmonic Analysis

Our method is based on harmonic analysis. The Idea: 1 Define a height pairing

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 3 Poisson formula 4 Compute Fourier Transforms 5 Get a meromorphic continuation 6 Tauberian Theorem

A Brief Roadmap to use Harmonic Analysis

Our method is based on harmonic analysis. The Idea: 1 Define a height pairing 2 Define the Height Zeta function

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 4 Compute Fourier Transforms 5 Get a meromorphic continuation 6 Tauberian Theorem

A Brief Roadmap to use Harmonic Analysis

Our method is based on harmonic analysis. The Idea: 1 Define a height pairing 2 Define the Height Zeta function 3 Poisson formula

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 5 Get a meromorphic continuation 6 Tauberian Theorem

A Brief Roadmap to use Harmonic Analysis

Our method is based on harmonic analysis. The Idea: 1 Define a height pairing 2 Define the Height Zeta function 3 Poisson formula 4 Compute Fourier Transforms

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 6 Tauberian Theorem

A Brief Roadmap to use Harmonic Analysis

Our method is based on harmonic analysis. The Idea: 1 Define a height pairing 2 Define the Height Zeta function 3 Poisson formula 4 Compute Fourier Transforms 5 Get a meromorphic continuation

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach A Brief Roadmap to use Harmonic Analysis

Our method is based on harmonic analysis. The Idea: 1 Define a height pairing 2 Define the Height Zeta function 3 Poisson formula 4 Compute Fourier Transforms 5 Get a meromorphic continuation 6 Tauberian Theorem

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach ∼ d Split:T = Gm,K ∗ Characters: Tˆ := Hom(T, K ). Duality:   ( ) discrete and continuous algebraic tori   ←→ Gal(K/K) − modules defined over a number field K  of finite rank over Z  given by T 7→ Tˆ and M 7→ T := Spec K[M]

Anisotropic: rank(TˆK ) = 0 e.g. An orthogonal group of a quadratic form non-vanishing / Q.

Algebraic torus

Torus: A linear algebraic group T over a field K such that ∼ d TL = Gm,L L: splitting field.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach ∗ Characters: Tˆ := Hom(T, K ). Duality:   ( ) discrete and continuous algebraic tori   ←→ Gal(K/K) − modules defined over a number field K  of finite rank over Z  given by T 7→ Tˆ and M 7→ T := Spec K[M]

Anisotropic: rank(TˆK ) = 0 e.g. An orthogonal group of a quadratic form non-vanishing / Q.

Algebraic torus

Torus: A linear algebraic group T over a field K such that ∼ d TL = Gm,L L: splitting field.

∼ d Split:T = Gm,K

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Duality:   ( ) discrete and continuous algebraic tori   ←→ Gal(K/K) − modules defined over a number field K  of finite rank over Z  given by T 7→ Tˆ and M 7→ T := Spec K[M]

Anisotropic: rank(TˆK ) = 0 e.g. An orthogonal group of a quadratic form non-vanishing / Q.

Algebraic torus

Torus: A linear algebraic group T over a field K such that ∼ d TL = Gm,L L: splitting field.

∼ d Split:T = Gm,K ∗ Characters: Tˆ := Hom(T, K ).

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Anisotropic: rank(TˆK ) = 0 e.g. An orthogonal group of a quadratic form non-vanishing / Q.

Algebraic torus

Torus: A linear algebraic group T over a field K such that ∼ d TL = Gm,L L: splitting field.

∼ d Split:T = Gm,K ∗ Characters: Tˆ := Hom(T, K ). Duality:   ( ) discrete and continuous algebraic tori   ←→ Gal(K/K) − modules defined over a number field K  of finite rank over Z  given by T 7→ Tˆ and M 7→ T := Spec K[M]

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach e.g. An orthogonal group of a quadratic form non-vanishing / Q.

Algebraic torus

Torus: A linear algebraic group T over a field K such that ∼ d TL = Gm,L L: splitting field.

∼ d Split:T = Gm,K ∗ Characters: Tˆ := Hom(T, K ). Duality:   ( ) discrete and continuous algebraic tori   ←→ Gal(K/K) − modules defined over a number field K  of finite rank over Z  given by T 7→ Tˆ and M 7→ T := Spec K[M]

Anisotropic: rank(TˆK ) = 0

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Algebraic torus

Torus: A linear algebraic group T over a field K such that ∼ d TL = Gm,L L: splitting field.

∼ d Split:T = Gm,K ∗ Characters: Tˆ := Hom(T, K ). Duality:   ( ) discrete and continuous algebraic tori   ←→ Gal(K/K) − modules defined over a number field K  of finite rank over Z  given by T 7→ Tˆ and M 7→ T := Spec K[M]

Anisotropic: rank(TˆK ) = 0 e.g. An orthogonal group of a quadratic form non-vanishing / Q.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach S Complete Fans: NR = σ σ∈Σ Regular Fans: if the generators of every σ ∈ Σ form part of a basis of N. Toric Varieties: [ XΣ := Uσ where Uσ := Spec(K[M ∩ σˇ]) σ∈Σ

Fans and toric varieties

Split Tori Data of a Fan: 1 M - free abelian group of finite rank, N := Hom(M, Z) - the dual abelian group, NR := N ⊗ R 2 a fan Σ = {σ} is a finite collection of cones in NR: 0 ∈ Σ ∀σ ∈ Σ, every face τ ⊂ σ is in Σ ∀σ, σ0 ∈ Σ, σ ∩ σ0 ∈ Σ and is a face of σ, σ0

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Toric Varieties: [ XΣ := Uσ where Uσ := Spec(K[M ∩ σˇ]) σ∈Σ

Fans and toric varieties

Split Tori Data of a Fan: 1 M - free abelian group of finite rank, N := Hom(M, Z) - the dual abelian group, NR := N ⊗ R 2 a fan Σ = {σ} is a finite collection of cones in NR: 0 ∈ Σ ∀σ ∈ Σ, every face τ ⊂ σ is in Σ ∀σ, σ0 ∈ Σ, σ ∩ σ0 ∈ Σ and is a face of σ, σ0 S Complete Fans: NR = σ σ∈Σ Regular Fans: if the generators of every σ ∈ Σ form part of a basis of N.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Fans and toric varieties

Split Tori Data of a Fan: 1 M - free abelian group of finite rank, N := Hom(M, Z) - the dual abelian group, NR := N ⊗ R 2 a fan Σ = {σ} is a finite collection of cones in NR: 0 ∈ Σ ∀σ ∈ Σ, every face τ ⊂ σ is in Σ ∀σ, σ0 ∈ Σ, σ ∩ σ0 ∈ Σ and is a face of σ, σ0 S Complete Fans: NR = σ σ∈Σ Regular Fans: if the generators of every σ ∈ Σ form part of a basis of N. Toric Varieties: [ XΣ := Uσ where Uσ := Spec(K[M ∩ σˇ]) σ∈Σ

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 1-dimensional generators e1,..., en of Σ correspond to T -invariant boundary divisors D1,..., Dn - irreducible components of XΣ \ T

Σ regular ⇒ XΣ smooth.

Toric Varieties: An Example and Properties

Example Input: Σ: a d-dimensional fan with 1 1-dimensional cones generated by

e1,..., ed , ed+1 P where ed+1 = − ei . 2 m-dimensional cones generated by m-subsets of them. d Output: The toric variety XΣ=P

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Toric Varieties: An Example and Properties

Example Input: Σ: a d-dimensional fan with 1 1-dimensional cones generated by

e1,..., ed , ed+1 P where ed+1 = − ei . 2 m-dimensional cones generated by m-subsets of them. d Output: The toric variety XΣ=P

1-dimensional generators e1,..., en of Σ correspond to T -invariant boundary divisors D1,..., Dn - irreducible components of XΣ \ T

Σ regular ⇒ XΣ smooth.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Height functions: Global heights are product of local heights: 1 Non-archimedean case:

HΣ,v (xv , φ) = exp(φ(¯xv ) log qv )

2 Archimedean case:

HΣ,v (xv , φ) = exp(φ(¯xv ))

Picard group and Heights

PL(Σ) - piecewise linear Z-valued functions φ on Σ determined by {mσ,φ}σ∈Σ, i.e., by its values φ(ej ), j = 1,..., n.

π 0 → M → PL(Σ) −→ Pic(XΣ) → 0

every divisor is equivalent to a linear combination of boundary divisors D1,..., Dn, and φ is determined by its values on e1,..., en and these are denoted as sj . relations come from characters of T

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Picard group and Heights

PL(Σ) - piecewise linear Z-valued functions φ on Σ determined by {mσ,φ}σ∈Σ, i.e., by its values φ(ej ), j = 1,..., n.

π 0 → M → PL(Σ) −→ Pic(XΣ) → 0

every divisor is equivalent to a linear combination of boundary divisors D1,..., Dn, and φ is determined by its values on e1,..., en and these are denoted as sj . relations come from characters of T Height functions: Global heights are product of local heights: 1 Non-archimedean case:

HΣ,v (xv , φ) = exp(φ(¯xv ) log qv )

2 Archimedean case:

HΣ,v (xv , φ) = exp(φ(¯xv ))

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach ( (| x0 | )2 if |x | ≥ |x | x1 v 0 v 1 v Hv (xv , φ) = (| x1 | )3 otherwise x0 v Y H(x) = Hv (x) v

An Example Height Computation

Example X = P1, PicT(P1) =< [0], [∞] >, s = (2, 3).

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Y H(x) = Hv (x) v

An Example Height Computation

Example X = P1, PicT(P1) =< [0], [∞] >, s = (2, 3). ( (| x0 | )2 if |x | ≥ |x | x1 v 0 v 1 v Hv (xv , φ) = (| x1 | )3 otherwise x0 v

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach An Example Height Computation

Example X = P1, PicT(P1) =< [0], [∞] >, s = (2, 3). ( (| x0 | )2 if |x | ≥ |x | x1 v 0 v 1 v Hv (xv , φ) = (| x1 | )3 otherwise x0 v Y H(x) = Hv (x) v

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Proposition (Batryev and Tschinkel)

ZΣ(φ)

1 Holomorphic for <(sj ) > 1

2 Has a meromorphic extension to the domain <(sj ) > 1 − δ with poles of order ≤ 1 along sj = 1

Zeta Functions of the Complexified Height and their Analytic Properties

Height zeta function: X −1 ZΣ(φ) := HΣ(x, φ) x∈T(K) Poisson Formula (Anisotropic T ): 1 X Z (φ) = Hˆ (χ, −φ) Σ b (T ) Σ S ∗ χ∈(T (AK )/T (K)) Z HˆΣ(χ, −φ) := HΣ(x, −φ)χ(x)ωΩ,S T(AK )

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Zeta Functions of the Complexified Height and their Analytic Properties

Height zeta function: X −1 ZΣ(φ) := HΣ(x, φ) x∈T(K) Poisson Formula (Anisotropic T ): 1 X Z (φ) = Hˆ (χ, −φ) Σ b (T ) Σ S ∗ χ∈(T (AK )/T (K)) Z HˆΣ(χ, −φ) := HΣ(x, −φ)χ(x)ωΩ,S T(AK ) Proposition (Batryev and Tschinkel)

ZΣ(φ)

1 Holomorphic for <(sj ) > 1

2 Has a meromorphic extension to the domain <(sj ) > 1 − δ with poles of order ≤ 1 along sj = 1

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach This theorem of B&T only uses a single metrized height function.

Peyre in Libert´eet Accumulation and Beyond heights: slopes and distribution of rational points provided two approaches: 1 Freeness 2 All the Heights Why useful? This may remove the influence of accumulating subvarieties in some cases.

Apply Peyre’s multiheights approach for toric varieties.

Multi-Height Variant of the Batryev-Tschinkel Theorem

Restricting ZΣ(φ) to ZΣ,L(s) verification of Manin’s Conjecture for Toric Varieties by B and T.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Peyre in Libert´eet Accumulation and Beyond heights: slopes and distribution of rational points provided two approaches: 1 Freeness 2 All the Heights Why useful? This may remove the influence of accumulating subvarieties in some cases.

Apply Peyre’s multiheights approach for toric varieties.

Multi-Height Variant of the Batryev-Tschinkel Theorem

Restricting ZΣ(φ) to ZΣ,L(s) verification of Manin’s Conjecture for Toric Varieties by B and T.

This theorem of B&T only uses a single metrized height function.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 1 Freeness 2 All the Heights Why useful? This may remove the influence of accumulating subvarieties in some cases.

Apply Peyre’s multiheights approach for toric varieties.

Multi-Height Variant of the Batryev-Tschinkel Theorem

Restricting ZΣ(φ) to ZΣ,L(s) verification of Manin’s Conjecture for Toric Varieties by B and T.

This theorem of B&T only uses a single metrized height function.

Peyre in Libert´eet Accumulation and Beyond heights: slopes and distribution of rational points provided two approaches:

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Why useful? This may remove the influence of accumulating subvarieties in some cases.

Apply Peyre’s multiheights approach for toric varieties.

Multi-Height Variant of the Batryev-Tschinkel Theorem

Restricting ZΣ(φ) to ZΣ,L(s) verification of Manin’s Conjecture for Toric Varieties by B and T.

This theorem of B&T only uses a single metrized height function.

Peyre in Libert´eet Accumulation and Beyond heights: slopes and distribution of rational points provided two approaches: 1 Freeness 2 All the Heights

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach This may remove the influence of accumulating subvarieties in some cases.

Apply Peyre’s multiheights approach for toric varieties.

Multi-Height Variant of the Batryev-Tschinkel Theorem

Restricting ZΣ(φ) to ZΣ,L(s) verification of Manin’s Conjecture for Toric Varieties by B and T.

This theorem of B&T only uses a single metrized height function.

Peyre in Libert´eet Accumulation and Beyond heights: slopes and distribution of rational points provided two approaches: 1 Freeness 2 All the Heights Why useful?

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Apply Peyre’s multiheights approach for toric varieties.

Multi-Height Variant of the Batryev-Tschinkel Theorem

Restricting ZΣ(φ) to ZΣ,L(s) verification of Manin’s Conjecture for Toric Varieties by B and T.

This theorem of B&T only uses a single metrized height function.

Peyre in Libert´eet Accumulation and Beyond heights: slopes and distribution of rational points provided two approaches: 1 Freeness 2 All the Heights Why useful? This may remove the influence of accumulating subvarieties in some cases.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Multi-Height Variant of the Batryev-Tschinkel Theorem

Restricting ZΣ(φ) to ZΣ,L(s) verification of Manin’s Conjecture for Toric Varieties by B and T.

This theorem of B&T only uses a single metrized height function.

Peyre in Libert´eet Accumulation and Beyond heights: slopes and distribution of rational points provided two approaches: 1 Freeness 2 All the Heights Why useful? This may remove the influence of accumulating subvarieties in some cases.

Apply Peyre’s multiheights approach for toric varieties.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Main Theorem (Demirhan and Takloo-Bighash)

There are positive real constants Pβ and θ such that

β β1+β2+...+βm −θ
N = N(B ) = B Pβ + O(B )

as B → ∞

Multi-Height Variant of the Batryev-Tschinkel Theorem II

Consider the behaviour of

βi N := card{x ∈ X (K): Hi (x) ≤ B for i = 1,..., m}

as B → ∞ where Hi ’s are metrized heights determined by T-invariant Di ’s yielding the ample divisor X si Di i Then we claim the following:

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Multi-Height Variant of the Batryev-Tschinkel Theorem II

Consider the behaviour of

βi N := card{x ∈ X (K): Hi (x) ≤ B for i = 1,..., m}

as B → ∞ where Hi ’s are metrized heights determined by T-invariant Di ’s yielding the ample divisor X si Di i Then we claim the following: Main Theorem (Demirhan and Takloo-Bighash)

There are positive real constants Pβ and θ such that

β β1+β2+...+βm −θ
N = N(B ) = B Pβ + O(B )

as B → ∞

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Key point: Any T -invariant divisor is itself a toric variety with a fan that is completely explicitly determined from the fan of the original toric variety.

This allows us to do an inductive argument on the dimension of the toric variety.

Sketch of the proof: Reduction Argument

It suffices to prove the theorem for X∆ replaced by T .

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Sketch of the proof: Reduction Argument

It suffices to prove the theorem for X∆ replaced by T .

Key point: Any T -invariant divisor is itself a toric variety with a fan that is completely explicitly determined from the fan of the original toric variety.

This allows us to do an inductive argument on the dimension of the toric variety.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Replace the data (X∆, T1, D1 ..., Dr ) with (D1, T1, D2,..., Dr ) Assume we show that

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ T (F ): HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ X∆ : HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

Sketch of the proof: Reduction Argument (cont.)

Consider D1 = T1 ∪ (D1 \ T1)

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach (D1, T1, D2,..., Dr ) Assume we show that

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ T (F ): HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ X∆ : HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

Sketch of the proof: Reduction Argument (cont.)

Consider D1 = T1 ∪ (D1 \ T1) Replace the data (X∆, T1, D1 ..., Dr ) with

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Assume we show that

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ T (F ): HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ X∆ : HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

Sketch of the proof: Reduction Argument (cont.)

Consider D1 = T1 ∪ (D1 \ T1) Replace the data (X∆, T1, D1 ..., Dr ) with (D1, T1, D2,..., Dr )

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ X∆ : HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

Sketch of the proof: Reduction Argument (cont.)

Consider D1 = T1 ∪ (D1 \ T1) Replace the data (X∆, T1, D1 ..., Dr ) with (D1, T1, D2,..., Dr ) Assume we show that

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ T (F ): HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ X∆ : HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

Sketch of the proof: Reduction Argument (cont.)

Consider D1 = T1 ∪ (D1 \ T1) Replace the data (X∆, T1, D1 ..., Dr ) with (D1, T1, D2,..., Dr ) Assume we show that

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ T (F ): HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Sketch of the proof: Reduction Argument (cont.)

Consider D1 = T1 ∪ (D1 \ T1) Replace the data (X∆, T1, D1 ..., Dr ) with (D1, T1, D2,..., Dr ) Assume we show that

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ T (F ): HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

a1 ar a1+a2+...+ar a1+a2+...+ar − {p ∈ X∆ : HD1 (P) ≤ B ,..., HDr (P) ≤ B } = CB + O(B )

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach a1 ar a1+...+ar − {p ∈ D1(F ): H1(P) ≤ B ,... Hr (P) ≤ B } = O(B )

a1 ar a1+...+ar − {p ∈ T1(F ): H2(P) ≤ B ,... Hr (P) ≤ B } = O(B )

∴ The problem of counting is reduced to counting rational points in a general tori.

Sketch of the proof: Reduction Argument (cont.)

X∆ \ T = D1 ∪ ... ∪ Dr

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach a1 ar a1+...+ar − {p ∈ T1(F ): H2(P) ≤ B ,... Hr (P) ≤ B } = O(B )

∴ The problem of counting is reduced to counting rational points in a general tori.

Sketch of the proof: Reduction Argument (cont.)

X∆ \ T = D1 ∪ ... ∪ Dr

a1 ar a1+...+ar − {p ∈ D1(F ): H1(P) ≤ B ,... Hr (P) ≤ B } = O(B )

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach ∴ The problem of counting is reduced to counting rational points in a general tori.

Sketch of the proof: Reduction Argument (cont.)

X∆ \ T = D1 ∪ ... ∪ Dr

a1 ar a1+...+ar − {p ∈ D1(F ): H1(P) ≤ B ,... Hr (P) ≤ B } = O(B )

a1 ar a1+...+ar − {p ∈ T1(F ): H2(P) ≤ B ,... Hr (P) ≤ B } = O(B )

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Sketch of the proof: Reduction Argument (cont.)

X∆ \ T = D1 ∪ ... ∪ Dr

a1 ar a1+...+ar − {p ∈ D1(F ): H1(P) ≤ B ,... Hr (P) ≤ B } = O(B )

a1 ar a1+...+ar − {p ∈ T1(F ): H2(P) ≤ B ,... Hr (P) ≤ B } = O(B )

∴ The problem of counting is reduced to counting rational points in a general tori.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Sketch of the proof: Anisotropic Case

We consider the arithmetic function f

f : C −→ N

(c1, c2,..., cm) 7−→ card{x ∈ X (K): Hi (x) = ci for each i, 1 ≤ i ≤ m}

then define Multi Height Zeta Function

X X f (c1,..., cm) F (s) := ··· s1 sm (1) c1 ... cm c1∈C cm∈C Then F satisfies nice properties:

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Proof. 1 F (s) a sum of the terms Q si up to O(1) 1≤i≤m Hi (x) 1 1 Q s =
Hi (x) i HPm x 1≤i≤m i=1 si Di

Di ’s are T-invariant divisors yielding a very ample divisor and si ’s are in Pic(X )C Taking the sum of both sides we obtain:

F (s) = ZΣ(φ)

So F (s) is absolutely convergent for <(s) > 1.

Sketch of the proof: Anisotropic Case (cont.)

1 F (s) is absolutely convergent for <(s) > 1 where 1 = (1,..., 1)

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 1 1 Q s =
Hi (x) i HPm x 1≤i≤m i=1 si Di

Di ’s are T-invariant divisors yielding a very ample divisor and si ’s are in Pic(X )C Taking the sum of both sides we obtain:

F (s) = ZΣ(φ)

So F (s) is absolutely convergent for <(s) > 1.

Sketch of the proof: Anisotropic Case (cont.)

1 F (s) is absolutely convergent for <(s) > 1 where 1 = (1,..., 1) Proof. 1 F (s) a sum of the terms Q si up to O(1) 1≤i≤m Hi (x)

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Di ’s are T-invariant divisors yielding a very ample divisor and si ’s are in Pic(X )C Taking the sum of both sides we obtain:

F (s) = ZΣ(φ)

So F (s) is absolutely convergent for <(s) > 1.

Sketch of the proof: Anisotropic Case (cont.)

1 F (s) is absolutely convergent for <(s) > 1 where 1 = (1,..., 1) Proof. 1 F (s) a sum of the terms Q si up to O(1) 1≤i≤m Hi (x) 1 1 Q s =
Hi (x) i HPm x 1≤i≤m i=1 si Di

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach So F (s) is absolutely convergent for <(s) > 1.

Sketch of the proof: Anisotropic Case (cont.)

1 F (s) is absolutely convergent for <(s) > 1 where 1 = (1,..., 1) Proof. 1 F (s) a sum of the terms Q si up to O(1) 1≤i≤m Hi (x) 1 1 Q s =
Hi (x) i HPm x 1≤i≤m i=1 si Di

Di ’s are T-invariant divisors yielding a very ample divisor and si ’s are in Pic(X )C Taking the sum of both sides we obtain:

F (s) = ZΣ(φ)

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Sketch of the proof: Anisotropic Case (cont.)

1 F (s) is absolutely convergent for <(s) > 1 where 1 = (1,..., 1) Proof. 1 F (s) a sum of the terms Q si up to O(1) 1≤i≤m Hi (x) 1 1 Q s =
Hi (x) i HPm x 1≤i≤m i=1 si Di

Di ’s are T-invariant divisors yielding a very ample divisor and si ’s are in Pic(X )C Taking the sum of both sides we obtain:

F (s) = ZΣ(φ)

So F (s) is absolutely convergent for <(s) > 1.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach where π(i)(s) is the

coordinate function. Then H(s) is holomorphic in the domain

1 D(δ) := {s ∈ Cm : <(π(i)(s)) > − ∀i} 2m

Proof.

F (s) = ZΣ(φ) and so by a Theorem in Batyrev and Tschinkel the claim follows.

Sketch of the proof: Anisotropic Case (cont.)

m Y (i) 2 Consider H(s) := F (s + 1) π (s) i=1

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 1 D(δ) := {s ∈ Cm : <(π(i)(s)) > − ∀i} 2m

Proof.

F (s) = ZΣ(φ) and so by a Theorem in Batyrev and Tschinkel the claim follows.

Sketch of the proof: Anisotropic Case (cont.)

m Y (i) (i) 2 Consider H(s) := F (s + 1) π (s) where π (s) is the i=1 coordinate function. Then H(s) is holomorphic in the domain

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Proof.

F (s) = ZΣ(φ) and so by a Theorem in Batyrev and Tschinkel the claim follows.

Sketch of the proof: Anisotropic Case (cont.)

m Y (i) (i) 2 Consider H(s) := F (s + 1) π (s) where π (s) is the i=1 coordinate function. Then H(s) is holomorphic in the domain

1 D(δ) := {s ∈ Cm : <(π(i)(s)) > − ∀i} 2m

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Sketch of the proof: Anisotropic Case (cont.)

m Y (i) (i) 2 Consider H(s) := F (s + 1) π (s) where π (s) is the i=1 coordinate function. Then H(s) is holomorphic in the domain

1 D(δ) := {s ∈ Cm : <(π(i)(s)) > − ∀i} 2m

Proof.

F (s) = ZΣ(φ) and so by a Theorem in Batyrev and Tschinkel the claim follows.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach |H(s)| =  n  Y (i) 1− 1 min(0,<(π(i)(s))) 
O (|=(π (s)| + 1) 3 ) ∗ 1 + ||=(s)1|| i=1

is uniformly valid in the region D(1/2m − 0) when Re(s) < 1

Sketch of the proof: Anisotropic Case (cont.)

0 3 For all ,  > 0

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach is uniformly valid in the region D(1/2m − 0) when Re(s) < 1

Sketch of the proof: Anisotropic Case (cont.)

0 3 For all ,  > 0

|H(s)| =  n  Y (i) 1− 1 min(0,<(π(i)(s))) 
O (|=(π (s)| + 1) 3 ) ∗ 1 + ||=(s)1|| i=1

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Sketch of the proof: Anisotropic Case (cont.)

0 3 For all ,  > 0

|H(s)| =  n  Y (i) 1− 1 min(0,<(π(i)(s))) 
O (|=(π (s)| + 1) 3 ) ∗ 1 + ||=(s)1|| i=1

is uniformly valid in the region D(1/2m − 0) when Re(s) < 1

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach The estimates in this case follow from standard bounds for L-functions and integration by parts as described in the paper Batyrev and Tschinkel on anisotropic tori.

Sketch of the proof: Anisotropic Case (cont.)

In the anisotropic case the Poisson summation formula applied to the height zeta function gives a discrete sum over a lattice of characters.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Sketch of the proof: Anisotropic Case (cont.)

In the anisotropic case the Poisson summation formula applied to the height zeta function gives a discrete sum over a lattice of characters.

The estimates in this case follow from standard bounds for L-functions and integration by parts as described in the paper Batyrev and Tschinkel on anisotropic tori.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 2 There exists a family L of non trivial linear forms `i where 1 ≤ i ≤ n and another finite such family hr (not necessarily + non-trivial ) in LRm(C) such that

n Y H(s) := F (s + α) `(i)(s) i=1

A Tauberian Theorem of R. de la Bret`eche

Theorem (R. de la Bret`eche) f : arithmetic function on Nm, F : its Dirichlet Series:

∞ ∞ X X f (d1 ..., dm) F (s) = ··· s1 sm d1 ... dm 1≤d1 1≤dm with the following properties: 1 F (s) is absolutely convergent for s with <(s) > α where α ∈ (R+)m

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach A Tauberian Theorem of R. de la Bret`eche

Theorem (R. de la Bret`eche) f : arithmetic function on Nm, F : its Dirichlet Series:

∞ ∞ X X f (d1 ..., dm) F (s) = ··· s1 sm d1 ... dm 1≤d1 1≤dm with the following properties: 1 F (s) is absolutely convergent for s with <(s) > α where α ∈ (R+)m

2 There exists a family L of non trivial linear forms `i where 1 ≤ i ≤ n and another finite such family hr (not necessarily + non-trivial ) in LRm(C) such that

n Y H(s) := F (s + α) `(i)(s) i=1 Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach 0 3 There exists a δ2 > 0 such that for all ,  > 0 the following is 0 0 satisfied in D(δ1 −  , δ3 −  ): n  i  Y i
1−δ2min(0,<(` (s)))   |H(s)| = O |=(` (s))|+1 1+k=(s)k1 i=1 Then X X <α,β> −θ
··· f (d1 ..., dm) = X Qβ(log X ) + O(X ) β β d1≤X 1 dm≤X m

where Qβ ∈ R[X ] and θ > 0

A Tauberian Theorem of R. de la Bret`eche(cont.)

Theorem (R. de la Bret`eche)cont. can be extended to a holomorphic function in the domain

m (i) (r)(s) D(δ1, δ3) := {s ∈ C : <(` (s)) > −δ1, <(h ) > −δ3 ∀i, r}

for some δ1, δ3 > 0

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach n  i  Y i
1−δ2min(0,<(` (s)))   |H(s)| = O |=(` (s))|+1 1+k=(s)k1 i=1 Then X X <α,β> −θ
··· f (d1 ..., dm) = X Qβ(log X ) + O(X ) β β d1≤X 1 dm≤X m

where Qβ ∈ R[X ] and θ > 0

A Tauberian Theorem of R. de la Bret`eche(cont.)

Theorem (R. de la Bret`eche)cont. can be extended to a holomorphic function in the domain

m (i) (r)(s) D(δ1, δ3) := {s ∈ C : <(` (s)) > −δ1, <(h ) > −δ3 ∀i, r}

for some δ1, δ3 > 0 0 3 There exists a δ2 > 0 such that for all ,  > 0 the following is 0 0 satisfied in D(δ1 −  , δ3 −  ):

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Then X X <α,β> −θ
··· f (d1 ..., dm) = X Qβ(log X ) + O(X ) β β d1≤X 1 dm≤X m

where Qβ ∈ R[X ] and θ > 0

A Tauberian Theorem of R. de la Bret`eche(cont.)

Theorem (R. de la Bret`eche)cont. can be extended to a holomorphic function in the domain

m (i) (r)(s) D(δ1, δ3) := {s ∈ C : <(` (s)) > −δ1, <(h ) > −δ3 ∀i, r}

for some δ1, δ3 > 0 0 3 There exists a δ2 > 0 such that for all ,  > 0 the following is 0 0 satisfied in D(δ1 −  , δ3 −  ): n  i  Y i
1−δ2min(0,<(` (s)))   |H(s)| = O |=(` (s))|+1 1+k=(s)k1 i=1

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach A Tauberian Theorem of R. de la Bret`eche(cont.)

Theorem (R. de la Bret`eche)cont. can be extended to a holomorphic function in the domain

m (i) (r)(s) D(δ1, δ3) := {s ∈ C : <(` (s)) > −δ1, <(h ) > −δ3 ∀i, r}

for some δ1, δ3 > 0 0 3 There exists a δ2 > 0 such that for all ,  > 0 the following is 0 0 satisfied in D(δ1 −  , δ3 −  ): n  i  Y i
1−δ2min(0,<(` (s)))   |H(s)| = O |=(` (s))|+1 1+k=(s)k1 i=1 Then X X <α,β> −θ
··· f (d1 ..., dm) = X Qβ(log X ) + O(X ) β β d1≤X 1 dm≤X m

where Qβ ∈ R[X ] and θ > 0

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Technical Set up: Sufficient Functions

Definition p E := R , EC: its complexification, V : a subspace in E.A non-negative real valued function c on V is called sufficient if 1 For any subspace U ⊂ V and v ∈ V , the function U → R defined by u 7→ c(v + u) is measurable on U and Z cU (v) := c(v + u)du U is finite. 2 For any subspace U ⊂ V and for each v ∈ V \ U we have

lim cU (τ · v) = 0 τ→±∞

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Definition f is called as distinguished with respect to the data consisting of V and linearly independent linear forms `1, . . . , `m if it satisfies 1 The function m Y `j (s) g(s) := f (s) `j (s) + 1 j=1

is holomorphic in TB .

2 For some c for all compact subsets K ⊂ TB , there is a constant κ with |g(s + iv)| ≤ κc(v)

Technical Set up: Distinguished Functions

`1, . . . , `m: linearly independent linear forms on E, B: convex and open neighborhood of 0 = (0,..., 0) in E such that `j (x) > −1, TB := B + iE, f ∈ M(TB )

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Technical Set up: Distinguished Functions

`1, . . . , `m: linearly independent linear forms on E, B: convex and open neighborhood of 0 = (0,..., 0) in E such that `j (x) > −1, TB := B + iE, f ∈ M(TB ) Definition f is called as distinguished with respect to the data consisting of V and linearly independent linear forms `1, . . . , `m if it satisfies 1 The function m Y `j (s) g(s) := f (s) `j (s) + 1 j=1

is holomorphic in TB .

2 For some c for all compact subsets K ⊂ TB , there is a constant κ with |g(s + iv)| ≤ κc(v)

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Z ˜ 1 fC (s) := d f (s + iv)dv (2π) V Proposition (Strauch and Tschinkel) Let f be distinguished function w.r.t some data. Then

1 f˜C : TC → C is holomorphic. 2 There is an open and convex neighborhood B˜ of 0 = (0,..., 0), containing C, and linear forms `ei (1 ≤ i ≤ m˜) vanishing on V such that

m˜ Y s 7→ f˜C (s) `˜j (s) j=1

has a holomorphic continuation to TB˜

A Crucial Property of Distinguished Functions

Let C be a connected component of B \ ∪Ker(`j ). On TC define:

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach A Crucial Property of Distinguished Functions

Let C be a connected component of B \ ∪Ker(`j ). On TC define: Z ˜ 1 fC (s) := d f (s + iv)dv (2π) V Proposition (Strauch and Tschinkel) Let f be distinguished function w.r.t some data. Then

1 f˜C : TC → C is holomorphic. 2 There is an open and convex neighborhood B˜ of 0 = (0,..., 0), containing C, and linear forms `ei (1 ≤ i ≤ m˜) vanishing on V such that

m˜ Y s 7→ f˜C (s) `˜j (s) j=1

has a holomorphic continuation to TB˜

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach General Case: ˜ 1 R Consider: fC (s) := (2π)d f (s + iv) dv where d = dim V V By using the method of Strauch and Tschinkel (1999), we do Induction on d.

Idea of the proof: General Case

Anisotropic Case Base Case.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach By using the method of Strauch and Tschinkel (1999), we do Induction on d.

Idea of the proof: General Case

Anisotropic Case Base Case.

General Case: ˜ 1 R Consider: fC (s) := (2π)d f (s + iv) dv where d = dim V V

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Idea of the proof: General Case

Anisotropic Case Base Case.

General Case: ˜ 1 R Consider: fC (s) := (2π)d f (s + iv) dv where d = dim V V By using the method of Strauch and Tschinkel (1999), we do Induction on d.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach References

V. Batyrev and Yu. Tschinkel, Manin’s conjecture for toric varieties, Journ. of Alg. Geometry 7, 1998. V. Batyrev and Yu. Tschinkel, Rational points of bounded height on compactifications of anisotropic tori, IMRN, 1995. R. de la Bret`eche, Compter des points d’une variete torique, 2000. R. de la Bret`eche, Estimation de sommes multiples de fonctions arithmetiques, 1998. D. Cox, J. Little, and H. Schenck Toric Varieties, 2010. W. Fulton, Introduction to toric varieties, Ann. of Math. Studies 131, Princeton Univ. Press, 1993.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach References (cont.)

E. Peyre, Beyond heights: slopes and distribution of rational points, dans Arakelov geometry and diophantine applications, Lecture Notes in Mathematics 2276 (2020), 1-67. E. Peyre, Libert´eet accumulation, Documenta Math. 22 (2017), 1615-1659. J-P Serre, Lectures on the Mordell-Weil theorem, translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. With a foreword by Brown and Serre. Third edition. Aspects of Mathematics. Friedr Vieweg and Sohn, Braunschweig, 1997. x+218 pp. M. Strauch, Yu. Tschinkel, Height zeta functions of toric bundles over flag varieties, Selecta Math. (N.S.) 5 (1999), no. 3, p. 325-396. G. Tenenbaum, Introduction `ala th´eorie analytique et probabiliste des nombres, 2`eme´ed.,Cours Sp´ecialis´es,No. 1, Soci´et´e Math´ematiquede France, 1995.

Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach