Categoricity of Shimura Varieties Sebastian Eterović University of Oxford PLS12, June 2019 S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 1 / 10
The Language of Algebraic Varieties Let V ( C ) ⊆ C n be an irreducible algebraic variety defined over a countable field F 0 . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 2 / 10
The Language of Algebraic Varieties Let V ( C ) ⊆ C n be an irreducible algebraic variety defined over a countable field F 0 . Let F be an algebraically closed field of characteristic 0 with an embedding F 0 → F . This defines a structure in the language L F 0 = { + , · , F 0 } which is the extension of the language of rings by a set of constants for F 0 . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 2 / 10
The Language of Algebraic Varieties Let V ( C ) ⊆ C n be an irreducible algebraic variety defined over a countable field F 0 . Let F be an algebraically closed field of characteristic 0 with an embedding F 0 → F . This defines a structure in the language L F 0 = { + , · , F 0 } which is the extension of the language of rings by a set of constants for F 0 . The set V ( F ) (consisting of the F -points of V ) can be interpreted over the L F 0 -structure F . For every m ≥ 1, every subvariety of V ( F ) m definable over F 0 is definable in the language L F 0 . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 2 / 10
The Language of Algebraic Varieties Let V ( C ) ⊆ C n be an irreducible algebraic variety defined over a countable field F 0 . Let F be an algebraically closed field of characteristic 0 with an embedding F 0 → F . This defines a structure in the language L F 0 = { + , · , F 0 } which is the extension of the language of rings by a set of constants for F 0 . The set V ( F ) (consisting of the F -points of V ) can be interpreted over the L F 0 -structure F . For every m ≥ 1, every subvariety of V ( F ) m definable over F 0 is definable in the language L F 0 . Let T be the complete first-order theory of V in this language. As V ( F ) is bi-interpretable with F , then T has the same model-theoretic properties as ACF 0 , in particular, it is uncountably categorical. S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 2 / 10
Quotient Varieties Let U ⊆ C n be a complex domain, and suppose that there is an action of a group Γ on U such that V ( C ) = Γ \ U is an algebraic variety. S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 3 / 10
Quotient Varieties Let U ⊆ C n be a complex domain, and suppose that there is an action of a group Γ on U such that V ( C ) = Γ \ U is an algebraic variety. Example Let H ⊂ C denote the upper half-plane. The group SL 2 ( Z ) acts on H through Möbius transformations. The quotient SL 2 ( Z ) \ H is an algebraic variety isomorphic to C . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 3 / 10
Quotient Varieties Let U ⊆ C n be a complex domain, and suppose that there is an action of a group Γ on U such that V ( C ) = Γ \ U is an algebraic variety. Example Let H ⊂ C denote the upper half-plane. The group SL 2 ( Z ) acts on H through Möbius transformations. The quotient SL 2 ( Z ) \ H is an algebraic variety isomorphic to C . Quotient varieties can have more interesting structures than just being algebraic varieties. In order to witness this with model theory, it helps to expand the language and make it two-sorted: q : U → V ( C ) V has the language of algebraic varieties, U at least has the structure of a Γ -action, q is a function symbol invariant under Γ . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 3 / 10
Shimura Varieties Let X + denote a complex domain and suppose that there is an algebraic group G defined over Q such that G ( R ) + acts on X + through biholomorphisms. Under certain axioms on this data, one can find discrete subgroups Γ < G ( Q ) + such that Γ \ X + is a variety. S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 4 / 10
Shimura Varieties Let X + denote a complex domain and suppose that there is an algebraic group G defined over Q such that G ( R ) + acts on X + through biholomorphisms. Under certain axioms on this data, one can find discrete subgroups Γ < G ( Q ) + such that Γ \ X + is a variety. Choosing the axioms to be the axioms of Shimura data , the quotient S ( C ) := Γ \ X + is called a (connected) Shimura variety . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 4 / 10
Shimura Varieties Let X + denote a complex domain and suppose that there is an algebraic group G defined over Q such that G ( R ) + acts on X + through biholomorphisms. Under certain axioms on this data, one can find discrete subgroups Γ < G ( Q ) + such that Γ \ X + is a variety. Choosing the axioms to be the axioms of Shimura data , the quotient S ( C ) := Γ \ X + is called a (connected) Shimura variety . By some very deep theorems, Shimura varieties are canonically defined over a corresponding number field E = E ( G , X + ) called the reflex field . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 4 / 10
Special Subvarieties Suppose V ⊆ S is a subvariety such that one can find a subdomain V ⊆ X + and an algebraic subgroup H < G defined over Q , so that X + H , X + � � also satisfies the axioms of Shimura data. V S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 5 / 10
Special Subvarieties Suppose V ⊆ S is a subvariety such that one can find a subdomain V ⊆ X + and an algebraic subgroup H < G defined over Q , so that X + H , X + � � also satisfies the axioms of Shimura data. V Let Γ H = Γ ∩ H ( Q ) . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 5 / 10
Special Subvarieties Suppose V ⊆ S is a subvariety such that one can find a subdomain V ⊆ X + and an algebraic subgroup H < G defined over Q , so that X + H , X + � � also satisfies the axioms of Shimura data. V Let Γ H = Γ ∩ H ( Q ) . If V ( C ) = Γ H \ X + V , then we call V a special subvariety of S . We also call X + V a special domain for V . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 5 / 10
Special Subvarieties Suppose V ⊆ S is a subvariety such that one can find a subdomain V ⊆ X + and an algebraic subgroup H < G defined over Q , so that X + H , X + � � also satisfies the axioms of Shimura data. V Let Γ H = Γ ∩ H ( Q ) . If V ( C ) = Γ H \ X + V , then we call V a special subvariety of S . We also call X + V a special domain for V . A 0-dimensional special subvariety of S is called a special point . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 5 / 10
Some Important Special Subvarieties Let p : X + → S ( C ) be a Shimura variety. Choose g 1 , . . . , g n ∈ G ( Q ) + . Define: X + S ( C ) n p g : → x �→ ( p ( g 1 x ) , . . . , p ( g n x )) . The image of p g is a special subvariety of S n which we denote Z g . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 6 / 10
Shimura Structures Let p : X + → S ( C ) be a Shimura variety. We interpret this as a Shimura structure q : D → S ( F ) using the language L consisting of: S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 7 / 10
Shimura Structures Let p : X + → S ( C ) be a Shimura variety. We interpret this as a Shimura structure q : D → S ( F ) using the language L consisting of: 1 S ( F ) is interpreted as an algebraic variety over F 0 = E (Σ) , where Σ is the set of coordinates of all special points of S ( C ) . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 7 / 10
Shimura Structures Let p : X + → S ( C ) be a Shimura variety. We interpret this as a Shimura structure q : D → S ( F ) using the language L consisting of: 1 S ( F ) is interpreted as an algebraic variety over F 0 = E (Σ) , where Σ is the set of coordinates of all special points of S ( C ) . 2 D is a set with an action of G ( Q ) + and also predicates D V ⊆ D m (for all m ≥ 1) interpreted as the special domains of a special subvariety V . S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 7 / 10
Shimura Structures Let p : X + → S ( C ) be a Shimura variety. We interpret this as a Shimura structure q : D → S ( F ) using the language L consisting of: 1 S ( F ) is interpreted as an algebraic variety over F 0 = E (Σ) , where Σ is the set of coordinates of all special points of S ( C ) . 2 D is a set with an action of G ( Q ) + and also predicates D V ⊆ D m (for all m ≥ 1) interpreted as the special domains of a special subvariety V . 3 q is a function. S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 7 / 10
Shimura Structures Let p : X + → S ( C ) be a Shimura variety. We interpret this as a Shimura structure q : D → S ( F ) using the language L consisting of: 1 S ( F ) is interpreted as an algebraic variety over F 0 = E (Σ) , where Σ is the set of coordinates of all special points of S ( C ) . 2 D is a set with an action of G ( Q ) + and also predicates D V ⊆ D m (for all m ≥ 1) interpreted as the special domains of a special subvariety V . 3 q is a function. Let Th ( p ) be the complete first-order theory of p : X + → S ( C ) in this language. S. Eterović (Oxford) Categoricity of Shimura Varieties June 2019 7 / 10
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