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 `.
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 ss GK y V ⊆ Het (XK , Q`(j))
Xiyuan Wang (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 ss GK y V ⊆ Het (XK , Q`(j)) Properties of V
Xiyuan Wang (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 ss GK y V ⊆ Het (XK , Q`(j)) Properties of V (a) V unramified almost everywhere.
Xiyuan Wang 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 ss GK y V ⊆ Het (XK , Q`(j)) Properties of V (a) V unramified almost everywhere. (b) V de Rham above `.
Xiyuan Wang 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 ss GK y V ⊆ 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).
Xiyuan Wang The Fontaine-Mazur conjecture If a Galois representation is geometric, it comes from algebraic geometry.
Met : Pure motives → Galois representations
• The FM conj. ⇔ EssImage(Met ) = {geometric Galois rep}
• The Tate conj. ⇔ Met is fully faithful
Slogan Geometric Galois representations are motivic!
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Fontaine-Mazur conjecture
Xiyuan Wang Met : Pure motives → Galois representations
• The FM conj. ⇔ EssImage(Met ) = {geometric Galois rep}
• The Tate conj. ⇔ Met is fully faithful
Slogan Geometric Galois representations are motivic!
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Fontaine-Mazur conjecture
The Fontaine-Mazur conjecture If a Galois representation is geometric, it comes from algebraic geometry.
Xiyuan Wang • The FM conj. ⇔ EssImage(Met ) = {geometric Galois rep}
• The Tate conj. ⇔ Met is fully faithful
Slogan Geometric Galois representations are motivic!
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Fontaine-Mazur conjecture
The Fontaine-Mazur conjecture If a Galois representation is geometric, it comes from algebraic geometry.
Met : Pure motives → Galois representations
Xiyuan Wang • The Tate conj. ⇔ Met is fully faithful
Slogan Geometric Galois representations are motivic!
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Fontaine-Mazur conjecture
The Fontaine-Mazur conjecture If a Galois representation is geometric, it comes from algebraic geometry.
Met : Pure motives → Galois representations
• The FM conj. ⇔ EssImage(Met ) = {geometric Galois rep}
Xiyuan Wang Slogan Geometric Galois representations are motivic!
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Fontaine-Mazur conjecture
The Fontaine-Mazur conjecture If a Galois representation is geometric, it comes from algebraic geometry.
Met : Pure motives → Galois representations
• The FM conj. ⇔ EssImage(Met ) = {geometric Galois rep}
• The Tate conj. ⇔ Met is fully faithful
Xiyuan Wang Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S The Fontaine-Mazur conjecture
The Fontaine-Mazur conjecture If a Galois representation is geometric, it comes from algebraic geometry.
Met : Pure motives → Galois representations
• The FM conj. ⇔ EssImage(Met ) = {geometric Galois rep}
• The Tate conj. ⇔ Met is fully faithful
Slogan Geometric Galois representations are motivic!
Xiyuan Wang n=1 Class field theory+ Classification of algebraic Hecke characters n=2 K = Q!
FM conj Geom. Galois reps Modular forms
FM conj Pure motives
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases, n = dim V
Xiyuan Wang n=2 K = Q!
FM conj Geom. Galois reps Modular forms
FM conj Pure motives
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases, n = dim V
n=1 Class field theory+ Classification of algebraic Hecke characters
Xiyuan Wang FM conj Geom. Galois reps Modular forms
FM conj Pure motives
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases, n = dim V
n=1 Class field theory+ Classification of algebraic Hecke characters n=2 K = Q!
Xiyuan Wang Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Known cases, n = dim V
n=1 Class field theory+ Classification of algebraic Hecke characters n=2 K = Q!
FM conj Geom. Galois reps Modular forms
FM conj Pure motives
Xiyuan Wang Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Modular forms of wt > 1 Distinct HT wts even Doesn0t exists! odd Modular form of wt = 1 Same HT wts even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Xiyuan Wang Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Modular forms of wt > 1 Distinct HT wts even Doesn0t exists! odd Modular form of wt = 1 Same HT wts even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation.
Xiyuan Wang odd Modular forms of wt > 1 Distinct HT wts even Doesn0t exists! odd Modular form of wt = 1 Same HT wts even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
Xiyuan Wang odd Modular forms of wt > 1 even Doesn0t exists! odd Modular form of wt = 1 Same HT wts even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
Distinct HT wts
Xiyuan Wang Modular forms of wt > 1 even Doesn0t exists! odd Modular form of wt = 1 Same HT wts even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Distinct HT wts
Xiyuan Wang even Doesn0t exists! odd Modular form of wt = 1 Same HT wts even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
> odd Modular forms of wt 1 Distinct HT wts
Xiyuan Wang Doesn0t exists! odd Modular form of wt = 1 Same HT wts even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Modular forms of wt > 1 Distinct HT wts even
Xiyuan Wang odd Modular form of wt = 1 Same HT wts even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Modular forms of wt > 1 Distinct HT wts even Doesn0t exists!
Xiyuan Wang odd Modular form of wt = 1 even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Modular forms of wt > 1 Distinct HT wts even Doesn0t exists! Same HT wts
Xiyuan Wang Modular form of wt = 1 even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Modular forms of wt > 1 Distinct HT wts even Doesn0t exists! odd Same HT wts
Xiyuan Wang even Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Modular forms of wt > 1 Distinct HT wts even Doesn0t exists! odd Modular form of wt = 1 Same HT wts
Xiyuan Wang Maass form with λ = 1/4
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Modular forms of wt > 1 Distinct HT wts even Doesn0t exists! odd Modular form of wt = 1 Same HT wts even
Xiyuan Wang Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S n = 2, Geometric Galois representations of GQ
Let c ∈ GQ be the complex conjugation. Definition (Odd)
A Galois representation r : GQ → GL2(Q`) is odd if det r(c) = −1.
odd Modular forms of wt > 1 Distinct HT wts even Doesn0t exists! odd Modular form of wt = 1 Same HT wts even Maass form with λ = 1/4
Xiyuan Wang (2) r|G is not a twist of a representation of the form Q` ε` ∗ 0 1
where ε` is the mod-` cyclotomic character, (3) r is not of dihedral type, (4) r is absolutely irreducible. In particular, r is odd.
Theorem (Calegari)
Let r : GQ → GL2(Q`) be a geometric Galois representation. Suppose that ` > 7, and, furthermore, that
(1) r|G has distinct Hodge-Tate weights, Q`
Then r is modular.
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Calegari’s theorem
Xiyuan Wang (2) r|G is not a twist of a representation of the form Q` ε` ∗ 0 1
where ε` is the mod-` cyclotomic character, (3) r is not of dihedral type, (4) r is absolutely irreducible. In particular, r is odd.
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Calegari’s theorem
Theorem (Calegari)
Let r : GQ → GL2(Q`) be a geometric Galois representation. Suppose that ` > 7, and, furthermore, that
(1) r|G has distinct Hodge-Tate weights, Q`
Then r is modular.
Xiyuan Wang In particular, r is odd.
Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Calegari’s theorem
Theorem (Calegari)
Let r : GQ → GL2(Q`) be a geometric Galois representation. Suppose that ` > 7, and, furthermore, that
(1) r|G has distinct Hodge-Tate weights, Q` (2) r|G is not a twist of a representation of the form Q` ε` ∗ 0 1
where ε` is the mod-` cyclotomic character, (3) r is not of dihedral type, (4) r is absolutely irreducible. Then r is modular.
Xiyuan Wang Two conjectures The Tate conjecture A family of elliptic surfaces The Fontaine-Mazur conjecture The Tate conjecture for S Calegari’s theorem
Theorem (Calegari)
Let r : GQ → GL2(Q`) be a geometric Galois representation. Suppose that ` > 7, and, furthermore, that
(1) r|G has distinct Hodge-Tate weights, Q` (2) r|G is not a twist of a representation of the form Q` ε` ∗ 0 1
where ε` is the mod-` cyclotomic character, (3) r is not of dihedral type, (4) r is absolutely irreducible. Then r is modular. In particular, r is odd.
Xiyuan Wang (a + 1) E : Y 2 = X (X 2 + 2( + a)X + 1), a ∈ Q a t2 Define Sa Xa Ea
y y 1 j 1 h 1 Pz Pu Pt
where j : z 7→ u = (z2 − 1)/z and h : u 7→ t = (u2 − 4)/4u. Main player
S := Sa
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Elliptic surfaces
Xiyuan Wang Define Sa Xa Ea
y y 1 j 1 h 1 Pz Pu Pt
where j : z 7→ u = (z2 − 1)/z and h : u 7→ t = (u2 − 4)/4u. Main player
S := Sa
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Elliptic surfaces
(a + 1) E : Y 2 = X (X 2 + 2( + a)X + 1), a ∈ Q a t2
Xiyuan Wang Sa Xa Ea
y y 1 j 1 h 1 Pz Pu Pt
where j : z 7→ u = (z2 − 1)/z and h : u 7→ t = (u2 − 4)/4u. Main player
S := Sa
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Elliptic surfaces
(a + 1) E : Y 2 = X (X 2 + 2( + a)X + 1), a ∈ Q a t2 Define
Xiyuan Wang where j : z 7→ u = (z2 − 1)/z and h : u 7→ t = (u2 − 4)/4u. Main player
S := Sa
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Elliptic surfaces
(a + 1) E : Y 2 = X (X 2 + 2( + a)X + 1), a ∈ Q a t2 Define Sa Xa Ea
y y 1 j 1 h 1 Pz Pu Pt
Xiyuan Wang Main player
S := Sa
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Elliptic surfaces
(a + 1) E : Y 2 = X (X 2 + 2( + a)X + 1), a ∈ Q a t2 Define Sa Xa Ea
y y 1 j 1 h 1 Pz Pu Pt
where j : z 7→ u = (z2 − 1)/z and h : u 7→ t = (u2 − 4)/4u.
Xiyuan Wang Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Elliptic surfaces
(a + 1) E : Y 2 = X (X 2 + 2( + a)X + 1), a ∈ Q a t2 Define Sa Xa Ea
y y 1 j 1 h 1 Pz Pu Pt
where j : z 7→ u = (z2 − 1)/z and h : u 7→ t = (u2 − 4)/4u. Main player
S := Sa
Xiyuan Wang S X E
y y 1 1 1 Pz Pu Pt
• Taking a good prime p, E → P1 has bad fibre at 0, ±i, and Fp q 1+a ± 1−a . • The elliptic surface X is K3 with Picard number 19 or 20 (depending on a). • dim H2(S) = 46, dim H2,0(S) = dim H0,2(S) = 3, and Picard 1 number ρ(SQ) = rank(NS (SQ)) = 37, 38, 39, or 40 (depending on a).
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Properties of S
Xiyuan Wang • Taking a good prime p, E → P1 has bad fibre at 0, ±i, and Fp q 1+a ± 1−a . • The elliptic surface X is K3 with Picard number 19 or 20 (depending on a). • dim H2(S) = 46, dim H2,0(S) = dim H0,2(S) = 3, and Picard 1 number ρ(SQ) = rank(NS (SQ)) = 37, 38, 39, or 40 (depending on a).
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Properties of S
S X E
y y 1 1 1 Pz Pu Pt
Xiyuan Wang • The elliptic surface X is K3 with Picard number 19 or 20 (depending on a). • dim H2(S) = 46, dim H2,0(S) = dim H0,2(S) = 3, and Picard 1 number ρ(SQ) = rank(NS (SQ)) = 37, 38, 39, or 40 (depending on a).
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Properties of S
S X E
y y 1 1 1 Pz Pu Pt
• Taking a good prime p, E → P1 has bad fibre at 0, ±i, and Fp q 1+a ± 1−a .
Xiyuan Wang • dim H2(S) = 46, dim H2,0(S) = dim H0,2(S) = 3, and Picard 1 number ρ(SQ) = rank(NS (SQ)) = 37, 38, 39, or 40 (depending on a).
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Properties of S
S X E
y y 1 1 1 Pz Pu Pt
• Taking a good prime p, E → P1 has bad fibre at 0, ±i, and Fp q 1+a ± 1−a . • The elliptic surface X is K3 with Picard number 19 or 20 (depending on a).
Xiyuan Wang and Picard 1 number ρ(SQ) = rank(NS (SQ)) = 37, 38, 39, or 40 (depending on a).
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Properties of S
S X E
y y 1 1 1 Pz Pu Pt
• Taking a good prime p, E → P1 has bad fibre at 0, ±i, and Fp q 1+a ± 1−a . • The elliptic surface X is K3 with Picard number 19 or 20 (depending on a). • dim H2(S) = 46, dim H2,0(S) = dim H0,2(S) = 3,
Xiyuan Wang Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Properties of S
S X E
y y 1 1 1 Pz Pu Pt
• Taking a good prime p, E → P1 has bad fibre at 0, ±i, and Fp q 1+a ± 1−a . • The elliptic surface X is K3 with Picard number 19 or 20 (depending on a). • dim H2(S) = 46, dim H2,0(S) = dim H0,2(S) = 3, and Picard 1 number ρ(SQ) = rank(NS (SQ)) = 37, 38, 39, or 40 (depending on a).
Xiyuan Wang 2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(S) ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W`
(Taking semisimplification!!!)
46 = 37 + 3 + (43 − 37)
By the construction of S, C4 y W`. We have ∼ W` = V` ⊕ V `,
where dim V` = dim V ` = 3.
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
Xiyuan Wang 2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(S) ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W`
46 = 37 + 3 + (43 − 37)
By the construction of S, C4 y W`. We have ∼ W` = V` ⊕ V `,
where dim V` = dim V ` = 3.
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
(Taking semisimplification!!!)
Xiyuan Wang ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(S) ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W` 46 = 37 + 3 + (43 − 37)
By the construction of S, C4 y W`. We have ∼ W` = V` ⊕ V `,
where dim V` = dim V ` = 3.
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
(Taking semisimplification!!!)
2 Het (SQ, Q`(1))
Xiyuan Wang Tran`(S) ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W` 46 = 37 + 3 + (43 − 37)
By the construction of S, C4 y W`. We have ∼ W` = V` ⊕ V `,
where dim V` = dim V ` = 3.
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
(Taking semisimplification!!!)
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕
Xiyuan Wang ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W` 46 = 37 + 3 + (43 − 37)
By the construction of S, C4 y W`. We have ∼ W` = V` ⊕ V `,
where dim V` = dim V ` = 3.
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
(Taking semisimplification!!!)
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(S)
Xiyuan Wang 46 = 37 + 3 + (43 − 37)
By the construction of S, C4 y W`. We have ∼ W` = V` ⊕ V `,
where dim V` = dim V ` = 3.
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
(Taking semisimplification!!!)
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(S) ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W`
Xiyuan Wang By the construction of S, C4 y W`. We have ∼ W` = V` ⊕ V `,
where dim V` = dim V ` = 3.
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
(Taking semisimplification!!!)
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(S) ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W` 46 = 37 + 3 + (43 − 37)
Xiyuan Wang We have ∼ W` = V` ⊕ V `,
where dim V` = dim V ` = 3.
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
(Taking semisimplification!!!)
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(S) ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W` 46 = 37 + 3 + (43 − 37)
By the construction of S, C4 y W`.
Xiyuan Wang where dim V` = dim V ` = 3.
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
(Taking semisimplification!!!)
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(S) ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W` 46 = 37 + 3 + (43 − 37)
By the construction of S, C4 y W`. We have ∼ W` = V` ⊕ V `,
Xiyuan Wang Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S 2 Het (SQ, Q`(1)), ρ(SQ) = 37
(Taking semisimplification!!!)
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(S) ∼ 1 = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ W` 46 = 37 + 3 + (43 − 37)
By the construction of S, C4 y W`. We have ∼ W` = V` ⊕ V `,
where dim V` = dim V ` = 3.
Xiyuan Wang Several properties of V` ∼ • V` = V ` 1,−1 0,0 −1,1 • h (V`) = h (V`) = h (V`) = 1 ∼ ∗ • V` = V` , i.e., V` self-dual
Conjecture (van Geemen and Top, 1995) For each a, ∼ 2 V`(−1) = δ Sym T`E for some elliptic curve E and some quadratic character δ.
Selfdual and non-selfdual 3-dimensional Galois representations, 1995
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Geemen and Top’s conjecture
Xiyuan Wang ∼ • V` = V ` 1,−1 0,0 −1,1 • h (V`) = h (V`) = h (V`) = 1 ∼ ∗ • V` = V` , i.e., V` self-dual
Conjecture (van Geemen and Top, 1995) For each a, ∼ 2 V`(−1) = δ Sym T`E for some elliptic curve E and some quadratic character δ.
Selfdual and non-selfdual 3-dimensional Galois representations, 1995
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Geemen and Top’s conjecture
Several properties of V`
Xiyuan Wang 1,−1 0,0 −1,1 • h (V`) = h (V`) = h (V`) = 1 ∼ ∗ • V` = V` , i.e., V` self-dual
Conjecture (van Geemen and Top, 1995) For each a, ∼ 2 V`(−1) = δ Sym T`E for some elliptic curve E and some quadratic character δ.
Selfdual and non-selfdual 3-dimensional Galois representations, 1995
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Geemen and Top’s conjecture
Several properties of V` ∼ • V` = V `
Xiyuan Wang ∼ ∗ • V` = V` , i.e., V` self-dual
Conjecture (van Geemen and Top, 1995) For each a, ∼ 2 V`(−1) = δ Sym T`E for some elliptic curve E and some quadratic character δ.
Selfdual and non-selfdual 3-dimensional Galois representations, 1995
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Geemen and Top’s conjecture
Several properties of V` ∼ • V` = V ` 1,−1 0,0 −1,1 • h (V`) = h (V`) = h (V`) = 1
Xiyuan Wang Conjecture (van Geemen and Top, 1995) For each a, ∼ 2 V`(−1) = δ Sym T`E for some elliptic curve E and some quadratic character δ.
Selfdual and non-selfdual 3-dimensional Galois representations, 1995
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Geemen and Top’s conjecture
Several properties of V` ∼ • V` = V ` 1,−1 0,0 −1,1 • h (V`) = h (V`) = h (V`) = 1 ∼ ∗ • V` = V` , i.e., V` self-dual
Xiyuan Wang Selfdual and non-selfdual 3-dimensional Galois representations, 1995
Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Geemen and Top’s conjecture
Several properties of V` ∼ • V` = V ` 1,−1 0,0 −1,1 • h (V`) = h (V`) = h (V`) = 1 ∼ ∗ • V` = V` , i.e., V` self-dual
Conjecture (van Geemen and Top, 1995) For each a, ∼ 2 V`(−1) = δ Sym T`E for some elliptic curve E and some quadratic character δ.
Xiyuan Wang Two conjectures Geemen and Top’s construction A family of elliptic surfaces Geemen and Top’s conjecture The Tate conjecture for S Geemen and Top’s conjecture
Several properties of V` ∼ • V` = V ` 1,−1 0,0 −1,1 • h (V`) = h (V`) = h (V`) = 1 ∼ ∗ • V` = V` , i.e., V` self-dual
Conjecture (van Geemen and Top, 1995) For each a, ∼ 2 V`(−1) = δ Sym T`E for some elliptic curve E and some quadratic character δ.
Selfdual and non-selfdual 3-dimensional Galois representations, 1995
Xiyuan Wang Theorem (Duan and W., 2019) If a ≡ 2, 3 mod 5 and none of 2(1 + a) or 2(1 − a) is a square in Q, then, for a density one subset of primes `, the Tate conjecture for S is true, i.e.,
∼ 1 = 2 GQ NS (S) ⊗ Q` −→ Het (SQ, Q`(1)) .
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Statement
Xiyuan Wang i.e.,
∼ 1 = 2 GQ NS (S) ⊗ Q` −→ Het (SQ, Q`(1)) .
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Statement
Theorem (Duan and W., 2019) If a ≡ 2, 3 mod 5 and none of 2(1 + a) or 2(1 − a) is a square in Q, then, for a density one subset of primes `, the Tate conjecture for S is true,
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Statement
Theorem (Duan and W., 2019) If a ≡ 2, 3 mod 5 and none of 2(1 + a) or 2(1 − a) is a square in Q, then, for a density one subset of primes `, the Tate conjecture for S is true, i.e.,
∼ 1 = 2 GQ NS (S) ⊗ Q` −→ Het (SQ, Q`(1)) .
Xiyuan Wang Up to semisimplification
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ V` ⊕ V `.
Taking GQ-invariant part
G G 2 GQ ∼ 1 Q Q Het (SQ, Q`(1)) = (NS (S) ⊗ Q`) ⊕ {0} ⊕ V` ⊕ V ` .
GQ V` = {0}?
Goal ss To prove ρ := V` is absolutely irreducible.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Tate classes are not transcendental
Xiyuan Wang 2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ V` ⊕ V `.
Taking GQ-invariant part
G G 2 GQ ∼ 1 Q Q Het (SQ, Q`(1)) = (NS (S) ⊗ Q`) ⊕ {0} ⊕ V` ⊕ V ` .
GQ V` = {0}?
Goal ss To prove ρ := V` is absolutely irreducible.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Tate classes are not transcendental
Up to semisimplification
Xiyuan Wang Taking GQ-invariant part
G G 2 GQ ∼ 1 Q Q Het (SQ, Q`(1)) = (NS (S) ⊗ Q`) ⊕ {0} ⊕ V` ⊕ V ` .
GQ V` = {0}?
Goal ss To prove ρ := V` is absolutely irreducible.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Tate classes are not transcendental
Up to semisimplification
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ V` ⊕ V `.
Xiyuan Wang G G 2 GQ ∼ 1 Q Q Het (SQ, Q`(1)) = (NS (S) ⊗ Q`) ⊕ {0} ⊕ V` ⊕ V ` .
GQ V` = {0}?
Goal ss To prove ρ := V` is absolutely irreducible.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Tate classes are not transcendental
Up to semisimplification
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ V` ⊕ V `.
Taking GQ-invariant part
Xiyuan Wang GQ V` = {0}?
Goal ss To prove ρ := V` is absolutely irreducible.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Tate classes are not transcendental
Up to semisimplification
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ V` ⊕ V `.
Taking GQ-invariant part
G G 2 GQ ∼ 1 Q Q Het (SQ, Q`(1)) = (NS (S) ⊗ Q`) ⊕ {0} ⊕ V` ⊕ V ` .
Xiyuan Wang Goal ss To prove ρ := V` is absolutely irreducible.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Tate classes are not transcendental
Up to semisimplification
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ V` ⊕ V `.
Taking GQ-invariant part
G G 2 GQ ∼ 1 Q Q Het (SQ, Q`(1)) = (NS (S) ⊗ Q`) ⊕ {0} ⊕ V` ⊕ V ` .
GQ V` = {0}?
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Tate classes are not transcendental
Up to semisimplification
2 ∼ 1 Het (SQ, Q`(1)) = (NS (SQ) ⊗ Q`) ⊕ Tran`(X ) ⊕ V` ⊕ V `.
Taking GQ-invariant part
G G 2 GQ ∼ 1 Q Q Het (SQ, Q`(1)) = (NS (S) ⊗ Q`) ⊕ {0} ⊕ V` ⊕ V ` .
GQ V` = {0}?
Goal ss To prove ρ := V` is absolutely irreducible.
Xiyuan Wang This is a contradiction. We are done!
What is ??? ?
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 ∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Xiyuan Wang This is a contradiction. We are done!
What is ??? ?
∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 ∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible.
Xiyuan Wang This is a contradiction. We are done!
What is ??? ?
∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3
Xiyuan Wang This is a contradiction. We are done!
What is ??? ?
r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 • (1 + 2) : ρ =∼ ψ ⊕ r,
Xiyuan Wang This is a contradiction. We are done!
What is ??? ?
Calegari’s FM Theorem odd
r
??? even
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 ∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual
Xiyuan Wang This is a contradiction. We are done!
What is ??? ?
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 ∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even
Xiyuan Wang This is a contradiction. We are done!
What is ??? ?
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 ∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even
Xiyuan Wang This is a contradiction. We are done!
What is ??? ?
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 ∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even
Xiyuan Wang We are done!
What is ??? ?
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 ∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even This is a contradiction.
Xiyuan Wang What is ??? ?
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 ∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even This is a contradiction. We are done!
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Odd and even
Assume that ρ : GQ → GL3(Q`) is not absolutely irreducible. ∼ • (1 + 1 + 1): ρ = χ1 ⊕ χ2 ⊕ χ3 ∼ • (1 + 2) : ρ = ψ ⊕ r, r : GQ → GL2(Q`) self-dual Calegari’s FM Theorem odd
r
??? even This is a contradiction. We are done!
What is ??? ?
Xiyuan Wang self-dual
self-dual
The FM conj ⇒ r is motivic ⇐⇒ r odd ⇐===⇒ det r 6= 1
The Tate conj ⇒ r is not motivic ⇐⇒ r even ⇐===⇒ det r = 1
Remember that r is self-dual.
??? = det r trivial
Main step To prove det r = 1. In summary,
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Xiyuan Wang self-dual
self-dual
The FM conj ⇒ r is motivic ⇐⇒ r odd ⇐===⇒ det r 6= 1
The Tate conj ⇒ r is not motivic ⇐⇒ r even ⇐===⇒ det r = 1
??? = det r trivial
Main step To prove det r = 1. In summary,
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
Xiyuan Wang self-dual
self-dual
The FM conj ⇒ r is motivic ⇐⇒ r odd ⇐===⇒ det r 6= 1
The Tate conj ⇒ r is not motivic ⇐⇒ r even ⇐===⇒ det r = 1
Main step To prove det r = 1. In summary,
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
??? = det r trivial
Xiyuan Wang self-dual
self-dual
The FM conj ⇒ r is motivic ⇐⇒ r odd ⇐===⇒ det r 6= 1
The Tate conj ⇒ r is not motivic ⇐⇒ r even ⇐===⇒ det r = 1
In summary,
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
??? = det r trivial
Main step To prove det r = 1.
Xiyuan Wang self-dual
self-dual
The FM conj ⇒ r is motivic ⇐⇒ r odd ⇐===⇒ det r 6= 1
The Tate conj ⇒ r is not motivic ⇐⇒ r even ⇐===⇒ det r = 1
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
??? = det r trivial
Main step To prove det r = 1. In summary,
Xiyuan Wang self-dual
self-dual
⇒ r is motivic ⇐⇒ r odd ⇐===⇒ det r 6= 1
⇒ r is not motivic ⇐⇒ r even ⇐===⇒ det r = 1
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
??? = det r trivial
Main step To prove det r = 1. In summary,
The FM conj
The Tate conj
Xiyuan Wang self-dual
self-dual
⇐⇒ r odd ⇐===⇒ det r 6= 1
⇒ r is not motivic ⇐⇒ r even ⇐===⇒ det r = 1
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
??? = det r trivial
Main step To prove det r = 1. In summary,
The FM conj ⇒ r is motivic
The Tate conj
Xiyuan Wang self-dual
self-dual
⇐⇒ r odd ⇐===⇒ det r 6= 1
⇐⇒ r even ⇐===⇒ det r = 1
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
??? = det r trivial
Main step To prove det r = 1. In summary,
The FM conj ⇒ r is motivic
The Tate conj ⇒ r is not motivic
Xiyuan Wang self-dual
self-dual
⇐===⇒ det r 6= 1
⇐===⇒ det r = 1
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
??? = det r trivial
Main step To prove det r = 1. In summary,
The FM conj ⇒ r is motivic ⇐⇒ r odd
The Tate conj ⇒ r is not motivic ⇐⇒ r even
Xiyuan Wang self-dual
self-dual
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
??? = det r trivial
Main step To prove det r = 1. In summary,
The FM conj ⇒ r is motivic ⇐⇒ r odd ⇐===⇒ det r 6= 1
The Tate conj ⇒ r is not motivic ⇐⇒ r even ⇐===⇒ det r = 1
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Elementary idea
Remember that r is self-dual.
??? = det r trivial
Main step To prove det r = 1. In summary,
The FM conj ⇒ r is motivic ⇐⇒ r odd ⇐self-dual===⇒ det r 6= 1
The Tate conj ⇒ r is not motivic ⇐⇒ r even ⇐self-dual===⇒ det r = 1
Xiyuan Wang Since r is self-dual, det r is a quadratic character. There is an integer D, such that D det r(Frob ) = ( ), p p
for any p - D. Goal To prove D = 1 (up to a square).
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant
Xiyuan Wang There is an integer D, such that D det r(Frob ) = ( ), p p
for any p - D. Goal To prove D = 1 (up to a square).
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant
Since r is self-dual, det r is a quadratic character.
Xiyuan Wang such that D det r(Frob ) = ( ), p p
for any p - D. Goal To prove D = 1 (up to a square).
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant
Since r is self-dual, det r is a quadratic character. There is an integer D,
Xiyuan Wang Goal To prove D = 1 (up to a square).
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant
Since r is self-dual, det r is a quadratic character. There is an integer D, such that D det r(Frob ) = ( ), p p
for any p - D.
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant
Since r is self-dual, det r is a quadratic character. There is an integer D, such that D det r(Frob ) = ( ), p p
for any p - D. Goal To prove D = 1 (up to a square).
Xiyuan Wang For g ∈ GQ,
det(λI − ρ(g)) = det(λI − ρ(g)∗).
Considering the eigenvalue {α, β, γ} of ρ(g), then
{α, β, γ} = {α−1, β−1, γ−1}.
• α = ±1, β = ±1, and γ = ±1. • αβ = 1(α 6= ±1), and γ = ±1.
Lemma If tr ρ(g 2) 6= 3 or tr ρ(g) 6= ±1, then det r(g) = 1.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant and trace
Xiyuan Wang Considering the eigenvalue {α, β, γ} of ρ(g), then
{α, β, γ} = {α−1, β−1, γ−1}.
• α = ±1, β = ±1, and γ = ±1. • αβ = 1(α 6= ±1), and γ = ±1.
Lemma If tr ρ(g 2) 6= 3 or tr ρ(g) 6= ±1, then det r(g) = 1.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant and trace
For g ∈ GQ,
det(λI − ρ(g)) = det(λI − ρ(g)∗).
Xiyuan Wang then
{α, β, γ} = {α−1, β−1, γ−1}.
• α = ±1, β = ±1, and γ = ±1. • αβ = 1(α 6= ±1), and γ = ±1.
Lemma If tr ρ(g 2) 6= 3 or tr ρ(g) 6= ±1, then det r(g) = 1.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant and trace
For g ∈ GQ,
det(λI − ρ(g)) = det(λI − ρ(g)∗).
Considering the eigenvalue {α, β, γ} of ρ(g),
Xiyuan Wang • α = ±1, β = ±1, and γ = ±1. • αβ = 1(α 6= ±1), and γ = ±1.
Lemma If tr ρ(g 2) 6= 3 or tr ρ(g) 6= ±1, then det r(g) = 1.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant and trace
For g ∈ GQ,
det(λI − ρ(g)) = det(λI − ρ(g)∗).
Considering the eigenvalue {α, β, γ} of ρ(g), then
{α, β, γ} = {α−1, β−1, γ−1}.
Xiyuan Wang • αβ = 1(α 6= ±1), and γ = ±1.
Lemma If tr ρ(g 2) 6= 3 or tr ρ(g) 6= ±1, then det r(g) = 1.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant and trace
For g ∈ GQ,
det(λI − ρ(g)) = det(λI − ρ(g)∗).
Considering the eigenvalue {α, β, γ} of ρ(g), then
{α, β, γ} = {α−1, β−1, γ−1}.
• α = ±1, β = ±1, and γ = ±1.
Xiyuan Wang Lemma If tr ρ(g 2) 6= 3 or tr ρ(g) 6= ±1, then det r(g) = 1.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant and trace
For g ∈ GQ,
det(λI − ρ(g)) = det(λI − ρ(g)∗).
Considering the eigenvalue {α, β, γ} of ρ(g), then
{α, β, γ} = {α−1, β−1, γ−1}.
• α = ±1, β = ±1, and γ = ±1. • αβ = 1(α 6= ±1), and γ = ±1.
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Determinant and trace
For g ∈ GQ,
det(λI − ρ(g)) = det(λI − ρ(g)∗).
Considering the eigenvalue {α, β, γ} of ρ(g), then
{α, β, γ} = {α−1, β−1, γ−1}.
• α = ±1, β = ±1, and γ = ±1. • αβ = 1(α 6= ±1), and γ = ±1.
Lemma If tr ρ(g 2) 6= 3 or tr ρ(g) 6= ±1, then det r(g) = 1.
Xiyuan Wang There is a trace formula
#S(Fpi ) − #X (Fpi ) tr ρe−1(Frobi ) = . ` p 2 The fibers are elliptic curves. Counting points fiber by fiber. • Need input from geometry. • Still very hard to obtain an explicit answer.
• A small observation: 8|#Smfiber(Fpi ) ! • Counting the point on bad fibers carefully.
Proposition 2(1+a) 2(1−a) 2 If p = p = −1, then tr(ρ(Frobp)) = −1 mod 8.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step i Trace of Frobp
Xiyuan Wang The fibers are elliptic curves. Counting points fiber by fiber. • Need input from geometry. • Still very hard to obtain an explicit answer.
• A small observation: 8|#Smfiber(Fpi ) ! • Counting the point on bad fibers carefully.
Proposition 2(1+a) 2(1−a) 2 If p = p = −1, then tr(ρ(Frobp)) = −1 mod 8.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step i Trace of Frobp
There is a trace formula
#S(Fpi ) − #X (Fpi ) tr ρe−1(Frobi ) = . ` p 2
Xiyuan Wang Counting points fiber by fiber. • Need input from geometry. • Still very hard to obtain an explicit answer.
• A small observation: 8|#Smfiber(Fpi ) ! • Counting the point on bad fibers carefully.
Proposition 2(1+a) 2(1−a) 2 If p = p = −1, then tr(ρ(Frobp)) = −1 mod 8.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step i Trace of Frobp
There is a trace formula
#S(Fpi ) − #X (Fpi ) tr ρe−1(Frobi ) = . ` p 2 The fibers are elliptic curves.
Xiyuan Wang • Need input from geometry. • Still very hard to obtain an explicit answer.
• A small observation: 8|#Smfiber(Fpi ) ! • Counting the point on bad fibers carefully.
Proposition 2(1+a) 2(1−a) 2 If p = p = −1, then tr(ρ(Frobp)) = −1 mod 8.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step i Trace of Frobp
There is a trace formula
#S(Fpi ) − #X (Fpi ) tr ρe−1(Frobi ) = . ` p 2 The fibers are elliptic curves. Counting points fiber by fiber.
Xiyuan Wang • Still very hard to obtain an explicit answer.
• A small observation: 8|#Smfiber(Fpi ) ! • Counting the point on bad fibers carefully.
Proposition 2(1+a) 2(1−a) 2 If p = p = −1, then tr(ρ(Frobp)) = −1 mod 8.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step i Trace of Frobp
There is a trace formula
#S(Fpi ) − #X (Fpi ) tr ρe−1(Frobi ) = . ` p 2 The fibers are elliptic curves. Counting points fiber by fiber. • Need input from geometry.
Xiyuan Wang • A small observation: 8|#Smfiber(Fpi ) ! • Counting the point on bad fibers carefully.
Proposition 2(1+a) 2(1−a) 2 If p = p = −1, then tr(ρ(Frobp)) = −1 mod 8.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step i Trace of Frobp
There is a trace formula
#S(Fpi ) − #X (Fpi ) tr ρe−1(Frobi ) = . ` p 2 The fibers are elliptic curves. Counting points fiber by fiber. • Need input from geometry. • Still very hard to obtain an explicit answer.
Xiyuan Wang • Counting the point on bad fibers carefully.
Proposition 2(1+a) 2(1−a) 2 If p = p = −1, then tr(ρ(Frobp)) = −1 mod 8.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step i Trace of Frobp
There is a trace formula
#S(Fpi ) − #X (Fpi ) tr ρe−1(Frobi ) = . ` p 2 The fibers are elliptic curves. Counting points fiber by fiber. • Need input from geometry. • Still very hard to obtain an explicit answer.
• A small observation: 8|#Smfiber(Fpi ) !
Xiyuan Wang Proposition 2(1+a) 2(1−a) 2 If p = p = −1, then tr(ρ(Frobp)) = −1 mod 8.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step i Trace of Frobp
There is a trace formula
#S(Fpi ) − #X (Fpi ) tr ρe−1(Frobi ) = . ` p 2 The fibers are elliptic curves. Counting points fiber by fiber. • Need input from geometry. • Still very hard to obtain an explicit answer.
• A small observation: 8|#Smfiber(Fpi ) ! • Counting the point on bad fibers carefully.
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step i Trace of Frobp
There is a trace formula
#S(Fpi ) − #X (Fpi ) tr ρe−1(Frobi ) = . ` p 2 The fibers are elliptic curves. Counting points fiber by fiber. • Need input from geometry. • Still very hard to obtain an explicit answer.
• A small observation: 8|#Smfiber(Fpi ) ! • Counting the point on bad fibers carefully.
Proposition 2(1+a) 2(1−a) 2 If p = p = −1, then tr(ρ(Frobp)) = −1 mod 8.
Xiyuan Wang Proposition + Lemma ⇒ det r(Frobp) = 1, for some primes p.
Assume D is not 1, in those primes, find a p such that
D det r(Frob ) = = −1. p p
This is a contradiction. We are done !
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step End of the proof
Xiyuan Wang Assume D is not 1, in those primes, find a p such that
D det r(Frob ) = = −1. p p
This is a contradiction. We are done !
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step End of the proof
Proposition + Lemma ⇒ det r(Frobp) = 1, for some primes p.
Xiyuan Wang This is a contradiction. We are done !
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step End of the proof
Proposition + Lemma ⇒ det r(Frobp) = 1, for some primes p.
Assume D is not 1, in those primes, find a p such that
D det r(Frob ) = = −1. p p
Xiyuan Wang We are done !
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step End of the proof
Proposition + Lemma ⇒ det r(Frobp) = 1, for some primes p.
Assume D is not 1, in those primes, find a p such that
D det r(Frob ) = = −1. p p
This is a contradiction.
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step End of the proof
Proposition + Lemma ⇒ det r(Frobp) = 1, for some primes p.
Assume D is not 1, in those primes, find a p such that
D det r(Frob ) = = −1. p p
This is a contradiction. We are done !
Xiyuan Wang • Checking that r satisfies the additional conditions in Calegari’s theorem is non-trivial. • An on-going project with Ariel and Lian about compatible systems of Galois representations and hypergeometric motives. • van Geemen and Top’s conjecture? Potential automorphy.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Final remarks
Xiyuan Wang • An on-going project with Ariel and Lian about compatible systems of Galois representations and hypergeometric motives. • van Geemen and Top’s conjecture? Potential automorphy.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Final remarks
• Checking that r satisfies the additional conditions in Calegari’s theorem is non-trivial.
Xiyuan Wang • van Geemen and Top’s conjecture? Potential automorphy.
Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Final remarks
• Checking that r satisfies the additional conditions in Calegari’s theorem is non-trivial. • An on-going project with Ariel and Lian about compatible systems of Galois representations and hypergeometric motives.
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step Final remarks
• Checking that r satisfies the additional conditions in Calegari’s theorem is non-trivial. • An on-going project with Ariel and Lian about compatible systems of Galois representations and hypergeometric motives. • van Geemen and Top’s conjecture? Potential automorphy.
Xiyuan Wang Two conjectures Main theorem A family of elliptic surfaces Idea of the proof The Tate conjecture for S Main step
Thank you for your listening!
Xiyuan Wang