Two conjectures A family of elliptic surfaces The Tate conjecture for S The Tate conjecture for a concrete family of elliptic surfaces Xiyuan Wang Johns Hopkins University November 14, 2020 (Joint work with Lian Duan) Xiyuan Wang Two conjectures A family of elliptic surfaces The Tate conjecture for S Outline 1. Two conjectures The Tate conjecture The Fontaine-Mazur conjecture 2. A family of elliptic surfaces Geemen and Top’s construction Geemen and Top’s conjecture 3. The Tate conjecture for S Main theorem Idea of the proof Main step Xiyuan Wang • K : number field (eg. K = Q) • GK : absolute Galois group Gal(K /K ) • X : smooth projective variety over K •Z i (X ) : free abelian group Zhcodimen i irred subvar of X i i i • NS (X ) : Neron-Severi group Z (X )/ ∼alg Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Set up Xiyuan Wang • GK : absolute Galois group Gal(K /K ) • X : smooth projective variety over K •Z i (X ) : free abelian group Zhcodimen i irred subvar of X i i i • NS (X ) : Neron-Severi group Z (X )/ ∼alg Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Set up • K : number field (eg. K = Q) Xiyuan Wang • X : smooth projective variety over K •Z i (X ) : free abelian group Zhcodimen i irred subvar of X i i i • NS (X ) : Neron-Severi group Z (X )/ ∼alg Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Set up • K : number field (eg. K = Q) • GK : absolute Galois group Gal(K /K ) Xiyuan Wang •Z i (X ) : free abelian group Zhcodimen i irred subvar of X i i i • NS (X ) : Neron-Severi group Z (X )/ ∼alg Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Set up • K : number field (eg. K = Q) • GK : absolute Galois group Gal(K /K ) • X : smooth projective variety over K Xiyuan Wang i i • NS (X ) : Neron-Severi group Z (X )/ ∼alg Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Set up • K : number field (eg. K = Q) • GK : absolute Galois group Gal(K /K ) • X : smooth projective variety over K •Z i (X ) : free abelian group Zhcodimen i irred subvar of X i Xiyuan Wang Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Set up • K : number field (eg. K = Q) • GK : absolute Galois group Gal(K /K ) • X : smooth projective variety over K •Z i (X ) : free abelian group Zhcodimen i irred subvar of X i i i • NS (X ) : Neron-Severi group Z (X )/ ∼alg Xiyuan Wang The cycle class map i i 2i GK c : NS (X ) ! Het (XK , Q`(i)) . The Tate conjecture The map i i 2i GK c ⊗ Q` : NS (X ) ⊗ Q` ! Het (XK , Q`(i)) is surjective. Slogan Tate classes are algebraic! Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Tate conjecture Xiyuan Wang The Tate conjecture The map i i 2i GK c ⊗ Q` : NS (X ) ⊗ Q` ! Het (XK , Q`(i)) is surjective. Slogan Tate classes are algebraic! Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Tate conjecture The cycle class map i i 2i GK c : NS (X ) ! Het (XK , Q`(i)) . Xiyuan Wang is surjective. Slogan Tate classes are algebraic! Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Tate conjecture The cycle class map i i 2i GK c : NS (X ) ! Het (XK , Q`(i)) . The Tate conjecture The map i i 2i GK c ⊗ Q` : NS (X ) ⊗ Q` ! Het (XK , Q`(i)) Xiyuan Wang Slogan Tate classes are algebraic! Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Tate conjecture The cycle class map i i 2i GK c : NS (X ) ! Het (XK , Q`(i)) . The Tate conjecture The map i i 2i GK c ⊗ Q` : NS (X ) ⊗ Q` ! Het (XK , Q`(i)) is surjective. Xiyuan Wang Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Tate conjecture The cycle class map i i 2i GK c : NS (X ) ! Het (XK , Q`(i)) . The Tate conjecture The map i i 2i GK c ⊗ Q` : NS (X ) ⊗ Q` ! Het (XK , Q`(i)) is surjective. Slogan Tate classes are algebraic! Xiyuan Wang Divisor case (i = 1) • Abelian varieties • K3 surfaces • Varieties with h2,0 = 1 • Hilbert modular surfaces • Elliptic modular surfaces • ... Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases Xiyuan Wang • Abelian varieties • K3 surfaces • Varieties with h2,0 = 1 • Hilbert modular surfaces • Elliptic modular surfaces • ... Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases Divisor case (i = 1) Xiyuan Wang • K3 surfaces • Varieties with h2,0 = 1 • Hilbert modular surfaces • Elliptic modular surfaces • ... Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases Divisor case (i = 1) • Abelian varieties Xiyuan Wang • Varieties with h2,0 = 1 • Hilbert modular surfaces • Elliptic modular surfaces • ... Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases Divisor case (i = 1) • Abelian varieties • K3 surfaces Xiyuan Wang • Hilbert modular surfaces • Elliptic modular surfaces • ... Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases Divisor case (i = 1) • Abelian varieties • K3 surfaces • Varieties with h2,0 = 1 Xiyuan Wang • Elliptic modular surfaces • ... Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases Divisor case (i = 1) • Abelian varieties • K3 surfaces • Varieties with h2,0 = 1 • Hilbert modular surfaces Xiyuan Wang • ... Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases Divisor case (i = 1) • Abelian varieties • K3 surfaces • Varieties with h2,0 = 1 • Hilbert modular surfaces • Elliptic modular surfaces Xiyuan Wang Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases Divisor case (i = 1) • Abelian varieties • K3 surfaces • Varieties with h2,0 = 1 • Hilbert modular surfaces • Elliptic modular surfaces • ... Xiyuan Wang • Let X be a K3 surface. The Kuga-Satake construction X ` A × A where A is an abelian variety. Rk: H2,0(X ) = 1 is a crucial input. • The Hodge conj. + The Mumford-Tate conj.) The Tate conj. Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Some remarks on the proofs Xiyuan Wang Rk: H2,0(X ) = 1 is a crucial input. • The Hodge conj. + The Mumford-Tate conj.) The Tate conj. Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Some remarks on the proofs • Let X be a K3 surface. The Kuga-Satake construction X ` A × A where A is an abelian variety. Xiyuan Wang • The Hodge conj. + The Mumford-Tate conj.) The Tate conj. Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Some remarks on the proofs • Let X be a K3 surface. The Kuga-Satake construction X ` A × A where A is an abelian variety. Rk: H2,0(X ) = 1 is a crucial input. Xiyuan Wang Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Some remarks on the proofs • Let X be a K3 surface. The Kuga-Satake construction X ` A × A where A is an abelian variety. Rk: H2,0(X ) = 1 is a crucial input. • The Hodge conj. + The Mumford-Tate conj.) The Tate conj. Xiyuan Wang V ⊆ ss Galois representations come from algebraic geometry. i GK y Het (XK , Q`(j)) Properties of V (a) V unramified almost everywhere. (b) V de Rham above `. Definition (Geometric) A Galois representation is geometric if it satisfies (a) and (b). Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Geometric Galois representations Xiyuan Wang V ⊆ ss i GK y Het (XK , Q`(j)) Properties of V (a) V unramified almost everywhere. (b) V de Rham above `. Definition (Geometric) A Galois representation is geometric if it satisfies (a) and (b). Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Geometric Galois representations Galois representations come from algebraic geometry. Xiyuan Wang V ⊆ ss Properties of V (a) V unramified almost everywhere. (b) V de Rham above `. Definition (Geometric) A Galois representation is geometric if it satisfies (a) and (b). Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Geometric Galois representations Galois representations come from algebraic geometry. i GK y Het (XK , Q`(j)) Xiyuan Wang Properties of V (a) V unramified almost everywhere. (b) V de Rham above `.