8月19日-报告9-莫毅明-Complex differential geometry in the solution of arithmetico-geometric problems over complex function fields.pdf

Complex Differential Geometry in the Solution of Arithmetico-Geometric Problems over Complex Function Fields Ngaiming Mok The University of Hong Kong Conference on Several Complex Variables Shanghai University Shanghai, China August 19, 2022 Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 1 / 40 Moduli Space of Elliptic Curves An elliptic curve is complex-analytically a compact Riemann surface S of genus 1. In other words, S := C/L for some lattice L ⊂ C. Replacing L by λL for some λ ∈ C − {0}, without loss of generality we may assume Lτ = Z + Zτ , Im(τ ) > 0, i.e., τ ∈ H, where H := τ ∈ C : Im(τ ) > 0 , the upper half plane. Write Sτ := C/Lτ . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 2 / 40 Moduli Space of Elliptic Curves An elliptic curve is complex-analytically a compact Riemann surface S of genus 1. In other words, S := C/L for some lattice L ⊂ C. Replacing L by λL for some λ ∈ C − {0}, without loss of generality we may assume Lτ = Z + Zτ , Im(τ ) > 0, i.e., τ ∈ H, where H := τ ∈ C : Im(τ ) > 0 , the upper half plane. Write Sτ := C/Lτ . For τ, τ 0 ∈ H, we have Sτ ∼ = Sτ 0 if and only if there exists λ ∈ C, aτ +b λ 6= 0 such that Lτ 0 = λLτ , i.e., if and only if τ 0 = cτ +d where ad − bc 6= 0. Thus, the set of equivalence classes of C/L is in one-to-one correspondence with X = X (1) := H/PSL(2, Z). PSL(2, Z) acts discretely on H with fixed points. We have the j-function ∼ = j : X (1) −→ C, and X (1) = P1 . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 2 / 40 Moduli Space of Elliptic Curves An elliptic curve is complex-analytically a compact Riemann surface S of genus 1. In other words, S := C/L for some lattice L ⊂ C. Replacing L by λL for some λ ∈ C − {0}, without loss of generality we may assume Lτ = Z + Zτ , Im(τ ) > 0, i.e., τ ∈ H, where H := τ ∈ C : Im(τ ) > 0 , the upper half plane. Write Sτ := C/Lτ . For τ, τ 0 ∈ H, we have Sτ ∼ = Sτ 0 if and only if there exists λ ∈ C, aτ +b λ 6= 0 such that Lτ 0 = λLτ , i.e., if and only if τ 0 = cτ +d where ad − bc 6= 0. Thus, the set of equivalence classes of C/L is in one-to-one correspondence with X = X (1) := H/PSL(2, Z). PSL(2, Z) acts discretely on H with fixed points. We have the j-function ∼ = j : X (1) −→ C, and X (1) = P1 . A suitable finite-index subgroup Γ ⊂ PSL(2, Z) acts on H without fixed points and XΓ := H/Γ can be compactified to a compact Riemann surface. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 2 / 40 The j-function On the upper half plane H = {τ : Im(τ ) > 0} define j(τ ) = 1728 where g2 (τ ) = 60 X g2 (τ )3 g2 (τ )3 = 1728 g2 (τ )3 − 27g3 (τ )2 ∆(τ ) (m+nτ )−4 ; g3 (τ ) = 140 X (m+nτ )−6 . (m,n)6=(0,0) (m,n)6=(0,0) and ∆(τ ) := g2 (τ )3 − 27g3 (τ )2 is the modular discriminant. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 3 / 40 The j-function On the upper half plane H = {τ : Im(τ ) > 0} define j(τ ) = 1728 where g2 (τ ) = 60 X g2 (τ )3 g2 (τ )3 = 1728 g2 (τ )3 − 27g3 (τ )2 ∆(τ ) (m+nτ )−4 ; g3 (τ ) = 140 X (m+nτ )−6 . (m,n)6=(0,0) (m,n)6=(0,0) and ∆(τ ) := g2 (τ )3 − 27g3 (τ )2 is the modular discriminant. ∼ = The j-function establishes a biholomorphism j : H/SL(2, Z) −→ C. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 3 / 40 Invariant Kähler metrics on H × C On π : H × C → H, there is the relative tangent bundle V = Tπ , and the horizontal real-analytic integrable subbundle H ⊂ T (H × C) whose leaves are images of horizontal sections w = a + bτ , a, b ∈ R. We have T (H × C) = V ⊕ H. There is a semi-Kähler form µ with kernel H so that, denoting by ω the Kähler form of the Poincaré metric on H, and defining νt := π ∗ ω + t 2 µ, t > 0, (H × C, νt ) is a Kähler form invariant under SL(2, R) n R2 . Let Γ ⊂ SL(2, Z) be a torsion-free finite index subgroup. Write XΓ0 := H/Γ, M0Γ = (H × C)/(Γ n Z2 ), π : MΓ → XΓ a compactification to a minimal elliptic surface over the projective curve XΓ . Verticality of a section Let σ : XΓ → MΓ be a holomorphic section and dσ : TXΓ → σ ∗ T (MΓ ) be its differential. Define the verticality of σ as ησ := ΠV ◦ dσ|T (X 0 ) : T (XΓ0 ) → σ ∗ V . Thus, ησ is a real-analytic section Γ of the holomorphic line bundle T ∗ (XΓ0 ) ⊗ σ ∗ V on XΓ0 . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 4 / 40 Shioda’s Theorem: A differential-geometric proof Proposition (geometric characterization of torsion sections) ησ ≡ 0 if and only if σ is a torsion section. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 5 / 40 Shioda’s Theorem: A differential-geometric proof Proposition (geometric characterization of torsion sections) ησ ≡ 0 if and only if σ is a torsion section. Shioda’s Theorem (diff.-geom. proof by Mok (1991)) The Mordell-Well group of the elliptic curve EΓ over C(XΓ ) is finite. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 5 / 40 Shioda’s Theorem: A differential-geometric proof Proposition (geometric characterization of torsion sections) ησ ≡ 0 if and only if σ is a torsion section. Shioda’s Theorem (diff.-geom. proof by Mok (1991)) The Mordell-Well group of the elliptic curve EΓ over C(XΓ ) is finite. Proof: Given a holomorphic section σ : XΓ → MΓ σ corresponds to f : H → C satisfying f (γτ ) = cγfτ(τ+d) γ + Aγ (γτ ) + Bγ for some integers a τ +b Aγ , Bγ , in which γ(τ ) = cγγ +dγγ . Then, f 00 (γτ ) = (cγ τ + dγ )3 f 00 (τ ) . 3 (Eichler) We discovered that ξσ := f 00 (τ )(dτ ) 2 is actually given by ξσ = ∇ησ . We have ∂ξσ = 0, hence ∂∇ησ = 0. Interchanging the order of ∗ differention we have ∇ ∇ησ = −ησ . Integrating by parts we get R R 2ω = − 2 kη k σ XΓ XΓ k∇ησ k ω , forcing ησ ≡ 0, hence σ is a torsion section. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 5 / 40 Betti coordinates and the Betti map of a section Betti coordinates On H × C, for a point (τ, w ), express w in terms of a basis of the lattice Lτ , e.g., w = β1 · 1 + β2 τ . The pair (β1 , β2 ) are Betti coordinates. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 6 / 40 Betti coordinates and the Betti map of a section Betti coordinates On H × C, for a point (τ, w ), express w in terms of a basis of the lattice Lτ , e.g., w = β1 · 1 + β2 τ . The pair (β1 , β2 ) are Betti coordinates. The Betti map associated to a holomorphic section σ For a holomorphic section σ : XΓ → MΓ , the local pullback β := (σ ∗ β1 , σ ∗ β2 ) is called the Betti map of σ. Since the construction of (β1 , β2 ) involves a choice of abelian logarithm on M0Γ , so does the Betti map β, but the vanishing order of β at any point b ∈ B 0 is independent of such choice and is intrinsic to the section σ. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 6 / 40 The Betti map The following definition is due to Corvaja-Demeio-Masser-Zannier. The Betti multiplicity of a Betti map at a finite point The multiplicity of a Betti map β at b is defined to be the smallest positive integer m(b) such that the partial derivatives of σ ∗ β1 , σ ∗ β2 at b vanish up to order m(b) − 1. We will also call m(b) the Betti multiplicity of σ at b. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 7 / 40 The Betti map The following definition is due to Corvaja-Demeio-Masser-Zannier. The Betti multiplicity of a Betti map at a finite point The multiplicity of a Betti map β at b is defined to be the smallest positive integer m(b) such that the partial derivatives of σ ∗ β1 , σ ∗ β2 at b vanish up to order m(b) − 1. We will also call m(b) the Betti multiplicity of σ at b. The Betti multiplicity of a Betti map at a cusp When a holomorphic section σ cuts over a base point c of bad reduction, i.e., corresponding to a cusp, we express the section σ locally near the cusp c in terms of toroidal compactification Σ(w ) = (ξ(w ), ζ(w )) If |ξ(0)| = 1, then we define the Betti multiplicity mc of σ at c to be the vanishing order of ξ(w ) − ξ(0) at w = 0. Otherwise, we define mc = 1. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 7 / 40 Betti Multiplicities for a Section of an Elliptic Surface Theorem (Ulmer-Ursúa IMRN 2021) Suppose π : E → B is a non-isotrivial minimal elliptic surface, with exactly δ singular fibers, and σ : B → E be a section of infinite order. Denote by g be the genus of B. Let O denote the zero section of E and denote by d the degree of the holomorphic line bundle O ∗ Ω1E|C , where Ω1E|C denotes the dual of the relative tangent bundle. Denote by S ⊂ B the set of base points of singular fibers, and write B 0 := B − S. Then, P b∈B 0 (mb − 1) ≤ 2g − 2 − d + δ . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 8 / 40 Betti Multiplicities for a Section of an Elliptic Surface Theorem (Ulmer-Ursúa IMRN 2021) Suppose π : E → B is a non-isotrivial minimal elliptic surface, with exactly δ singular fibers, and σ : B → E be a section of infinite order. Denote by g be the genus of B. Let O denote the zero section of E and denote by d the degree of the holomorphic line bundle O ∗ Ω1E|C , where Ω1E|C denotes the dual of the relative tangent bundle. Denote by S ⊂ B the set of base points of singular fibers, and write B 0 := B − S. Then, P b∈B 0 (mb − 1) ≤ 2g − 2 − d + δ . (a) The finiteness of points of B 0 with multiplicities ≥ 2 was due to Corvaja-Demeio-Masser-Zannier (Crelles 2022) (b) Multiplities mc at cusps were defined algebraically and using the Kodaira classification of elliptic surfaces, and the analytic definition of Mok-Ng using toroidal coordinates agree with the algebraic definition. Equality was proven when the sum on the left hand side is replaced by taking all b ∈ B, including the cusps. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 8 / 40 Diff.-geom. proof for estimates on Betti multiplicities Theorem (Mok-Ng 2022) Let E → B be an elliptic surface over a projective curve B with a classifying map f : B → X of degree d, where X = XΓ(k) for some k ≥ 3. Let σ be a non-torsion section of E and mb be the Betti multiplicity of σ at b, then Z X X d (mb − 1) = (rb − 1) + ω, 2π X 0 b∈B b∈B\S 0 where X 0 = XΓ(k) and S = f −1 (X \ X 0 ); rb is the ramification index of f at b and ω is the Kähler form on X 0 descending from the invariant form −i∂ ∂¯ log Imτ on H. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 9 / 40 Diff.-geom. proof for estimates on Betti multiplicities Theorem (Mok-Ng 2022) Let E → B be an elliptic surface over a projective curve B with a classifying map f : B → X of degree d, where X = XΓ(k) for some k ≥ 3. Let σ be a non-torsion section of E and mb be the Betti multiplicity of σ at b, then Z X X d (mb − 1) = (rb − 1) + ω, 2π X 0 b∈B b∈B\S 0 where X 0 = XΓ(k) and S = f −1 (X \ X 0 ); rb is the ramification index of f at b and ω is the Kähler form on X 0 descending from the invariant form −i∂ ∂¯ log Imτ on H. The general case can be reduced to the case with classifying maps. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 9 / 40 Diff.-geom. proof for estimates on Betti multiplicities Theorem (Mok-Ng 2022) Let E → B be an elliptic surface over a projective curve B with a classifying map f : B → X of degree d, where X = XΓ(k) for some k ≥ 3. Let σ be a non-torsion section of E and mb be the Betti multiplicity of σ at b, then Z X X d (mb − 1) = (rb − 1) + ω, 2π X 0 b∈B b∈B\S 0 where X 0 = XΓ(k) and S = f −1 (X \ X 0 ); rb is the ramification index of f at b and ω is the Kähler form on X 0 descending from the invariant form −i∂ ∂¯ log Imτ on H. The general case can be reduced to the case with classifying maps. Corollary Denote by Bσ the divisor of points on B 0 over which the Betti multiplicity mb ≥ 2, with weight mb − 1 at each of these points. We have 1 |Bσ | ≤ 2g − 2 − deg(f ∗ (KX ⊗ SX ) 2 )) + |S| , where g is the genus of B. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 9 / 40 Mordell-Weil Groups Complex Function Fields Main Theorem (Mok-To (Crelles 1991)) Let π : AΓ → XΓ be a Kuga family of polarized abelian varieties without locally constant parts, π : AΓ → XΓ be a projective compactification which is a geometic model for the associated modular polarized abelian variety AΓ over C(XΓ . Then, there are at most a finite number of meromorphic sections of AΓ over XΓ , i.e., rankZ (AΓ (C(XΓ ))) = 0 for the Mordell-Weil group AΓ (C(XΓ ). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 10 / 40 Mordell-Weil Groups Complex Function Fields Main Theorem (Mok-To (Crelles 1991)) Let π : AΓ → XΓ be a Kuga family of polarized abelian varieties without locally constant parts, π : AΓ → XΓ be a projective compactification which is a geometic model for the associated modular polarized abelian variety AΓ over C(XΓ . Then, there are at most a finite number of meromorphic sections of AΓ over XΓ , i.e., rankZ (AΓ (C(XΓ ))) = 0 for the Mordell-Weil group AΓ (C(XΓ ). Mordell-Weil group for f : B → XΓ dominant and equidimensional Theorem(Mok 1991) Let Γ ⊂ Sp(g , Z) be torsion-free. Suppose dim(B) = dim(XΓ ) and f : B → XΓ is a dominant classifying map. Denote by Af the elliptic curve over C(B) obtained by pulling back the universal abelian variety AΓ over C(XΓ ) by the classifying map f . Then, rankZ Af (C(B)) ≤ C · Volume(Rf , ω) , where ω is the Kähler-Einstein (1,1)-form on XΓ , C is a universal constant depending only on XΓ , and Rf is the ramification divisor f : B → XΓ . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 10 / 40 Shimura varieties: An example The Siegel upper half-plane Hg L ⊂ Cn lattice, C/L = A ∼ = S 1 × · · · × S 1 (2g copies), H 1 (A, R) ∼ = R2g first de Rham cohomology group. A is called an Abelian variety if A ,→ PN is (projective)-algebraic. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 11 / 40 Shimura varieties: An example The Siegel upper half-plane Hg L ⊂ Cn lattice, C/L = A ∼ = S 1 × · · · × S 1 (2g copies), H 1 (A, R) ∼ = R2g first de Rham cohomology group. A is called an Abelian variety if A ,→ PN is (projective)-algebraic. A (principally polarized) Abelian variety corresponds to an n-by-n matrix τ obeying Riemann bilinear relations (a) τ is symmetric, (b) Im(τ ) > 0. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 11 / 40 Shimura varieties: An example The Siegel upper half-plane Hg L ⊂ Cn lattice, C/L = A ∼ = S 1 × · · · × S 1 (2g copies), H 1 (A, R) ∼ = R2g first de Rham cohomology group. A is called an Abelian variety if A ,→ PN is (projective)-algebraic. A (principally polarized) Abelian variety corresponds to an n-by-n matrix τ obeying Riemann bilinear relations (a) τ is symmetric, (b) Im(τ ) > 0. Lτ ⊂ Cg is spanned by basis vectors e1 , · · · , eg and column vectors τ1 , · · · τg of τ , Aτ := Cg /Lτ . Hg := {τ ∈ Mg (C) : τ t = τ ; Im(τ ) > 0}. The Cayley transform κ(τ ) = (τ − ıIg ) (τ + ıIg )−1 gives a biholomorphism ∼ = κ : Hg −→ DgIII = {Z ∈ Mg (C) : Z t = Z , I − Z Z > 0} with a BSD. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 11 / 40 Shimura varieties: An example The Siegel upper half-plane Hg L ⊂ Cn lattice, C/L = A ∼ = S 1 × · · · × S 1 (2g copies), H 1 (A, R) ∼ = R2g first de Rham cohomology group. A is called an Abelian variety if A ,→ PN is (projective)-algebraic. A (principally polarized) Abelian variety corresponds to an n-by-n matrix τ obeying Riemann bilinear relations (a) τ is symmetric, (b) Im(τ ) > 0. Lτ ⊂ Cg is spanned by basis vectors e1 , · · · , eg and column vectors τ1 , · · · τg of τ , Aτ := Cg /Lτ . Hg := {τ ∈ Mg (C) : τ t = τ ; Im(τ ) > 0}. The Cayley transform κ(τ ) = (τ − ıIg ) (τ + ıIg )−1 gives a biholomorphism ∼ = κ : Hg −→ DgIII = {Z ∈ Mg (C) : Z t = Z , I − Z Z > 0} with a BSD. We have a Hodge decomposition H 1 (A, C) = H 0 (A, ΩA ) ⊕ H 1 (A, OA ) in terms of ∂-cohomology and harmonic forms. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 11 / 40 Shimura varieties: An example The Siegel upper half-plane Hg L ⊂ Cn lattice, C/L = A ∼ = S 1 × · · · × S 1 (2g copies), H 1 (A, R) ∼ = R2g first de Rham cohomology group. A is called an Abelian variety if A ,→ PN is (projective)-algebraic. A (principally polarized) Abelian variety corresponds to an n-by-n matrix τ obeying Riemann bilinear relations (a) τ is symmetric, (b) Im(τ ) > 0. Lτ ⊂ Cg is spanned by basis vectors e1 , · · · , eg and column vectors τ1 , · · · τg of τ , Aτ := Cg /Lτ . Hg := {τ ∈ Mg (C) : τ t = τ ; Im(τ ) > 0}. The Cayley transform κ(τ ) = (τ − ıIg ) (τ + ıIg )−1 gives a biholomorphism ∼ = κ : Hg −→ DgIII = {Z ∈ Mg (C) : Z t = Z , I − Z Z > 0} with a BSD. We have a Hodge decomposition H 1 (A, C) = H 0 (A, ΩA ) ⊕ H 1 (A, OA ) in terms of ∂-cohomology and harmonic forms. Sp(g ; R) acts on Hg as hol. isometries. The arithmetic subgroup Sp(g ; Z) ⊂ Sp(g ; R) acts on Hg as a discrete group. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 11 / 40 Shimura varieties: An example The Siegel upper half-plane Hg L ⊂ Cn lattice, C/L = A ∼ = S 1 × · · · × S 1 (2g copies), H 1 (A, R) ∼ = R2g first de Rham cohomology group. A is called an Abelian variety if A ,→ PN is (projective)-algebraic. A (principally polarized) Abelian variety corresponds to an n-by-n matrix τ obeying Riemann bilinear relations (a) τ is symmetric, (b) Im(τ ) > 0. Lτ ⊂ Cg is spanned by basis vectors e1 , · · · , eg and column vectors τ1 , · · · τg of τ , Aτ := Cg /Lτ . Hg := {τ ∈ Mg (C) : τ t = τ ; Im(τ ) > 0}. The Cayley transform κ(τ ) = (τ − ıIg ) (τ + ıIg )−1 gives a biholomorphism ∼ = κ : Hg −→ DgIII = {Z ∈ Mg (C) : Z t = Z , I − Z Z > 0} with a BSD. We have a Hodge decomposition H 1 (A, C) = H 0 (A, ΩA ) ⊕ H 1 (A, OA ) in terms of ∂-cohomology and harmonic forms. Sp(g ; R) acts on Hg as hol. isometries. The arithmetic subgroup Sp(g ; Z) ⊂ Sp(g ; R) acts on Hg as a discrete group. Ag := Hg /Sp(g ; Z) is called the Siegel modular variety. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 11 / 40 Shimura varieties: An example The Siegel upper half-plane Hg L ⊂ Cn lattice, C/L = A ∼ = S 1 × · · · × S 1 (2g copies), H 1 (A, R) ∼ = R2g first de Rham cohomology group. A is called an Abelian variety if A ,→ PN is (projective)-algebraic. A (principally polarized) Abelian variety corresponds to an n-by-n matrix τ obeying Riemann bilinear relations (a) τ is symmetric, (b) Im(τ ) > 0. Lτ ⊂ Cg is spanned by basis vectors e1 , · · · , eg and column vectors τ1 , · · · τg of τ , Aτ := Cg /Lτ . Hg := {τ ∈ Mg (C) : τ t = τ ; Im(τ ) > 0}. The Cayley transform κ(τ ) = (τ − ıIg ) (τ + ıIg )−1 gives a biholomorphism ∼ = κ : Hg −→ DgIII = {Z ∈ Mg (C) : Z t = Z , I − Z Z > 0} with a BSD. We have a Hodge decomposition H 1 (A, C) = H 0 (A, ΩA ) ⊕ H 1 (A, OA ) in terms of ∂-cohomology and harmonic forms. Sp(g ; R) acts on Hg as hol. isometries. The arithmetic subgroup Sp(g ; Z) ⊂ Sp(g ; R) acts on Hg as a discrete group. Ag := Hg /Sp(g ; Z) is called the Siegel modular variety. In general, for Ω a BSD and an arithmetic subgroup Γ ⊂ Aut(Ω), XΓ := Ω/Γ is called a Shimura variety. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 11 / 40 Irreducible Bounded Symmetric Domains The rank-1 case The complex unit ball Bn := z ∈ Cn : kzk2 < 1 Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 12 / 40 Irreducible Bounded Symmetric Domains The rank-1 case The complex unit ball Bn := z ∈ Cn : kzk2 < 1 Classical domains in general t D I (p, q) = {Z ∈ M(p, q, C) : I − Z Z > 0} , I DnII (n, n) = {Z ∈ Dn,n : Z t = −Z } , p, q ≥ 1 n≥2 I DnIII = {Z ∈ Dn,n : Zt = Z} , n ≥ 3 DnIV = (z1 , . . . , zn ) ∈ Cn : kzk2 < 2 ; n X 2 2 1 kzk < 1 + 2 , n≥3. zi2 i=1 Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 12 / 40 Irreducible Bounded Symmetric Domains The rank-1 case The complex unit ball Bn := z ∈ Cn : kzk2 < 1 Classical domains in general t D I (p, q) = {Z ∈ M(p, q, C) : I − Z Z > 0} , I DnII (n, n) = {Z ∈ Dn,n : Z t = −Z } , p, q ≥ 1 n≥2 I DnIII = {Z ∈ Dn,n : Zt = Z} , n ≥ 3 DnIV = (z1 , . . . , zn ) ∈ Cn : kzk2 < 2 ; n X 2 2 1 kzk < 1 + 2 , n≥3. zi2 i=1 Exceptional domains D V , dim 16, type E6 ; D VI , dim 27, type E7 Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 12 / 40 The André-Oort Conjecture A point τ ∈ H such that τ, j(τ ) ∈ Q is called a special point (in which case [Q(τ ) : Q] = 2 by Schneider). The notion of special points is defined for any Shimura variety XΓ = Ω/Γ, and the André-Oort Conjecture ascertains that the Zariski closure of any set of special points on XΓ is a finite union of Shimura subvarieties XΓ0 0 ,→ XΓ . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 13 / 40 The André-Oort Conjecture A point τ ∈ H such that τ, j(τ ) ∈ Q is called a special point (in which case [Q(τ ) : Q] = 2 by Schneider). The notion of special points is defined for any Shimura variety XΓ = Ω/Γ, and the André-Oort Conjecture ascertains that the Zariski closure of any set of special points on XΓ is a finite union of Shimura subvarieties XΓ0 0 ,→ XΓ . The Pila-Zannier strategy Pila-Zannier [PZ10] proposed strategy for finiteness and characterization problems concerning distinguished points in different arithmetic contexts (e.g. torsion points on Abelian varieties, special points on Shimura varieties). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 13 / 40 The André-Oort Conjecture A point τ ∈ H such that τ, j(τ ) ∈ Q is called a special point (in which case [Q(τ ) : Q] = 2 by Schneider). The notion of special points is defined for any Shimura variety XΓ = Ω/Γ, and the André-Oort Conjecture ascertains that the Zariski closure of any set of special points on XΓ is a finite union of Shimura subvarieties XΓ0 0 ,→ XΓ . The Pila-Zannier strategy Pila-Zannier [PZ10] proposed strategy for finiteness and characterization problems concerning distinguished points in different arithmetic contexts (e.g. torsion points on Abelian varieties, special points on Shimura varieties). For the André-Oort Conjecture on a Shimura variety XΓ = Ω/Γ, π : Ω → XΓ , it breaks down into (a) an arithmetic component consisting of lower estimates on the size of Galois orbits of special points and Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 13 / 40 The André-Oort Conjecture A point τ ∈ H such that τ, j(τ ) ∈ Q is called a special point (in which case [Q(τ ) : Q] = 2 by Schneider). The notion of special points is defined for any Shimura variety XΓ = Ω/Γ, and the André-Oort Conjecture ascertains that the Zariski closure of any set of special points on XΓ is a finite union of Shimura subvarieties XΓ0 0 ,→ XΓ . The Pila-Zannier strategy Pila-Zannier [PZ10] proposed strategy for finiteness and characterization problems concerning distinguished points in different arithmetic contexts (e.g. torsion points on Abelian varieties, special points on Shimura varieties). For the André-Oort Conjecture on a Shimura variety XΓ = Ω/Γ, π : Ω → XΓ , it breaks down into (a) an arithmetic component consisting of lower estimates on the size of Galois orbits of special points and (b) a geometric component consisting of the characterization of Zariski closures of π(S) ⊂ XΓ for an algebraic subset S ⊂ Ω. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 13 / 40 Theorem of Gelfond-Schneider Lang’s general formulation Main Theorem (Lang 1966) Let K be a number field, f1 , · · · , fN be meromorphic functions on C of order ρ, Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 14 / 40 Theorem of Gelfond-Schneider Lang’s general formulation Main Theorem (Lang 1966) Let K be a number field, f1 , · · · , fN be meromorphic functions on C of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ 2, ∂ and D = ∂z : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 14 / 40 Theorem of Gelfond-Schneider Lang’s general formulation Main Theorem (Lang 1966) Let K be a number field, f1 , · · · , fN be meromorphic functions on C of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ 2, ∂ and D = ∂z : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ). Let x1 , ..., xm be distinct complex numbers outside the union of pole sets of f1 , · · · , fN such that fi (xν ) ∈ K for 1 ≤ i ≤ N, 1 ≤ ν ≤ m. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 14 / 40 Theorem of Gelfond-Schneider Lang’s general formulation Main Theorem (Lang 1966) Let K be a number field, f1 , · · · , fN be meromorphic functions on C of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ 2, ∂ and D = ∂z : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ). Let x1 , ..., xm be distinct complex numbers outside the union of pole sets of f1 , · · · , fN such that fi (xν ) ∈ K for 1 ≤ i ≤ N, 1 ≤ ν ≤ m. Then, m ≤ 20ρ[K : Q]. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 14 / 40 Theorem of Gelfond-Schneider Lang’s general formulation Main Theorem (Lang 1966) Let K be a number field, f1 , · · · , fN be meromorphic functions on C of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ 2, ∂ and D = ∂z : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ). Let x1 , ..., xm be distinct complex numbers outside the union of pole sets of f1 , · · · , fN such that fi (xν ) ∈ K for 1 ≤ i ≤ N, 1 ≤ ν ≤ m. Then, m ≤ 20ρ[K : Q]. Hermite-Lindemann (1882) Corollary Let α 6= 0 be an algebraic number. Then, e α ∈ / Q. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 14 / 40 Theorem of Gelfond-Schneider Lang’s general formulation Main Theorem (Lang 1966) Let K be a number field, f1 , · · · , fN be meromorphic functions on C of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ 2, ∂ and D = ∂z : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ). Let x1 , ..., xm be distinct complex numbers outside the union of pole sets of f1 , · · · , fN such that fi (xν ) ∈ K for 1 ≤ i ≤ N, 1 ≤ ν ≤ m. Then, m ≤ 20ρ[K : Q]. Hermite-Lindemann (1882) Corollary Let α 6= 0 be an algebraic number. Then, e α ∈ / Q. α α Proof. Assume e algebraic. Put K = Q(α, e ); f (z) = z, g (z) = e z . Main Theorem applies but f , g take values in K for xk = kα, k ∈ N, contradiction! Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 14 / 40 Theorem of Gelfond-Schneider Lang’s general formulation Main Theorem (Lang 1966) Let K be a number field, f1 , · · · , fN be meromorphic functions on C of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ 2, ∂ and D = ∂z : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ). Let x1 , ..., xm be distinct complex numbers outside the union of pole sets of f1 , · · · , fN such that fi (xν ) ∈ K for 1 ≤ i ≤ N, 1 ≤ ν ≤ m. Then, m ≤ 20ρ[K : Q]. Hermite-Lindemann (1882) Corollary Let α 6= 0 be an algebraic number. Then, e α ∈ / Q. α α Proof. Assume e algebraic. Put K = Q(α, e ); f (z) = z, g (z) = e z . Main Theorem applies but f , g take values in K for xk = kα, k ∈ N, contradiction! Hence, e = e 1 ∈ / Q; e 2πı = 1 ∈ Q ⇒ π ∈ / Q. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 14 / 40 Theorem of Gelfond-Schneider Lang’s general formulation Main Theorem (Lang 1966) Let K be a number field, f1 , · · · , fN be meromorphic functions on C of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ 2, ∂ and D = ∂z : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ). Let x1 , ..., xm be distinct complex numbers outside the union of pole sets of f1 , · · · , fN such that fi (xν ) ∈ K for 1 ≤ i ≤ N, 1 ≤ ν ≤ m. Then, m ≤ 20ρ[K : Q]. Hermite-Lindemann (1882) Corollary Let α 6= 0 be an algebraic number. Then, e α ∈ / Q. α α Proof. Assume e algebraic. Put K = Q(α, e ); f (z) = z, g (z) = e z . Main Theorem applies but f , g take values in K for xk = kα, k ∈ N, contradiction! Hence, e = e 1 ∈ / Q; e 2πı = 1 ∈ Q ⇒ π ∈ / Q. Gelfond-Schneider (1934) Corollary Let α, β ∈ Q, α 6= 0, 1 and β ∈ / Q. Then, αβ ∈ /Q. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 14 / 40 Theorem of Gelfond-Schneider Lang’s general formulation Main Theorem (Lang 1966) Let K be a number field, f1 , · · · , fN be meromorphic functions on C of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ 2, ∂ and D = ∂z : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ). Let x1 , ..., xm be distinct complex numbers outside the union of pole sets of f1 , · · · , fN such that fi (xν ) ∈ K for 1 ≤ i ≤ N, 1 ≤ ν ≤ m. Then, m ≤ 20ρ[K : Q]. Hermite-Lindemann (1882) Corollary Let α 6= 0 be an algebraic number. Then, e α ∈ / Q. α α Proof. Assume e algebraic. Put K = Q(α, e ); f (z) = z, g (z) = e z . Main Theorem applies but f , g take values in K for xk = kα, k ∈ N, contradiction! Hence, e = e 1 ∈ / Q; e 2πı = 1 ∈ Q ⇒ π ∈ / Q. Gelfond-Schneider (1934) Corollary Let α, β ∈ Q, α 6= 0, 1 and β ∈ / Q. Then, αβ ∈ /Q. Proof. Assume αβ ∈ Q. Put K = Q(α, β, αβ ); f (z) = e z , g (z) = e βz ; xk = k log α, k ∈ N to get a contradiction. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 14 / 40 Lindemann-Weierstrass Theorem and Schanuel Conjecture Lindemann-Weiersrstrass Theorem (1882) Suppose α1 , · · · , αn ∈ Q are Q-linearly independent. Then, e α1 , · · · , e αn are algebraically independent. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 15 / 40 Lindemann-Weierstrass Theorem and Schanuel Conjecture Lindemann-Weiersrstrass Theorem (1882) Suppose α1 , · · · , αn ∈ Q are Q-linearly independent. Then, e α1 , · · · , e αn are algebraically independent. Schanuel Conjecture (1960s) Suppose α1 , · · · , αn ∈ C are Q-linearly independent. Then, trans.deg.Q Q (α1 , · · · , αn ; e α1 , · · · , e αn ) ≥ n. LWT answers the special case of SC where α1 , · · · , αn ∈ Q Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 15 / 40 Lindemann-Weierstrass Theorem and Schanuel Conjecture Lindemann-Weiersrstrass Theorem (1882) Suppose α1 , · · · , αn ∈ Q are Q-linearly independent. Then, e α1 , · · · , e αn are algebraically independent. Schanuel Conjecture (1960s) Suppose α1 , · · · , αn ∈ C are Q-linearly independent. Then, trans.deg.Q Q (α1 , · · · , αn ; e α1 , · · · , e αn ) ≥ n. LWT answers the special case of SC where α1 , · · · , αn ∈ Q Baker’s Theorem (1975) Suppose x1 , · · · , xn ∈ Q, and log(x1 ), · · · log(xn ) are linearly independent over Q. Then 1, log(x1 ), · · · , log(xn ) are linearly independent over Q Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 15 / 40 Algebraic Diff. Eqns. in Several Complex Variables Algebraic diff. eqns. in SCV (Bombieri, Invent. Math. 1970) Theorem Let K be a number field, f1 , · · · , fN be meromorphic functions on Cd of order ρ, Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 16 / 40 Algebraic Diff. Eqns. in Several Complex Variables Algebraic diff. eqns. in SCV (Bombieri, Invent. Math. 1970) Theorem Let K be a number field, f1 , · · · , fN be meromorphic functions on Cd of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ d + 1, and D = ∂z∂α : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ) for 1 ≤ α ≤ d. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 16 / 40 Algebraic Diff. Eqns. in Several Complex Variables Algebraic diff. eqns. in SCV (Bombieri, Invent. Math. 1970) Theorem Let K be a number field, f1 , · · · , fN be meromorphic functions on Cd of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ d + 1, and D = ∂z∂α : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ) for 1 ≤ α ≤ d. Then, the set of ζ ∈ Cd lying outside poles of f1 , · · · , fN and obeying f (ζ) ∈ K N must lie in an alg. hypersurface of degree ≤ d(d + 1)ρ[K : Q] + 2d. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 16 / 40 Algebraic Diff. Eqns. in Several Complex Variables Algebraic diff. eqns. in SCV (Bombieri, Invent. Math. 1970) Theorem Let K be a number field, f1 , · · · , fN be meromorphic functions on Cd of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ d + 1, and D = ∂z∂α : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ) for 1 ≤ α ≤ d. Then, the set of ζ ∈ Cd lying outside poles of f1 , · · · , fN and obeying f (ζ) ∈ K N must lie in an alg. hypersurface of degree ≤ d(d + 1)ρ[K : Q] + 2d. Closed positive (p, p)-currents (Lelong 1964) A C ∞ positive (1,1)-form ω means ı Ngaiming Mok (HKU) P ωi j̄ (z)dz i ∧ dz j , ωi j̄ (z) > 0. Complex Function Fields August 19, 2022 16 / 40 Algebraic Diff. Eqns. in Several Complex Variables Algebraic diff. eqns. in SCV (Bombieri, Invent. Math. 1970) Theorem Let K be a number field, f1 , · · · , fN be meromorphic functions on Cd of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ d + 1, and D = ∂z∂α : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ) for 1 ≤ α ≤ d. Then, the set of ζ ∈ Cd lying outside poles of f1 , · · · , fN and obeying f (ζ) ∈ K N must lie in an alg. hypersurface of degree ≤ d(d + 1)ρ[K : Q] + 2d. Closed positive (p, p)-currents (Lelong 1964) P A C ∞ positive (1,1)-form ω means ı ωi j̄ (z)dz i ∧ dz j , ωi j̄ (z) > 0. A (p, p)-current T is positive ⇔ T ∧ ω1 ∧ · · · ∧ ωn−p ≥ 0 as a measure. (ωi like ω). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 16 / 40 Algebraic Diff. Eqns. in Several Complex Variables Algebraic diff. eqns. in SCV (Bombieri, Invent. Math. 1970) Theorem Let K be a number field, f1 , · · · , fN be meromorphic functions on Cd of order ρ, trans.deg.K K (f1 , · · · , fN ) ≥ d + 1, and D = ∂z∂α : K (f1 , · · · , fN ) ,→ K (f1 , · · · , fN ) for 1 ≤ α ≤ d. Then, the set of ζ ∈ Cd lying outside poles of f1 , · · · , fN and obeying f (ζ) ∈ K N must lie in an alg. hypersurface of degree ≤ d(d + 1)ρ[K : Q] + 2d. Closed positive (p, p)-currents (Lelong 1964) P A C ∞ positive (1,1)-form ω means ı ωi j̄ (z)dz i ∧ dz j , ωi j̄ (z) > 0. A (p, p)-current T is positive ⇔ T ∧ ω1 ∧ · · · ∧ ωn−p ≥ 0 as a measure. (ωi ∞ like ω). A (d-)closed 2 positive C (1, 1)-form is locally T = ı∂∂ϕ where ϕ ϕ ∈ C ∞ and ∂z∂i ∂z > 0. Locally a closed positive (1,1)-current j T = ı∂∂ϕ where ϕ is weakly psh. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 16 / 40 Techniques from complex geometry Monotonicity of weighted mass of T over concentric Euclidean balls Assume T defined on Bn (0; R). For 0 < r < R denote by m(T ; 0; r ) the m(T ,0;r ) integral of T ∧ (ı∂∂kzk2 )n−p over Bn (0; r ); ν(T , 0; r ) := Vol(B n−p (0;R)) . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 17 / 40 Techniques from complex geometry Monotonicity of weighted mass of T over concentric Euclidean balls Assume T defined on Bn (0; R). For 0 < r < R denote by m(T ; 0; r ) the m(T ,0;r ) integral of T ∧ (ı∂∂kzk2 )n−p over Bn (0; r ); ν(T , 0; r ) := Vol(B n−p (0;R)) . Lelong proved that ν(T ; 0; r ) is decreasing as r → 0; the limit as r → 0 is now called the Lelong number ν(T ; 0) at 0. E.g., T := [S], the integral current of a pure (n − p)-dimensional complex analytic subvariety S ⊂ Bn (0; R), where ν([S]; 0) = mult0 (S) ∈ N is the multiplicity of S at 0. Recovering complex analytic subvarieties from density conditions Theorem (Siu [Invent. Math. (1970)]) Let X be a complex manifold, dimC (X ) =: n, 1 ≤ p < n, and T be a closed positive (p, p)-current on X . Let c > 0. Put Ec (T ) := {x ∈ X : ν(T ; x) ≥ c}. Then, Ec (T ) ⊂ X is a complex analytic subvariety where each irreducible subvariety is of complex codimension ≥ p. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 17 / 40 The Ax-Lindemann Theorem on XΓ = Ω/Γ After Ullmo-Yafaev [UY14] in the case of cocompact lattices, and Pila-Tsimerman [PT14] in the case of Siegel modular varieties, we have Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 18 / 40 The Ax-Lindemann Theorem on XΓ = Ω/Γ After Ullmo-Yafaev [UY14] in the case of cocompact lattices, and Pila-Tsimerman [PT14] in the case of Siegel modular varieties, we have Theorem (Klingler-Ullmo-Yafaev [KUY16]) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, Γ ⊂ Aut(Ω) be an arithmetic torsion-free lattice. Write XΓ := Ω/Γ, π : Ω → XΓ for the uniformization map. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 18 / 40 The Ax-Lindemann Theorem on XΓ = Ω/Γ After Ullmo-Yafaev [UY14] in the case of cocompact lattices, and Pila-Tsimerman [PT14] in the case of Siegel modular varieties, we have Theorem (Klingler-Ullmo-Yafaev [KUY16]) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, Γ ⊂ Aut(Ω) be an arithmetic torsion-free lattice. Write XΓ := Ω/Γ, π : Ω → XΓ for the uniformization map. Let Z ⊂ Ω be an Zar irreducible algebraic subset and denote by Z = π(Z ) ⊂ XΓ the Zariski closure of image of Z under the uniformization map in the quasi-projective variety XΓ . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 18 / 40 The Ax-Lindemann Theorem on XΓ = Ω/Γ After Ullmo-Yafaev [UY14] in the case of cocompact lattices, and Pila-Tsimerman [PT14] in the case of Siegel modular varieties, we have Theorem (Klingler-Ullmo-Yafaev [KUY16]) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, Γ ⊂ Aut(Ω) be an arithmetic torsion-free lattice. Write XΓ := Ω/Γ, π : Ω → XΓ for the uniformization map. Let Z ⊂ Ω be an Zar irreducible algebraic subset and denote by Z = π(Z ) ⊂ XΓ the Zariski closure of image of Z under the uniformization map in the quasi-projective variety XΓ . Then, Z ⊂ XΓ is a totally geodesic subset. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 18 / 40 The Ax-Lindemann Theorem on XΓ = Ω/Γ After Ullmo-Yafaev [UY14] in the case of cocompact lattices, and Pila-Tsimerman [PT14] in the case of Siegel modular varieties, we have Theorem (Klingler-Ullmo-Yafaev [KUY16]) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, Γ ⊂ Aut(Ω) be an arithmetic torsion-free lattice. Write XΓ := Ω/Γ, π : Ω → XΓ for the uniformization map. Let Z ⊂ Ω be an Zar irreducible algebraic subset and denote by Z = π(Z ) ⊂ XΓ the Zariski closure of image of Z under the uniformization map in the quasi-projective variety XΓ . Then, Z ⊂ XΓ is a totally geodesic subset. Key arguments are from model theory (counting theorem Pila-Wilkie) and complex differential geometry (volume estimates of Hwang-To). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 18 / 40 The Ax-Lindemann Theorem on XΓ = Ω/Γ After Ullmo-Yafaev [UY14] in the case of cocompact lattices, and Pila-Tsimerman [PT14] in the case of Siegel modular varieties, we have Theorem (Klingler-Ullmo-Yafaev [KUY16]) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, Γ ⊂ Aut(Ω) be an arithmetic torsion-free lattice. Write XΓ := Ω/Γ, π : Ω → XΓ for the uniformization map. Let Z ⊂ Ω be an Zar irreducible algebraic subset and denote by Z = π(Z ) ⊂ XΓ the Zariski closure of image of Z under the uniformization map in the quasi-projective variety XΓ . Then, Z ⊂ XΓ is a totally geodesic subset. Key arguments are from model theory (counting theorem Pila-Wilkie) and complex differential geometry (volume estimates of Hwang-To). Using the above Tsimerman [Ts18] has proven the André-Oort Conjecture for Siegel modular varieties Ag = Hg /Sp(g ; Z). Recently, Pila-ShankarTsimerman has announced a solution of the full André-Oort Conjecture. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 18 / 40 Counting points on definable sets For a rational point x = qp ; p, q ∈ Z, q 6= 0, where |p| and |q| are coprime, we define the height H(x) = max(|p|, |q|). For x = (x1 , · · · , xn ) ∈ Qn ) we define H(x) = max(H(x1 ), · · · , H(xn )). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 19 / 40 Counting points on definable sets For a rational point x = qp ; p, q ∈ Z, q 6= 0, where |p| and |q| are coprime, we define the height H(x) = max(|p|, |q|). For x = (x1 , · · · , xn ) ∈ Qn ) we define H(x) = max(H(x1 ), · · · , H(xn )). For Z ⊂ Rn , and for T > 0 we define the counting function N(Z , T ) := {x ∈ Z ∩ Qn : H(x) ≤ T } . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 19 / 40 Counting points on definable sets For a rational point x = qp ; p, q ∈ Z, q 6= 0, where |p| and |q| are coprime, we define the height H(x) = max(|p|, |q|). For x = (x1 , · · · , xn ) ∈ Qn ) we define H(x) = max(H(x1 ), · · · , H(xn )). For Z ⊂ Rn , and for T > 0 we define the counting function N(Z , T ) := {x ∈ Z ∩ Qn : H(x) ≤ T } . Model Theory: o-minimal structures on Rn A structure S on {Rn : n ∈ N} consists of Boolean algebras of subsets n Sn ⊂ 2R closed under taking Cartesian products and coordinate projections, s.t. Diag(R × R) ∈ S2 , and, Graph(+), Graph(×) ∈ S3 . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 19 / 40 Counting points on definable sets For a rational point x = qp ; p, q ∈ Z, q 6= 0, where |p| and |q| are coprime, we define the height H(x) = max(|p|, |q|). For x = (x1 , · · · , xn ) ∈ Qn ) we define H(x) = max(H(x1 ), · · · , H(xn )). For Z ⊂ Rn , and for T > 0 we define the counting function N(Z , T ) := {x ∈ Z ∩ Qn : H(x) ≤ T } . Model Theory: o-minimal structures on Rn A structure S on {Rn : n ∈ N} consists of Boolean algebras of subsets n Sn ⊂ 2R closed under taking Cartesian products and coordinate projections, s.t. Diag(R × R) ∈ S2 , and, Graph(+), Graph(×) ∈ S3 . S is called o-minimal if S1 consists of finite unions of intervals and points. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 19 / 40 Counting points on definable sets For a rational point x = qp ; p, q ∈ Z, q 6= 0, where |p| and |q| are coprime, we define the height H(x) = max(|p|, |q|). For x = (x1 , · · · , xn ) ∈ Qn ) we define H(x) = max(H(x1 ), · · · , H(xn )). For Z ⊂ Rn , and for T > 0 we define the counting function N(Z , T ) := {x ∈ Z ∩ Qn : H(x) ≤ T } . Model Theory: o-minimal structures on Rn A structure S on {Rn : n ∈ N} consists of Boolean algebras of subsets n Sn ⊂ 2R closed under taking Cartesian products and coordinate projections, s.t. Diag(R × R) ∈ S2 , and, Graph(+), Graph(×) ∈ S3 . S is called o-minimal if S1 consists of finite unions of intervals and points. Ran,exp is the minimal S including subanalytic sets and Graph(exp), and it is o-minimal (Dries-Miller 1994). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 19 / 40 Counting points on definable sets For a rational point x = qp ; p, q ∈ Z, q 6= 0, where |p| and |q| are coprime, we define the height H(x) = max(|p|, |q|). For x = (x1 , · · · , xn ) ∈ Qn ) we define H(x) = max(H(x1 ), · · · , H(xn )). For Z ⊂ Rn , and for T > 0 we define the counting function N(Z , T ) := {x ∈ Z ∩ Qn : H(x) ≤ T } . Model Theory: o-minimal structures on Rn A structure S on {Rn : n ∈ N} consists of Boolean algebras of subsets n Sn ⊂ 2R closed under taking Cartesian products and coordinate projections, s.t. Diag(R × R) ∈ S2 , and, Graph(+), Graph(×) ∈ S3 . S is called o-minimal if S1 consists of finite unions of intervals and points. Ran,exp is the minimal S including subanalytic sets and Graph(exp), and it is o-minimal (Dries-Miller 1994). Any member (called definable set) in an o-minimal S has finitely many connected components. Theorem (Pila-Wilkie, Duke J. 2006) Let Z ⊂ Rn be a definable subset in a given o-minimal structure. Then, N(Z − Z alg , T ) = T o(1) , i.e., Z − Z alg grows subpolynomially. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 19 / 40 A generalized Lelong monotonicity formula Proposition Let ϕ be an unbounded C ∞ strictly psh exhaustion fct on a Stein manifold X . Let F : R → R be strictly increasing s.t. ψ := F ◦ ϕ is weakly psh. Let S be a closed positive (p, p)-current on X , 0 < p < dim(X ). Then, Z √ 0 n−p hS,ϕ (T ) := F (T ) S ∧ ( −1∂∂ϕ)n−p {ϕ 0, passing through x0 . Then, ∃λ = λΩ and C = CΩ > 0 such that Volume (B(x0 ; r )) ≥ Ce λr . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 20 / 40 Geometric applications of Lelong formulas 1 Lelong’s original formula was for closed positive (p, p)-currents on Cn , in which one considers the psh function ϕ = kzk2 . In this case ψ := log ϕ = log kzk2 is weakly psh, F (T ) = log(T ). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 21 / 40 Geometric applications of Lelong formulas 1 2 Lelong’s original formula was for closed positive (p, p)-currents on Cn , in which one considers the psh function ϕ = kzk2 . In this case ψ := log ϕ = log kzk2 is weakly psh, F (T ) = log(T ). In the case Bn with potential function ϕ = −(n + 1) log 1 − kzk2 for the Bergman metric d(0; z) ∼ ϕ(z), ∃c2 > c1 > 0 such that {ϕ < c1 r } ⊂ B(0; r ) ⊂ {ϕ < c2 r }. Take F (T ) = −e −αT √. We can check that ∃α > 0 such that for ψ := F ◦ ϕ = −e −αϕ , −1∂∂ψ ≥ 0. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 21 / 40 Geometric applications of Lelong formulas 1 2 Lelong’s original formula was for closed positive (p, p)-currents on Cn , in which one considers the psh function ϕ = kzk2 . In this case ψ := log ϕ = log kzk2 is weakly psh, F (T ) = log(T ). In the case Bn with potential function ϕ = −(n + 1) log 1 − kzk2 for the Bergman metric d(0; z) ∼ ϕ(z), ∃c2 > c1 > 0 such that {ϕ < c1 r } ⊂ B(0; r ) ⊂ {ϕ < c2 r }. Take F (T ) = −e −αT √. We can check that ∃α > 0√such that for ψ := F ◦ ϕ = −e −αϕ , −1∂∂ψ ≥ 0. For this we check −1∂∂ϕ ≥ α∂ϕ ∧ ∂ϕ. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 21 / 40 Geometric applications of Lelong formulas 1 2 Lelong’s original formula was for closed positive (p, p)-currents on Cn , in which one considers the psh function ϕ = kzk2 . In this case ψ := log ϕ = log kzk2 is weakly psh, F (T ) = log(T ). In the case Bn with potential function ϕ = −(n + 1) log 1 − kzk2 for the Bergman metric d(0; z) ∼ ϕ(z), ∃c2 > c1 > 0 such that {ϕ < c1 r } ⊂ B(0; r ) ⊂ {ϕ < c2 r }. Take F (T ) = −e −αT √. We can check that ∃α > 0√such that for ψ := F ◦ ϕ = −e −αϕ , −1∂∂ψ ≥ 0. For this we check −1∂∂ϕ ≥ α∂ϕ ∧ ∂ϕ. For a BSD Ω, one uses ϕ(z) = log KΩ (z, z), KΩ = Bergman Kernal of Ω. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 21 / 40 Ax-Lindemann Theorem for Rank-1 Lattices Theorem (Mok [Mo19, Compositio Math.]) Let n ≥ 2 and Γ ⊂ Aut(Bn ) be a not necessarily arithmetic torsion-free lattice. Write XΓ := Bn /Γ, π : Ω → XΓ for the uniformization map. Let Z ⊂ Ω be an irreducible algebraic subset and denote by Zar Z = π(Z ) ⊂ XΓ be the Zariski closure of image of Z under the uniformization map in the quasi-projective variety XΓ . Then, Z ⊂ XΓ is a totally geodesic subset. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 22 / 40 Ax-Lindemann Theorem for Rank-1 Lattices Theorem (Mok [Mo19, Compositio Math.]) Let n ≥ 2 and Γ ⊂ Aut(Bn ) be a not necessarily arithmetic torsion-free lattice. Write XΓ := Bn /Γ, π : Ω → XΓ for the uniformization map. Let Z ⊂ Ω be an irreducible algebraic subset and denote by Zar Z = π(Z ) ⊂ XΓ be the Zariski closure of image of Z under the uniformization map in the quasi-projective variety XΓ . Then, Z ⊂ XΓ is a totally geodesic subset. b ⊂ Pn (a) We have Bn ⊂ Pn , Z as an open subset of an algebraic Z b ] as a member of an irreducible component K of the Consider [Z Chow scheme Chow(Pn ), with associated fiber bundle µ : U → Pn . Restrict U to Bn and take quotients wrt Γ to get µΓ : UΓ → XΓ . Prove that UΓ is algebraic by means of L2 -estimates of ∂. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 22 / 40 AL Theorem for Rank-1 Lattices (cont.) (b) Let Zf be an irreducible component of πΓ−1 (Z ). Then, at a good point b ∈ ∂ Zf, Zf extends across b as the union of an analytic family of algebraic subvarieties of Pn . Let D be a germ of complex submanifold at b grafted to extend Zf analytically across b. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 23 / 40 AL Theorem for Rank-1 Lattices (cont.) (b) Let Zf be an irreducible component of πΓ−1 (Z ). Then, at a good point b ∈ ∂ Zf, Zf extends across b as the union of an analytic family of algebraic subvarieties of Pn . Let D be a germ of complex submanifold at b grafted to extend Zf analytically across b. (c) D ∩ Bn is a local strictly peudoconvex manifold with smooth boundary, and by Klembeck [Kl87] D ∩ Bn is asymptotically of constant holomorphic sectional curvature −2, hence asymptotically totally geodesic. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 23 / 40 AL Theorem for Rank-1 Lattices (cont.) (b) Let Zf be an irreducible component of πΓ−1 (Z ). Then, at a good point b ∈ ∂ Zf, Zf extends across b as the union of an analytic family of algebraic subvarieties of Pn . Let D be a germ of complex submanifold at b grafted to extend Zf analytically across b. (c) D ∩ Bn is a local strictly peudoconvex manifold with smooth boundary, and by Klembeck [Kl87] D ∩ Bn is asymptotically of constant holomorphic sectional curvature −2, hence asymptotically totally geodesic. (d) By rescaling using elements γ ∈ π1 (Z ) ,→ π1 (XΓ ) ∼ = Γ, it follows that Π is of constant holomorphic sectional curvature −2, hence totally geodesic. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 23 / 40 Compactification Theorem by L2 -estimates of ∂ Theorem (Mok-Zhong [MZ89, Ann. Math.]) Let (X , g ) be a complete Kähler manifold. Assume that Vol(X , g ) < ∞, kSectional Curvature(X , g )k < ∞, and that X has finite topology. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 24 / 40 Compactification Theorem by L2 -estimates of ∂ Theorem (Mok-Zhong [MZ89, Ann. Math.]) Let (X , g ) be a complete Kähler manifold. Assume that Vol(X , g ) < ∞, kSectional Curvature(X , g )k < ∞, and that X has finite topology. Suppose there exists a Hermitian holomorphic line bundle (E , h) of pinched positive curvature. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 24 / 40 Compactification Theorem by L2 -estimates of ∂ Theorem (Mok-Zhong [MZ89, Ann. Math.]) Let (X , g ) be a complete Kähler manifold. Assume that Vol(X , g ) < ∞, kSectional Curvature(X , g )k < ∞, and that X has finite topology. Suppose there exists a Hermitian holomorphic line bundle (E , h) of pinched positive curvature. For k > 0, denote by N (X , E k ) the space of holomorphic sections s ∈ Γ(X , E k ) of the Nevanlinna class, i.e., s R satisfies X max(log kskhk , 0) < ∞. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 24 / 40 Compactification Theorem by L2 -estimates of ∂ Theorem (Mok-Zhong [MZ89, Ann. Math.]) Let (X , g ) be a complete Kähler manifold. Assume that Vol(X , g ) < ∞, kSectional Curvature(X , g )k < ∞, and that X has finite topology. Suppose there exists a Hermitian holomorphic line bundle (E , h) of pinched positive curvature. For k > 0, denote by N (X , E k ) the space of holomorphic sections s ∈ Γ(X , E k ) of the Nevanlinna class, i.e., s R satisfies X max(log kskhk , 0) < ∞. Then, dim(N (X , E k )) < ∞ for all k ≥ 0. Moreover, there exists some positive integer k such that N (X , E k ) has no base points and it embeds X into P N (X , E k )∗ ) realizing X as a quasi-projective manifold. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 24 / 40 Earlier Ax-Schanuel-type results Ax-Schanuel Theorem Theorem (Ax71, Annals) Let f1 , · · · , fn ∈ C[[z1 , · · · , zm ]] be Q-linearly independent formal power series with no constant terms. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 25 / 40 Earlier Ax-Schanuel-type results Ax-Schanuel Theorem Theorem (Ax71, Annals) Let f1 , · · · , fn ∈ C[[z1 , · · · , zm ]] be Q-linearly independent formal power series with no constant terms. Then, ∂fi trans.deg.C C f1 , · · · , fn ; e 2πıf1 , · · · , e 2πıfn ≥ n + rank ∂z . j Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 25 / 40 Earlier Ax-Schanuel-type results Ax-Schanuel Theorem Theorem (Ax71, Annals) Let f1 , · · · , fn ∈ C[[z1 , · · · , zm ]] be Q-linearly independent formal power series with no constant terms. Then, ∂fi trans.deg.C C f1 , · · · , fn ; e 2πıf1 , · · · , e 2πıfn ≥ n + rank ∂z . j 1 The case of formal power series is reducible to that of convergent power series, by Seidenberg, hence to considering the restriction of functions to a germ of complex submanifold (V ; 0) ⊂ (Cm ; 0). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 25 / 40 Earlier Ax-Schanuel-type results Ax-Schanuel Theorem Theorem (Ax71, Annals) Let f1 , · · · , fn ∈ C[[z1 , · · · , zm ]] be Q-linearly independent formal power series with no constant terms. Then, ∂fi trans.deg.C C f1 , · · · , fn ; e 2πıf1 , · · · , e 2πıfn ≥ n + rank ∂z . j 1 2 The case of formal power series is reducible to that of convergent power series, by Seidenberg, hence to considering the restriction of functions to a germ of complex submanifold (V ; 0) ⊂ (Cm ; 0). Let U ⊂ Cn × (C∗ )n be the graph of V above under the exponential map. The hypothesis implies that the projection of U to (C∗ )n is not contained in any proper algebraic subgroup. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 25 / 40 Earlier Ax-Schanuel-type results Ax-Schanuel Theorem Theorem (Ax71, Annals) Let f1 , · · · , fn ∈ C[[z1 , · · · , zm ]] be Q-linearly independent formal power series with no constant terms. Then, ∂fi trans.deg.C C f1 , · · · , fn ; e 2πıf1 , · · · , e 2πıfn ≥ n + rank ∂z . j 1 2 The case of formal power series is reducible to that of convergent power series, by Seidenberg, hence to considering the restriction of functions to a germ of complex submanifold (V ; 0) ⊂ (Cm ; 0). Let U ⊂ Cn × (C∗ )n be the graph of V above under the exponential map. The hypothesis implies that the projection of U to (C∗ )n is not contained in any proper algebraic subgroup. Ax-Schanuel for the j-function Pila-Tsimerman [PT16] proved an analogue of Ax-Schanuel for the Cartesian product Hn of upper half-planes, replacing the exponential function by j : H → C, thus considering C(f1 , · · · , fn ; j ◦ f1 , · · · , j ◦ fn ). They also proved an analogue involving at the same time j 0 and j 00 . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 25 / 40 Ax-Schanuel Theorem on Shimura varieties Theorem (Mok-Pila-Tsimerman ([MPT19, Annals]) Let Ω b CN be a bounded symmetric domain, Γ ⊂ Aut(Ω) be an arithmetic lattice, and write XΓ := Ω/Γ, as a quasi-projective variety. Let W ⊂ Ω × XΓ be an algebraic subvariety. Let D ⊂ Ω × XΓ be the graph of the uniformization map πΓ : Ω → XΓ , and U be an irreducible component of W ∩ D whose dimension is larger than expected, Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 26 / 40 Ax-Schanuel Theorem on Shimura varieties Theorem (Mok-Pila-Tsimerman ([MPT19, Annals]) Let Ω b CN be a bounded symmetric domain, Γ ⊂ Aut(Ω) be an arithmetic lattice, and write XΓ := Ω/Γ, as a quasi-projective variety. Let W ⊂ Ω × XΓ be an algebraic subvariety. Let D ⊂ Ω × XΓ be the graph of the uniformization map πΓ : Ω → XΓ , and U be an irreducible component of W ∩ D whose dimension is larger than expected, i.e., codimU < codim(W ) + codim(D), the codimensions being in Ω × XΓ , or, equivalently, dim(U) > dim(W ) − dim(XΓ ). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 26 / 40 Ax-Schanuel Theorem on Shimura varieties Theorem (Mok-Pila-Tsimerman ([MPT19, Annals]) Let Ω b CN be a bounded symmetric domain, Γ ⊂ Aut(Ω) be an arithmetic lattice, and write XΓ := Ω/Γ, as a quasi-projective variety. Let W ⊂ Ω × XΓ be an algebraic subvariety. Let D ⊂ Ω × XΓ be the graph of the uniformization map πΓ : Ω → XΓ , and U be an irreducible component of W ∩ D whose dimension is larger than expected, i.e., codimU < codim(W ) + codim(D), the codimensions being in Ω × XΓ , or, equivalently, dim(U) > dim(W ) − dim(XΓ ). Then, the projection of U to XΓ is contained in a totally geodesic subvariety Y ( XΓ . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 26 / 40 Ax-Schanuel of MPT in terms of functional transcendence Fix a torsion-free lattice Γ ⊂ Aut(Ω), π : Ω → XΓ . Modular functions are Γ-invariant meromorphic functions descending to rational functions on XΓ . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 27 / 40 Ax-Schanuel of MPT in terms of functional transcendence Fix a torsion-free lattice Γ ⊂ Aut(Ω), π : Ω → XΓ . Modular functions are Γ-invariant meromorphic functions descending to rational functions on XΓ . Theorem (Mok-Pila-Tsimerman ([MPT19, Annals]) Let V ⊂ Ω be an irreducible complex analytic subvariety, not contained in any weakly special subvariety E ( Ω. Let (zi )1≤i≤n be algebraic coordinates on Ω, {ϕ1 , . . . , ϕN } be a basis of modular functions. Then, trans.deg.C C {zi }, {φj } ≥ n + dim V , where all φj are assumed defined at some point on V and restricted to V . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 27 / 40 Ax-Schanuel of MPT in terms of functional transcendence Fix a torsion-free lattice Γ ⊂ Aut(Ω), π : Ω → XΓ . Modular functions are Γ-invariant meromorphic functions descending to rational functions on XΓ . Theorem (Mok-Pila-Tsimerman ([MPT19, Annals]) Let V ⊂ Ω be an irreducible complex analytic subvariety, not contained in any weakly special subvariety E ( Ω. Let (zi )1≤i≤n be algebraic coordinates on Ω, {ϕ1 , . . . , ϕN } be a basis of modular functions. Then, trans.deg.C C {zi }, {φj } ≥ n + dim V , where all φj are assumed defined at some point on V and restricted to V . 1 2 We may take the algebraic coordinates (z1 , · · · , zn ) to be the b Harish-Chandra coordinates on Ω b Cn ⊂ Ω. Here a weakly special subvariety E ⊂ Ω is a totally geodesic submanifold E ⊂ Ω such that π(E ) ⊂ XΓ is quasi-projective. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 27 / 40 Tame complex geometry The Definable Remmert-Stein Theorem Theorem (Peterzil-Starchenko [Proc. ICM 2010]) Let M be a definable complex manifold and E a definable complex analytic subset of M. Let A be a definable, complex analytic subset of M − E . Then, its topological closure A is a complex analytic subset of M. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 28 / 40 Tame complex geometry The Definable Remmert-Stein Theorem Theorem (Peterzil-Starchenko [Proc. ICM 2010]) Let M be a definable complex manifold and E a definable complex analytic subset of M. Let A be a definable, complex analytic subset of M − E . Then, its topological closure A is a complex analytic subset of M. The Definable Chow Theorem Theorem (Peterzil-Starchenko, variation of [Proc. ICM 2010]) Let Y be a quasi-projective algebraic variety. Let A ⊂ Y be definable, complex analytic, and closed in Y . Then, A is algebraic. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 28 / 40 Ax-Schanuel for variations of Hodge structures Theorem (Bakker-Tsimerman, Invent. Math. 2019) Let X be a nonsingular quasi-projective manifold underlying a polarized integral variation of Hodge structures, D be the associated period domain, D ⊂ Ď the standard embedding of D into its dual Ď, which is a rational homogeneous manifold. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 29 / 40 Ax-Schanuel for variations of Hodge structures Theorem (Bakker-Tsimerman, Invent. Math. 2019) Let X be a nonsingular quasi-projective manifold underlying a polarized integral variation of Hodge structures, D be the associated period domain, D ⊂ Ď the standard embedding of D into its dual Ď, which is a rational homogeneous manifold. Let W be the graph of the period map ϕ : X → D/Γ, where Γ ⊂ Aut(D) is the image of the monodromy representation of π1 (X ), assumed to be torsion-free. Let V ⊂ X × Ď be an algebraic subset and U be an irreducible component of V ∩ W satisfying codimX ×Ď (U) < codimX ×Ď (V ) + codimX ×Ď (W ). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 29 / 40 Ax-Schanuel for variations of Hodge structures Theorem (Bakker-Tsimerman, Invent. Math. 2019) Let X be a nonsingular quasi-projective manifold underlying a polarized integral variation of Hodge structures, D be the associated period domain, D ⊂ Ď the standard embedding of D into its dual Ď, which is a rational homogeneous manifold. Let W be the graph of the period map ϕ : X → D/Γ, where Γ ⊂ Aut(D) is the image of the monodromy representation of π1 (X ), assumed to be torsion-free. Let V ⊂ X × Ď be an algebraic subset and U be an irreducible component of V ∩ W satisfying codimX ×Ď (U) < codimX ×Ď (V ) + codimX ×Ď (W ). Then, the canonical projection of U to X is contained in a proper weak Mumford-Tate subvariety. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 29 / 40 A key ingredient for the generalization of Ax-Schanuel in the context of variations of Hodge structures was a volume growth estimate established by Bakker-Tsimerman for subvarieites generalizing that of Hwang-To. They achieved this by adapting the Lelong monotonicity formula. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 30 / 40 A key ingredient for the generalization of Ax-Schanuel in the context of variations of Hodge structures was a volume growth estimate established by Bakker-Tsimerman for subvarieites generalizing that of Hwang-To. They achieved this by adapting the Lelong monotonicity formula. Ax-Schanuel for the rank-1 case (Baldi-Ullmo) Ax-Schanuel for the rank-1 case was recently proven by BaldiUllmo. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 30 / 40 A key ingredient for the generalization of Ax-Schanuel in the context of variations of Hodge structures was a volume growth estimate established by Bakker-Tsimerman for subvarieites generalizing that of Hwang-To. They achieved this by adapting the Lelong monotonicity formula. Ax-Schanuel for the rank-1 case (Baldi-Ullmo) Ax-Schanuel for the rank-1 case was recently proven by BaldiUllmo. Given any torsion-free lattice Γ ⊂ Aut(Bn ), n ≥ 2, the lattice, though not necessarily arithmetic, must be integral in some precise way, and they embed XΓ = Bn /Γ via some period map into D/Γ and exploit atypcial intersection on D/Γ0 , proving Ax-Schanuel for XΓ by means of Bakker-Tsimerman’s Ax-Schanuel Theorem for period domains. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 30 / 40 A key ingredient for the generalization of Ax-Schanuel in the context of variations of Hodge structures was a volume growth estimate established by Bakker-Tsimerman for subvarieites generalizing that of Hwang-To. They achieved this by adapting the Lelong monotonicity formula. Ax-Schanuel for the rank-1 case (Baldi-Ullmo) Ax-Schanuel for the rank-1 case was recently proven by BaldiUllmo. Given any torsion-free lattice Γ ⊂ Aut(Bn ), n ≥ 2, the lattice, though not necessarily arithmetic, must be integral in some precise way, and they embed XΓ = Bn /Γ via some period map into D/Γ and exploit atypcial intersection on D/Γ0 , proving Ax-Schanuel for XΓ by means of Bakker-Tsimerman’s Ax-Schanuel Theorem for period domains. For finite-volume quotients of reducible bounded symmetric domains Ω = Ω1 × · · · × Ωm Ax-Schanuel remains unsolved. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 30 / 40 A key ingredient for the generalization of Ax-Schanuel in the context of variations of Hodge structures was a volume growth estimate established by Bakker-Tsimerman for subvarieites generalizing that of Hwang-To. They achieved this by adapting the Lelong monotonicity formula. Ax-Schanuel for the rank-1 case (Baldi-Ullmo) Ax-Schanuel for the rank-1 case was recently proven by BaldiUllmo. Given any torsion-free lattice Γ ⊂ Aut(Bn ), n ≥ 2, the lattice, though not necessarily arithmetic, must be integral in some precise way, and they embed XΓ = Bn /Γ via some period map into D/Γ and exploit atypcial intersection on D/Γ0 , proving Ax-Schanuel for XΓ by means of Bakker-Tsimerman’s Ax-Schanuel Theorem for period domains. For finite-volume quotients of reducible bounded symmetric domains Ω = Ω1 × · · · × Ωm Ax-Schanuel remains unsolved. Especially, when there exist 1-dimensional factors Ωi in general the counting argument of Pila-Wilkie no longer works. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 30 / 40 Algebraic subsets of a BSD invariant under cocompact Γ̌ Proposition 1 (Chan-Mok, JDG 2021) Let D and Ω be BSD, Φ : Aut0 (D) → Aut0 (Ω) be a group homomorphism, F : D → Ω be a Φ-equivariant holomorphic map. Then, F is totally geodesic. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 31 / 40 Algebraic subsets of a BSD invariant under cocompact Γ̌ Proposition 1 (Chan-Mok, JDG 2021) Let D and Ω be BSD, Φ : Aut0 (D) → Aut0 (Ω) be a group homomorphism, F : D → Ω be a Φ-equivariant holomorphic map. Then, F is totally geodesic. Theorem (Chan-Mok, JDG 2021) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, and Z ⊂ Ω be an algebraic subset. Suppose there exists a torsion-free discrete subgroup Γ̌ ⊂ Aut(Ω) such that Γ̌ stabilizes Z and Z /Γ̌ is compact. Then, Z ⊂ Ω is totally geodesic. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 31 / 40 Algebraic subsets of a BSD invariant under cocompact Γ̌ Proposition 1 (Chan-Mok, JDG 2021) Let D and Ω be BSD, Φ : Aut0 (D) → Aut0 (Ω) be a group homomorphism, F : D → Ω be a Φ-equivariant holomorphic map. Then, F is totally geodesic. Theorem (Chan-Mok, JDG 2021) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, and Z ⊂ Ω be an algebraic subset. Suppose there exists a torsion-free discrete subgroup Γ̌ ⊂ Aut(Ω) such that Γ̌ stabilizes Z and Z /Γ̌ is compact. Then, Z ⊂ Ω is totally geodesic. Corollary (Chan-Mok, JDG 2021) Let Γ ⊂ Aut(Ω) be a torsion-free cocompact lattice acting on Ω b CN , XΓ := Ω/Γ, π : Ω → XΓ the uniformization map. Let Y ⊂ XΓ be an irreducible subvariety, and Z ⊂ Ω be an irreducible component of π −1 (Y ). Suppose Z ⊂ Ω is an algebraic subset. Then, Z ⊂ Ω is totally geodesic. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 31 / 40 Asymptotic Total Geodesy of Embedded Poincaré Disks Theorem (Chan-Mok [CM21], JDG) 2 ) → (Ω, ds 2 ) be a holomorphic isometric embedding, Let f : (∆, λds∆ Ω where λ is a positive real constant and Ω b CN is a bounded symmetric domain in its Harish-Chandra realization. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 32 / 40 Asymptotic Total Geodesy of Embedded Poincaré Disks Theorem (Chan-Mok [CM21], JDG) 2 ) → (Ω, ds 2 ) be a holomorphic isometric embedding, Let f : (∆, λds∆ Ω where λ is a positive real constant and Ω b CN is a bounded symmetric domain in its Harish-Chandra realization. Then, f is asymptotically totally geodesic at a general point b ∈ ∂∆. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 32 / 40 Asymptotic Total Geodesy of Embedded Poincaré Disks Theorem (Chan-Mok [CM21], JDG) 2 ) → (Ω, ds 2 ) be a holomorphic isometric embedding, Let f : (∆, λds∆ Ω where λ is a positive real constant and Ω b CN is a bounded symmetric domain in its Harish-Chandra realization. Then, f is asymptotically totally geodesic at a general point b ∈ ∂∆. Theorem implies Proposition 1 by slicing D by totally geodesic disks. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 32 / 40 Asymptotic Total Geodesy of Embedded Poincaré Disks Theorem (Chan-Mok [CM21], JDG) 2 ) → (Ω, ds 2 ) be a holomorphic isometric embedding, Let f : (∆, λds∆ Ω where λ is a positive real constant and Ω b CN is a bounded symmetric domain in its Harish-Chandra realization. Then, f is asymptotically totally geodesic at a general point b ∈ ∂∆. Theorem implies Proposition 1 by slicing D by totally geodesic disks. Embedded Poincaré Disks with Aut(Ω)-equiv. Tangents 2 ) → (Ω, ds 2 ) be a holomorphic Proposition 2 Let f0 : (∆, λ ds∆ Ω isometric embedding. Suppose Z0 := f0 (∆) ⊂ Ω is not asymptotically totally geodesic at a generic point b ∈ ∂Z0 . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 32 / 40 Asymptotic Total Geodesy of Embedded Poincaré Disks Theorem (Chan-Mok [CM21], JDG) 2 ) → (Ω, ds 2 ) be a holomorphic isometric embedding, Let f : (∆, λds∆ Ω where λ is a positive real constant and Ω b CN is a bounded symmetric domain in its Harish-Chandra realization. Then, f is asymptotically totally geodesic at a general point b ∈ ∂∆. Theorem implies Proposition 1 by slicing D by totally geodesic disks. Embedded Poincaré Disks with Aut(Ω)-equiv. Tangents 2 ) → (Ω, ds 2 ) be a holomorphic Proposition 2 Let f0 : (∆, λ ds∆ Ω isometric embedding. Suppose Z0 := f0 (∆) ⊂ Ω is not asymptotically totally geodesic at a generic point b ∈ ∂Z0 . Then, ∃ a holomorphic 2 ) → (Ω, ds 2 ), f (∆) =: Z such that isometric embedding f : (∆, λ ds∆ Ω (†) All tangent lines Tx (Z ), x ∈ Z , are equivalent under Aut(Ω). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 32 / 40 Asymptotic Total Geodesy of Embedded Poincaré Disks Theorem (Chan-Mok [CM21], JDG) 2 ) → (Ω, ds 2 ) be a holomorphic isometric embedding, Let f : (∆, λds∆ Ω where λ is a positive real constant and Ω b CN is a bounded symmetric domain in its Harish-Chandra realization. Then, f is asymptotically totally geodesic at a general point b ∈ ∂∆. Theorem implies Proposition 1 by slicing D by totally geodesic disks. Embedded Poincaré Disks with Aut(Ω)-equiv. Tangents 2 ) → (Ω, ds 2 ) be a holomorphic Proposition 2 Let f0 : (∆, λ ds∆ Ω isometric embedding. Suppose Z0 := f0 (∆) ⊂ Ω is not asymptotically totally geodesic at a generic point b ∈ ∂Z0 . Then, ∃ a holomorphic 2 ) → (Ω, ds 2 ), f (∆) =: Z such that isometric embedding f : (∆, λ ds∆ Ω (†) All tangent lines Tx (Z ), x ∈ Z , are equivalent under Aut(Ω). Proof by rescaling: Compose with γi ∈ Aut(Ω) and take limits. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 32 / 40 Total Geodesy of Certain Curves on Tube Domains Proposition 3 Let Ω be an irreducible bounded symmetric domain of tube type and of rank r ; Z ⊂ Ω be a local holomorphic curve with Aut(Ω)-equivalent tangent planes spanned by vectors of rank r . Then, Z ⊂ Ω is totally geodesic and of rank r (i.e. of diagonal type). Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 33 / 40 Total Geodesy of Certain Curves on Tube Domains Proposition 3 Let Ω be an irreducible bounded symmetric domain of tube type and of rank r ; Z ⊂ Ω be a local holomorphic curve with Aut(Ω)-equivalent tangent planes spanned by vectors of rank r . Then, Z ⊂ Ω is totally geodesic and of rank r (i.e. of diagonal type). Proof. π : PTΩ → Ω, L → PTΩ tautological line bundle. [S ] ∼ = L−r ⊗ π ∗ E 2 , E dual to O(1) on the compact dual M of Ω. √ (2π)−1 −1∂∂ log ksk2 = rc1 (L, ĝ0 ) − 2c1 (π ∗ E , π ∗ h0 ), where ĝ0 and h0 are canonical metrics. ks(x)k only depends on the Aut(Ω)-isomorphism type of Tx (Ω). Thus, ksk = constant on Z . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 33 / 40 Total Geodesy of Certain Curves on Tube Domains Proposition 3 Let Ω be an irreducible bounded symmetric domain of tube type and of rank r ; Z ⊂ Ω be a local holomorphic curve with Aut(Ω)-equivalent tangent planes spanned by vectors of rank r . Then, Z ⊂ Ω is totally geodesic and of rank r (i.e. of diagonal type). Proof. π : PTΩ → Ω, L → PTΩ tautological line bundle. [S ] ∼ = L−r ⊗ π ∗ E 2 , E dual to O(1) on the compact dual M of Ω. √ (2π)−1 −1∂∂ log ksk2 = rc1 (L, ĝ0 ) − 2c1 (π ∗ E , π ∗ h0 ), where ĝ0 and h0 are canonical metrics. ks(x)k only depends on the Aut(Ω)-isomorphism type of Tx (Ω). Thus, ksk = constant on Z . Hence, 0 = rc1 (L, ĝ0 ) − 2c1 (π ∗ E , π ∗ h0 ). ⇔ Gauss curvature K (x) = −2/r , and σ ≡ 0. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 33 / 40 Bi-algebraicity by means of Nadel’s Theorem Maps inducing the representation θ : Γ̌ ,→ H0 ⊂ G0 = Aut0 (Ω) Without loss of generality assume Ω ⊃ Z smallest BSD containing Z , ı : Y ,→ ZΓ̌ , θ := ı∗ π1 (Y ) = Γ̌ ⊂ H0 . By the proof of Nadel’s Theorem, H0 is a semisimple Lie group without compact factors acting on Ω. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 34 / 40 Bi-algebraicity by means of Nadel’s Theorem Maps inducing the representation θ : Γ̌ ,→ H0 ⊂ G0 = Aut0 (Ω) Without loss of generality assume Ω ⊃ Z smallest BSD containing Z , ı : Y ,→ ZΓ̌ , θ := ı∗ π1 (Y ) = Γ̌ ⊂ H0 . By the proof of Nadel’s Theorem, H0 is a semisimple Lie group without compact factors acting on Ω. Write L ⊂ H0 for a maximal compact subgroup. Let f : Y → Γ̌\H0 /L =: SΓ̌ ,→ XΓ̌ be any smooth map inducing θ. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 34 / 40 Bi-algebraicity by means of Nadel’s Theorem Maps inducing the representation θ : Γ̌ ,→ H0 ⊂ G0 = Aut0 (Ω) Without loss of generality assume Ω ⊃ Z smallest BSD containing Z , ı : Y ,→ ZΓ̌ , θ := ı∗ π1 (Y ) = Γ̌ ⊂ H0 . By the proof of Nadel’s Theorem, H0 is a semisimple Lie group without compact factors acting on Ω. Write L ⊂ H0 for a maximal compact subgroup. Let f : Y → Γ̌\H0 /L =: SΓ̌ ,→ XΓ̌ be any smooth map inducing θ. Since (Ω, dsΩ2 ) is a Cartan-Hadamard manifold, i.e., a simply connected complete Riemannian manifold of nonpositive sectional curvature, the center of gravity argument gives a point x ∈ Ω fixed by L. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 34 / 40 Bi-algebraicity by means of Nadel’s Theorem Maps inducing the representation θ : Γ̌ ,→ H0 ⊂ G0 = Aut0 (Ω) Without loss of generality assume Ω ⊃ Z smallest BSD containing Z , ı : Y ,→ ZΓ̌ , θ := ı∗ π1 (Y ) = Γ̌ ⊂ H0 . By the proof of Nadel’s Theorem, H0 is a semisimple Lie group without compact factors acting on Ω. Write L ⊂ H0 for a maximal compact subgroup. Let f : Y → Γ̌\H0 /L =: SΓ̌ ,→ XΓ̌ be any smooth map inducing θ. Since (Ω, dsΩ2 ) is a Cartan-Hadamard manifold, i.e., a simply connected complete Riemannian manifold of nonpositive sectional curvature, the center of gravity argument gives a point x ∈ Ω fixed by L. Regard H0 /L as the orbit H0 x ⊂ Ω = G0 /K , L ⊂ K = Isotx (Ω, dsΩ2 ), hence SΓ̌ ,→ XΓ̌ := Γ̌\Ω = Γ̌\G /K as a real analytic submanifold. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 34 / 40 Proposition 1 ⇒ Total Geodesy of Z ⊂ Ω Since XΓ̌ is a K (π, 1), the two smooth maps ı, f : Y → XΓ̌ inducing the representation θ are homotopic to each other. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 35 / 40 Proposition 1 ⇒ Total Geodesy of Z ⊂ Ω Since XΓ̌ is a K (π, 1), the two smooth maps ı, f : Y → XΓ̌ inducing the representation θ are homotopic to each other. Denote by ω the Kähler form of the canonical KE metric on XΓ̌ . H0 acts on Ω. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 35 / 40 Proposition 1 ⇒ Total Geodesy of Z ⊂ Ω Since XΓ̌ is a K (π, 1), the two smooth maps ı, f : Y → XΓ̌ inducing the representation θ are homotopic to each other. Denote by ω the Kähler form of the canonical KE metric on XΓ̌ . H0 acts on Ω. For any x ∈ X , we have dimR (SΓ̌ ) ≤ dimR (H0 x) ≤ dimR Z = dimR Y := 2m. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 35 / 40 Proposition 1 ⇒ Total Geodesy of Z ⊂ Ω Since XΓ̌ is a K (π, 1), the two smooth maps ı, f : Y → XΓ̌ inducing the representation θ are homotopic to each other. Denote by ω the Kähler form of the canonical KE metric on XΓ̌ . H0 acts on Ω. For any x ∈ X , we have dimR (SΓ̌ ) ≤ dimR (H0 x) ≤ dimR Z = dimR Y := 2m. R R By homotopy Y (ı∗ ω)m = Y (f ∗ ω)m . The first integral gives m!Vol(Y , ω|Y ) > 0. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 35 / 40 Proposition 1 ⇒ Total Geodesy of Z ⊂ Ω Since XΓ̌ is a K (π, 1), the two smooth maps ı, f : Y → XΓ̌ inducing the representation θ are homotopic to each other. Denote by ω the Kähler form of the canonical KE metric on XΓ̌ . H0 acts on Ω. For any x ∈ X , we have dimR (SΓ̌ ) ≤ dimR (H0 x) ≤ dimR Z = dimR Y := 2m. R R By homotopy Y (ı∗ ω)m = Y (f ∗ ω)m . The first integral gives m!Vol(Y , ω|Y ) > 0. A contradiction would arise if we had strict inequality of dimensions. Hence, equality holds, Z is homogeneous under H0 , and H0 is of Hermitian type. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 35 / 40 Proposition 1 ⇒ Total Geodesy of Z ⊂ Ω Since XΓ̌ is a K (π, 1), the two smooth maps ı, f : Y → XΓ̌ inducing the representation θ are homotopic to each other. Denote by ω the Kähler form of the canonical KE metric on XΓ̌ . H0 acts on Ω. For any x ∈ X , we have dimR (SΓ̌ ) ≤ dimR (H0 x) ≤ dimR Z = dimR Y := 2m. R R By homotopy Y (ı∗ ω)m = Y (f ∗ ω)m . The first integral gives m!Vol(Y , ω|Y ) > 0. A contradiction would arise if we had strict inequality of dimensions. Hence, equality holds, Z is homogeneous under H0 , and H0 is of Hermitian type. Thus, Z ⊂ Ω is the image of an equivariant holomorphic map between bounded symmetric domains. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 35 / 40 Proposition 1 ⇒ Total Geodesy of Z ⊂ Ω Since XΓ̌ is a K (π, 1), the two smooth maps ı, f : Y → XΓ̌ inducing the representation θ are homotopic to each other. Denote by ω the Kähler form of the canonical KE metric on XΓ̌ . H0 acts on Ω. For any x ∈ X , we have dimR (SΓ̌ ) ≤ dimR (H0 x) ≤ dimR Z = dimR Y := 2m. R R By homotopy Y (ı∗ ω)m = Y (f ∗ ω)m . The first integral gives m!Vol(Y , ω|Y ) > 0. A contradiction would arise if we had strict inequality of dimensions. Hence, equality holds, Z is homogeneous under H0 , and H0 is of Hermitian type. Thus, Z ⊂ Ω is the image of an equivariant holomorphic map between bounded symmetric domains. By Proposition 1, Z ⊂ Ω is totally geodesic. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 35 / 40 Existential Closedness Problem The original Existential Closedness Problem, raised by Zilber, asks for a minimal set of conditions on an algebraic subvariety of V ⊂ Cn × (C ∗ )n to guarantee that V ∩ Graph(exp) is Zariski dense in V . It ties up with the André-Oort and the Zilber-Pink conjectures in Diophantine geometry. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 36 / 40 Existential Closedness Problem The original Existential Closedness Problem, raised by Zilber, asks for a minimal set of conditions on an algebraic subvariety of V ⊂ Cn × (C ∗ )n to guarantee that V ∩ Graph(exp) is Zariski dense in V . It ties up with the André-Oort and the Zilber-Pink conjectures in Diophantine geometry. ECP for Shimura Varieties Theorem (Eterovic-Zhao 2021) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, and Γ ⊂ Aut(Ω) be a torsionfree arithmetic lattice. Write XΓ := Ω/Γ, identified as a Zariski open subset XΓ ⊂ XΓ of its minimal (projective) compactification XΓ , and denote by q : Ω → XΓ the uniformization map. Write π1 : CN × XΓ → CN for the canonical projection map onto the first Cartesian factor. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 36 / 40 Existential Closedness Problem The original Existential Closedness Problem, raised by Zilber, asks for a minimal set of conditions on an algebraic subvariety of V ⊂ Cn × (C ∗ )n to guarantee that V ∩ Graph(exp) is Zariski dense in V . It ties up with the André-Oort and the Zilber-Pink conjectures in Diophantine geometry. ECP for Shimura Varieties Theorem (Eterovic-Zhao 2021) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, and Γ ⊂ Aut(Ω) be a torsionfree arithmetic lattice. Write XΓ := Ω/Γ, identified as a Zariski open subset XΓ ⊂ XΓ of its minimal (projective) compactification XΓ , and denote by q : Ω → XΓ the uniformization map. Write π1 : CN × XΓ → CN for the canonical projection map onto the first Cartesian factor. Let now V ⊂ CN × XΓ be an irreducible algebraic subvariety such that π1 (V ) is Zariski dense in CN . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 36 / 40 Existential Closedness Problem The original Existential Closedness Problem, raised by Zilber, asks for a minimal set of conditions on an algebraic subvariety of V ⊂ Cn × (C ∗ )n to guarantee that V ∩ Graph(exp) is Zariski dense in V . It ties up with the André-Oort and the Zilber-Pink conjectures in Diophantine geometry. ECP for Shimura Varieties Theorem (Eterovic-Zhao 2021) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, and Γ ⊂ Aut(Ω) be a torsionfree arithmetic lattice. Write XΓ := Ω/Γ, identified as a Zariski open subset XΓ ⊂ XΓ of its minimal (projective) compactification XΓ , and denote by q : Ω → XΓ the uniformization map. Write π1 : CN × XΓ → CN for the canonical projection map onto the first Cartesian factor. Let now V ⊂ CN × XΓ be an irreducible algebraic subvariety such that π1 (V ) is Zariski dense in CN . Then, π1 (V ∩ Graph(q)) is Zariski dense in CN , and V ∩ Graph(q) is Zariski dense in V . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 36 / 40 Existential Closedness Problem The original Existential Closedness Problem, raised by Zilber, asks for a minimal set of conditions on an algebraic subvariety of V ⊂ Cn × (C ∗ )n to guarantee that V ∩ Graph(exp) is Zariski dense in V . It ties up with the André-Oort and the Zilber-Pink conjectures in Diophantine geometry. ECP for Shimura Varieties Theorem (Eterovic-Zhao 2021) Let Ω b CN be a bounded symmetric domain in its Harish-Chandra realization, and Γ ⊂ Aut(Ω) be a torsionfree arithmetic lattice. Write XΓ := Ω/Γ, identified as a Zariski open subset XΓ ⊂ XΓ of its minimal (projective) compactification XΓ , and denote by q : Ω → XΓ the uniformization map. Write π1 : CN × XΓ → CN for the canonical projection map onto the first Cartesian factor. Let now V ⊂ CN × XΓ be an irreducible algebraic subvariety such that π1 (V ) is Zariski dense in CN . Then, π1 (V ∩ Graph(q)) is Zariski dense in CN , and V ∩ Graph(q) is Zariski dense in V . The proof involves studying the Shilov boundary of Ω b CN . Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 36 / 40 Bibliography Bakker, J.; Tsimerman, J.: The Ax-Schanuel conjecture for variations of Hodge structures, Invent. Math. 217 (2019), 77–94. 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Pila, J.; Wilkie, A.J.: The rational points of a definable set, Duke Math. J. 133 (2006), 591–616. Pila, J.; Zannier, U.: Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19 (2008), 149–162. Tsimerman, J.: The André-Oort conjecture for Ag , Ann. of Math. 187 (2018), 379–390. Tsimerman, J.: Functional transcendence and arithmetic applications, Proceedings of the ICM–Rio de Janeiro 2018 , Volume II, Invited lectures, 435–454, World Sci. Publ., Hackensack, NJ, 2018. Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 39 / 40 THANK YOU!! Ngaiming Mok (HKU) Complex Function Fields August 19, 2022 40 / 40