Sato–Tate groups of abelian threefolds Francesc Fit´ e (IAS), Kiran S. Kedlaya (UCSD), A.V. Sutherland (MIT) Arithmetic of low-dimensional abelian varieties. ICERM, 5th June 2019. Fit´ e, Kedlaya, Sutherland 1 / 22
Sato–Tate groups of elliptic curves k a number field. E / k an elliptic curve. The Sato–Tate group ST( E ) is defined as: ◮ SU(2) if E does not have CM. �� u � � 0 ◮ U(1) = : u ∈ C , | u | = 1 if E has CM by M ⊆ k . 0 u ◮ N SU(2) (U(1)) if E has CM by M �⊆ k . Note that Tr: ST( E ) → [ − 2 , 2]. Denote µ = Tr ∗ ( µ Haar ). SU(2) U(1) N (U(1)) Fit´ e, Kedlaya, Sutherland 2 / 22
The Sato–Tate conjecture for elliptic curves Let p be a prime of good reduction for E . The normalized Frobenius trace satisfies a p = N ( p ) + 1 − # E ( F p ) = Tr(Frob p | V ℓ ( E )) ∈ [ − 2 , 2] (for p ∤ ℓ ) � � N ( p ) N ( p ) Sato–Tate conjecture The sequence { a p } p is equidistributed on [ − 2 , 2] w.r.t µ . If ST( E ) = U(1) or N (U(1)): Known in full generality (Hecke, Deuring). Known if ST( E ) = SU(2) and k is totally real. (Barnet-Lamb, Geraghty, Harris, Shepherd-Barron, Taylor); Known if ST( E ) = SU(2) and k is a CM field (Allen,Calegary,Caraiani,Gee,Helm,LeHung,Newton,Scholze,Taylor,Thorne). Fit´ e, Kedlaya, Sutherland 3 / 22
The Sato–Tate conjecture for elliptic curves Let p be a prime of good reduction for E . The normalized Frobenius trace satisfies a p = N ( p ) + 1 − # E ( F p ) = Tr(Frob p | V ℓ ( E )) ∈ [ − 2 , 2] (for p ∤ ℓ ) � � N ( p ) N ( p ) Sato–Tate conjecture The sequence { a p } p is equidistributed on [ − 2 , 2] w.r.t µ . If ST( E ) = U(1) or N (U(1)): Known in full generality (Hecke, Deuring). Known if ST( E ) = SU(2) and k is totally real. (Barnet-Lamb, Geraghty, Harris, Shepherd-Barron, Taylor); Known if ST( E ) = SU(2) and k is a CM field (Allen,Calegary,Caraiani,Gee,Helm,LeHung,Newton,Scholze,Taylor,Thorne). Fit´ e, Kedlaya, Sutherland 3 / 22
The Sato–Tate conjecture for elliptic curves Let p be a prime of good reduction for E . The normalized Frobenius trace satisfies a p = N ( p ) + 1 − # E ( F p ) = Tr(Frob p | V ℓ ( E )) ∈ [ − 2 , 2] (for p ∤ ℓ ) � � N ( p ) N ( p ) Sato–Tate conjecture The sequence { a p } p is equidistributed on [ − 2 , 2] w.r.t µ . If ST( E ) = U(1) or N (U(1)): Known in full generality (Hecke, Deuring). Known if ST( E ) = SU(2) and k is totally real. (Barnet-Lamb, Geraghty, Harris, Shepherd-Barron, Taylor); Known if ST( E ) = SU(2) and k is a CM field (Allen,Calegary,Caraiani,Gee,Helm,LeHung,Newton,Scholze,Taylor,Thorne). Fit´ e, Kedlaya, Sutherland 3 / 22
Toward the Sato–Tate group: the ℓ -adic image Let A / k be an abelian variety of dimension g ≥ 1. Consider the ℓ -adic representation attached to A ̺ A ,ℓ : G k → Aut ψ ℓ ( V ℓ ( A )) ≃ GSp 2 g ( Q ℓ ) . Serre defines ST( A ) in terms of G ℓ = ̺ A ,ℓ ( G k ) Zar . For g ≤ 3, Banaszak and Kedlaya describe ST( A ) in terms of endomorphisms . By Faltings, there is a G k -equivariant isomorphism ℓ ( Q 2 g End( A Q ) ⊗ Q ℓ ≃ End G 0 ℓ ) . Therefore → { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = α for all α ∈ End( A Q ) } . G 0 ℓ ֒ Fit´ e, Kedlaya, Sutherland 4 / 22
Toward the Sato–Tate group: the ℓ -adic image Let A / k be an abelian variety of dimension g ≥ 1. Consider the ℓ -adic representation attached to A ̺ A ,ℓ : G k → Aut ψ ℓ ( V ℓ ( A )) ≃ GSp 2 g ( Q ℓ ) . Serre defines ST( A ) in terms of G ℓ = ̺ A ,ℓ ( G k ) Zar . For g ≤ 3, Banaszak and Kedlaya describe ST( A ) in terms of endomorphisms . By Faltings, there is a G k -equivariant isomorphism ℓ ( Q 2 g End( A Q ) ⊗ Q ℓ ≃ End G 0 ℓ ) . Therefore → { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = α for all α ∈ End( A Q ) } . G 0 ℓ ֒ Fit´ e, Kedlaya, Sutherland 4 / 22
Toward the Sato–Tate group: the ℓ -adic image Let A / k be an abelian variety of dimension g ≥ 1. Consider the ℓ -adic representation attached to A ̺ A ,ℓ : G k → Aut ψ ℓ ( V ℓ ( A )) ≃ GSp 2 g ( Q ℓ ) . Serre defines ST( A ) in terms of G ℓ = ̺ A ,ℓ ( G k ) Zar . For g ≤ 3, Banaszak and Kedlaya describe ST( A ) in terms of endomorphisms . By Faltings, there is a G k -equivariant isomorphism ℓ ( Q 2 g End( A Q ) ⊗ Q ℓ ≃ End G 0 ℓ ) . Therefore → { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = α for all α ∈ End( A Q ) } . G 0 ℓ ֒ Fit´ e, Kedlaya, Sutherland 4 / 22
Toward the Sato–Tate group: the ℓ -adic image Let A / k be an abelian variety of dimension g ≥ 1. Consider the ℓ -adic representation attached to A ̺ A ,ℓ : G k → Aut ψ ℓ ( V ℓ ( A )) ≃ GSp 2 g ( Q ℓ ) . Serre defines ST( A ) in terms of G ℓ = ̺ A ,ℓ ( G k ) Zar . For g ≤ 3, Banaszak and Kedlaya describe ST( A ) in terms of endomorphisms . By Faltings, there is a G k -equivariant isomorphism ℓ ( Q 2 g End( A Q ) ⊗ Q ℓ ≃ End G 0 ℓ ) . Therefore → { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = α for all α ∈ End( A Q ) } . G 0 ℓ ֒ Fit´ e, Kedlaya, Sutherland 4 / 22
Toward the Sato–Tate group: the ℓ -adic image Let A / k be an abelian variety of dimension g ≥ 1. Consider the ℓ -adic representation attached to A ̺ A ,ℓ : G k → Aut ψ ℓ ( V ℓ ( A )) ≃ GSp 2 g ( Q ℓ ) . Serre defines ST( A ) in terms of G ℓ = ̺ A ,ℓ ( G k ) Zar . For g ≤ 3, Banaszak and Kedlaya describe ST( A ) in terms of endomorphisms . By Faltings, there is a G k -equivariant isomorphism ℓ ( Q 2 g End( A Q ) ⊗ Q ℓ ≃ End G 0 ℓ ) . Therefore → { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = α for all α ∈ End( A Q ) } . G 0 ℓ ֒ Fit´ e, Kedlaya, Sutherland 4 / 22
The twisted Lefschetz group More accurately { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = σ ( α ) for all α ∈ End( A Q ) } . � G ℓ ֒ → σ ∈ G k For g = 4, Mumford has constructed A / k such that End( A Q ) ≃ Z and G ℓ � GSp 2 g ( Q ℓ ) . For g ≤ 3, one has { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = σ ( α ) for all α ∈ End( A Q ) } . � G ℓ ≃ σ ∈ G k Definition The Twisted Lefschetz group is defined as { γ ∈ Sp 2 g / Q | γαγ − 1 = σ ( α ) for all α ∈ End( A F ) } . � TL( A ) = σ ∈ G k Fit´ e, Kedlaya, Sutherland 5 / 22
The twisted Lefschetz group More accurately { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = σ ( α ) for all α ∈ End( A Q ) } . � G ℓ ֒ → σ ∈ G k For g = 4, Mumford has constructed A / k such that End( A Q ) ≃ Z and G ℓ � GSp 2 g ( Q ℓ ) . For g ≤ 3, one has { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = σ ( α ) for all α ∈ End( A Q ) } . � G ℓ ≃ σ ∈ G k Definition The Twisted Lefschetz group is defined as { γ ∈ Sp 2 g / Q | γαγ − 1 = σ ( α ) for all α ∈ End( A F ) } . � TL( A ) = σ ∈ G k Fit´ e, Kedlaya, Sutherland 5 / 22
The twisted Lefschetz group More accurately { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = σ ( α ) for all α ∈ End( A Q ) } . � G ℓ ֒ → σ ∈ G k For g = 4, Mumford has constructed A / k such that End( A Q ) ≃ Z and G ℓ � GSp 2 g ( Q ℓ ) . For g ≤ 3, one has { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = σ ( α ) for all α ∈ End( A Q ) } . � G ℓ ≃ σ ∈ G k Definition The Twisted Lefschetz group is defined as { γ ∈ Sp 2 g / Q | γαγ − 1 = σ ( α ) for all α ∈ End( A F ) } . � TL( A ) = σ ∈ G k Fit´ e, Kedlaya, Sutherland 5 / 22
The twisted Lefschetz group More accurately { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = σ ( α ) for all α ∈ End( A Q ) } . � G ℓ ֒ → σ ∈ G k For g = 4, Mumford has constructed A / k such that End( A Q ) ≃ Z and G ℓ � GSp 2 g ( Q ℓ ) . For g ≤ 3, one has { γ ∈ GSp 2 g ( Q ℓ ) | γαγ − 1 = σ ( α ) for all α ∈ End( A Q ) } . � G ℓ ≃ σ ∈ G k Definition The Twisted Lefschetz group is defined as { γ ∈ Sp 2 g / Q | γαγ − 1 = σ ( α ) for all α ∈ End( A F ) } . � TL( A ) = σ ∈ G k Fit´ e, Kedlaya, Sutherland 5 / 22
The Sato–Tate group when g ≤ 3 From now on, assume g ≤ 3. Definition ST( A ) ⊆ USp(2 g ) is a maximal compact subgroup of TL( A ) ⊗ Q C . Note that ST( A ) / ST( A ) 0 ≃ TL( A ) / TL( A ) 0 ≃ Gal( F / k ) . where F / k is the minimal extension such that End( A F ) ≃ End( A Q ). We call F the endomorphism field of A . To each prime p of good reduction for A , one can attach a conjugacy � ̺ A ,ℓ (Frob p ) � class x p ∈ X = Conj (ST( A )) s.t. Char ( x p ) = Char . √ N p Sato–Tate conjecture for abelian varieties The sequence { x p } p is equidistributed on X w.r.t the push forward of the Haar measure of ST( A ). Fit´ e, Kedlaya, Sutherland 6 / 22
The Sato–Tate group when g ≤ 3 From now on, assume g ≤ 3. Definition ST( A ) ⊆ USp(2 g ) is a maximal compact subgroup of TL( A ) ⊗ Q C . Note that ST( A ) / ST( A ) 0 ≃ TL( A ) / TL( A ) 0 ≃ Gal( F / k ) . where F / k is the minimal extension such that End( A F ) ≃ End( A Q ). We call F the endomorphism field of A . To each prime p of good reduction for A , one can attach a conjugacy � ̺ A ,ℓ (Frob p ) � class x p ∈ X = Conj (ST( A )) s.t. Char ( x p ) = Char . √ N p Sato–Tate conjecture for abelian varieties The sequence { x p } p is equidistributed on X w.r.t the push forward of the Haar measure of ST( A ). Fit´ e, Kedlaya, Sutherland 6 / 22
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