Additive groups
Thomas Scanlon Additive groups Introduction
Additive groups in difference and differential Thomas Scanlon fields Drinfeld University of California, Berkeley modules Isaac Newton Institute for Mathematical Sciences Cambridge, England 15 July 2005
Thomas Scanlon University of California, Berkeley Additive groups Thanks to the conference organizers
Additive groups
Thomas Scanlon
Introduction
Additive groups in difference and differential fields
Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Thanks to the program organizers
Additive groups
Thomas Scanlon
Introduction
Additive groups in difference and differential fields
Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Adding is easy
Additive groups “If you can add, you can integrate.” Thomas Scanlon -Paul Sally Introduction
Additive groups in difference and differential fields
Drinfeld modules
Z pn−1 −n X fdµHaar = p f (i) Zp i=0 n if f is constant on cosets of p Zp
Thomas Scanlon University of California, Berkeley Additive groups Adding is easy
Additive groups
Thomas Scanlon
Introduction
Additive groups in difference and differential you can add fields
Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon
Introduction
Additive groups in difference and differential fields
Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon
Introduction
Additive groups in $ 2 . 1 9 difference and differential + $ 1 . 8 3 fields
Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon
Introduction
Additive groups in $ 2 . 1 9 difference and differential + $ 1 . 8 3 fields
Drinfeld modules 2
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon
Introduction 1 Additive groups in $ 2 . 1 9 difference and differential + $ 1 . 8 3 fields
Drinfeld modules 2
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon
Introduction 1 Additive groups in $ 2 . 1 9 difference and differential + $ 1 . 8 3 fields
Drinfeld modules 0 2
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon
Introduction 1 Additive groups in $ 2 . 1 9 difference and differential + $ 1 . 8 3 fields
Drinfeld modules . 0 2
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon
Introduction 1 1 Additive groups in $ 2 . 1 9 difference and differential + $ 1 . 8 3 fields
Drinfeld modules . 0 2
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon
Introduction 1 1 Additive groups in $ 2 . 1 9 difference and differential + $ 1 . 8 3 fields
Drinfeld modules 4 . 0 2
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon
Introduction 1 1 Additive groups in $ 2 . 1 9 difference and differential + $ 1 . 8 3 fields
Drinfeld modules $ 4 . 0 2
Thomas Scanlon University of California, Berkeley Additive groups Is adding easy?
Additive groups
Thomas Scanlon 1 1 Introduction $ 2 . 1 9 Additive + $ 1 . 8 3 groups in difference and differential fields $ 4 . 0 2 Drinfeld modules
0 −−−−→ Z/10Z −−−−→ Z/100Z −−−−→ Z/10Z −−−−→ 0
is nonsplit
Thomas Scanlon University of California, Berkeley Additive groups What else makes adding difficult?
Additive groups
Thomas Scanlon If K is an infinite field, then the multiplicative group of K gives an infinite definable family of additive Introduction automorphisms. Additive groups in difference and The field K is a vector space over its prime field. So, differential fields abstractly we have a complete classification of the additive
Drinfeld subgroups, but there can be many of them and they can modules have a complicated interaction with the multiplicative structure. If K has charactertistic p > 0, then there are additional definable endomorphisms coming from the Frobenius.
Thomas Scanlon University of California, Berkeley Additive groups What else makes adding difficult?
Additive groups
Thomas Scanlon If K is an infinite field, then the multiplicative group of K gives an infinite definable family of additive Introduction automorphisms. Additive groups in difference and The field K is a vector space over its prime field. So, differential fields abstractly we have a complete classification of the additive
Drinfeld subgroups, but there can be many of them and they can modules have a complicated interaction with the multiplicative structure. If K has charactertistic p > 0, then there are additional definable endomorphisms coming from the Frobenius.
Thomas Scanlon University of California, Berkeley Additive groups What else makes adding difficult?
Additive groups
Thomas Scanlon If K is an infinite field, then the multiplicative group of K gives an infinite definable family of additive Introduction automorphisms. Additive groups in difference and The field K is a vector space over its prime field. So, differential fields abstractly we have a complete classification of the additive
Drinfeld subgroups, but there can be many of them and they can modules have a complicated interaction with the multiplicative structure. If K has charactertistic p > 0, then there are additional definable endomorphisms coming from the Frobenius.
Thomas Scanlon University of California, Berkeley Additive groups What else makes adding difficult?
Additive groups
Thomas Scanlon If K is an infinite field, then the multiplicative group of K gives an infinite definable family of additive Introduction automorphisms. Additive groups in difference and The field K is a vector space over its prime field. So, differential fields abstractly we have a complete classification of the additive
Drinfeld subgroups, but there can be many of them and they can modules have a complicated interaction with the multiplicative structure. If K has charactertistic p > 0, then there are additional definable endomorphisms coming from the Frobenius.
Thomas Scanlon University of California, Berkeley Additive groups Theories of differential and difference fields
Additive groups
Thomas Scanlon
Introduction Definition Additive groups in A difference field is a field K given together with a difference and differential distinguished field automorphism σ : K → K.A differential fields field is a field K given together with a derivation ∂ : K → K, a Drinfeld modules map satisfying ∂(x + y) = ∂(x) + ∂(y) and ∂(xy) = x∂(y) + y∂(x) universally
Thomas Scanlon University of California, Berkeley Additive groups Theories of differential and difference fields
Additive groups
Thomas Definition Scanlon A difference field is a field K given together with a Introduction distinguished field automorphism σ : K → K.A differential Additive groups in field is a field K given together with a derivation ∂ : K → K, a difference and differential map satisfying ∂(x + y) = ∂(x) + ∂(y) and fields ∂(xy) = x∂(y) + y∂(x) universally Drinfeld modules Theorem (A. Robinson; Macintyre-van den Dries-Wood) The theory of differential fields of characteristic zero has a model completion, DCF0, and the theory of difference fields has a model companion ACFA.
Thomas Scanlon University of California, Berkeley Additive groups Additive groups in DCF0
Additive groups Proposition Thomas If (K, ∂) is a differentially closed field of characteristic zero and Scanlon G ≤ (K, +) is a definable subgroup of the additive group, then Introduction n G is definably isomorphic to (K, +) or to (CK , +) where Additive groups in CK := {x ∈ K | ∂(x) = 0} is the field of constants for some difference and differential natural number n. fields Drinfeld The stabilizer L := {λ ∈ K | λG ≤ G} is a definable modules subring of K. By quantifier elimination, L is either CK or K. By a rank calculation, dimL G is finite. If L = K, then G = K. If L = CK and b1,..., bn ∈ G is a P basis, then the map (λ1, . . . , λn) 7→ λi bi is an n isomorphism between CK and G.
Thomas Scanlon University of California, Berkeley Additive groups Additive groups in DCF0
Additive groups Proposition Thomas If (K, ∂) is a differentially closed field of characteristic zero and Scanlon G ≤ (K, +) is a definable subgroup of the additive group, then Introduction n G is definably isomorphic to (K, +) or to (CK , +) where Additive groups in CK := {x ∈ K | ∂(x) = 0} is the field of constants for some difference and differential natural number n. fields
Drinfeld Proof sketch: modules The stabilizer L := {λ ∈ K | λG ≤ G} is a definable subring of K. By quantifier elimination, L is either CK or K. By a rank calculation, dimL G is finite. If L = K, then G = K. If L = CK and b1,..., bn ∈ G is a P basis, then the map (λ1, . . . , λn) 7→ λi bi is an n isomorphism between CK and G. Thomas Scanlon University of California, Berkeley Additive groups Additive groups in DCF0
Additive groups Proposition Thomas If (K, ∂) is a differentially closed field of characteristic zero and Scanlon G ≤ (K, +) is a definable subgroup of the additive group, then Introduction n G is definably isomorphic to (K, +) or to (CK , +) where Additive groups in CK := {x ∈ K | ∂(x) = 0} is the field of constants for some difference and differential natural number n. fields
Drinfeld Proof sketch: modules The stabilizer L := {λ ∈ K | λG ≤ G} is a definable subring of K. By quantifier elimination, L is either CK or K. By a rank calculation, dimL G is finite. If L = K, then G = K. If L = CK and b1,..., bn ∈ G is a P basis, then the map (λ1, . . . , λn) 7→ λi bi is an n isomorphism between CK and G. Thomas Scanlon University of California, Berkeley Additive groups Additive groups in DCF0
Additive groups Proposition Thomas If (K, ∂) is a differentially closed field of characteristic zero and Scanlon G ≤ (K, +) is a definable subgroup of the additive group, then Introduction n G is definably isomorphic to (K, +) or to (CK , +) where Additive groups in CK := {x ∈ K | ∂(x) = 0} is the field of constants for some difference and differential natural number n. fields
Drinfeld Proof sketch: modules The stabilizer L := {λ ∈ K | λG ≤ G} is a definable subring of K. By quantifier elimination, L is either CK or K. By a rank calculation, dimL G is finite. If L = K, then G = K. If L = CK and b1,..., bn ∈ G is a P basis, then the map (λ1, . . . , λn) 7→ λi bi is an n isomorphism between CK and G. Thomas Scanlon University of California, Berkeley Additive groups Additive groups in DCF0
Additive groups Proposition Thomas If (K, ∂) is a differentially closed field of characteristic zero and Scanlon G ≤ (K, +) is a definable subgroup of the additive group, then Introduction n G is definably isomorphic to (K, +) or to (CK , +) where Additive groups in CK := {x ∈ K | ∂(x) = 0} is the field of constants for some difference and differential natural number n. fields
Drinfeld Proof sketch: modules The stabilizer L := {λ ∈ K | λG ≤ G} is a definable subring of K. By quantifier elimination, L is either CK or K. By a rank calculation, dimL G is finite. If L = K, then G = K. If L = CK and b1,..., bn ∈ G is a P basis, then the map (λ1, . . . , λn) 7→ λi bi is an n isomorphism between CK and G. Thomas Scanlon University of California, Berkeley Additive groups Additive groups in DCF0
Additive groups Proposition Thomas If (K, ∂) is a differentially closed field of characteristic zero and Scanlon G ≤ (K, +) is a definable subgroup of the additive group, then Introduction n G is definably isomorphic to (K, +) or to (CK , +) where Additive groups in CK := {x ∈ K | ∂(x) = 0} is the field of constants for some difference and differential natural number n. fields
Drinfeld Proof sketch: modules The stabilizer L := {λ ∈ K | λG ≤ G} is a definable subring of K. By quantifier elimination, L is either CK or K. By a rank calculation, dimL G is finite. If L = K, then G = K. If L = CK and b1,..., bn ∈ G is a P basis, then the map (λ1, . . . , λn) 7→ λi bi is an n isomorphism between CK and G. Thomas Scanlon University of California, Berkeley Additive groups “Interesting” differential algebraic groups
Additive groups Let (K, ∂) |= DCF0 be a differentially closed field of
Thomas characteristic zero. Scanlon If G is a commutative algebraic group of dimension d over Introduction CK , then there is a definable surjective homomorphism Additive d groups in ∂ logG : G(K) → (K , +), the logarithmic derivative for difference and d differential G. So, for any definable vector group V ≤ K , there is a fields group Ve := ∂ log−1 V ≤ G(K) fitting into an exact Drinfeld modules sequence 0 −−−−→ G(CK ) −−−−→ Ve −−−−→ V −−−−→ 0. If A is an abelian variety of dimension d defined over K, then there is a surjective definable homomorphism µ : A(K) → (K d , +). If A is sufficiently general, the kernel A] has a very simple induced structure.
Thomas Scanlon University of California, Berkeley Additive groups “Interesting” differential algebraic groups
Additive groups Let (K, ∂) |= DCF0 be a differentially closed field of
Thomas characteristic zero. Scanlon If G is a commutative algebraic group of dimension d over Introduction CK , then there is a definable surjective homomorphism Additive d groups in ∂ logG : G(K) → (K , +), the logarithmic derivative for difference and d differential G. So, for any definable vector group V ≤ K , there is a fields group Ve := ∂ log−1 V ≤ G(K) fitting into an exact Drinfeld modules sequence 0 −−−−→ G(CK ) −−−−→ Ve −−−−→ V −−−−→ 0. If A is an abelian variety of dimension d defined over K, then there is a surjective definable homomorphism µ : A(K) → (K d , +). If A is sufficiently general, the kernel A] has a very simple induced structure.
Thomas Scanlon University of California, Berkeley Additive groups “Interesting” differential algebraic groups
Additive groups Let (K, ∂) |= DCF0 be a differentially closed field of
Thomas characteristic zero. Scanlon If G is a commutative algebraic group of dimension d over Introduction CK , then there is a definable surjective homomorphism Additive d groups in ∂ logG : G(K) → (K , +), the logarithmic derivative for difference and d differential G. So, for any definable vector group V ≤ K , there is a fields group Ve := ∂ log−1 V ≤ G(K) fitting into an exact Drinfeld modules sequence 0 −−−−→ G(CK ) −−−−→ Ve −−−−→ V −−−−→ 0. If A is an abelian variety of dimension d defined over K, then there is a surjective definable homomorphism µ : A(K) → (K d , +). If A is sufficiently general, the kernel A] has a very simple induced structure.
Thomas Scanlon University of California, Berkeley Additive groups Partial differential additive groups
Additive groups
Thomas Scanlon If (K, ∂1, . . . , ∂n) is a differentially closed field of characteristic Introduction zero in n > 1 commuting derivations, then there are many Additive groups in definable subgroups of the additive group. difference and differential fields For any sequence λ1, . . . , λn from K, the set Pn Drinfeld {x ∈ K | i=1 λi ∂i (x) = 0} is a definable subfield. modules In fact, the zero set of any finite system of linear differential operators is a definable additive group.
Thomas Scanlon University of California, Berkeley Additive groups Partial differential additive groups
Additive groups
Thomas Scanlon If (K, ∂1, . . . , ∂n) is a differentially closed field of characteristic Introduction zero in n > 1 commuting derivations, then there are many Additive groups in definable subgroups of the additive group. difference and differential fields For any sequence λ1, . . . , λn from K, the set Pn Drinfeld {x ∈ K | i=1 λi ∂i (x) = 0} is a definable subfield. modules In fact, the zero set of any finite system of linear differential operators is a definable additive group.
Thomas Scanlon University of California, Berkeley Additive groups Partial differential additive groups
Additive groups
Thomas Scanlon If (K, ∂1, . . . , ∂n) is a differentially closed field of characteristic Introduction zero in n > 1 commuting derivations, then there are many Additive groups in definable subgroups of the additive group. difference and differential fields For any sequence λ1, . . . , λn from K, the set Pn Drinfeld {x ∈ K | i=1 λi ∂i (x) = 0} is a definable subfield. modules In fact, the zero set of any finite system of linear differential operators is a definable additive group.
Thomas Scanlon University of California, Berkeley Additive groups Partial differential additive groups
Additive groups If (K, ∂1, . . . , ∂n) is a differentially closed field of characteristic Thomas zero in n > 1 commuting derivations, then there are many Scanlon definable subgroups of the additive group. Introduction
Additive For any sequence λ1, . . . , λn from K, the set groups in Pn difference and {x ∈ K | i=1 λi ∂i (x) = 0} is a definable subfield. differential fields In fact, the zero set of any finite system of linear Drinfeld differential operators is a definable additive group. modules Conjecture If G ≤ (K, +) is a definable subgroup of the additive group with a regular generic type g, then either g is locally modular or G is definably isomorphic to a definable field.
Thomas Scanlon University of California, Berkeley Additive groups Positive characteristic additive groups
Additive groups The description of the definable groups in ACFA0, the theory Thomas of difference closed fields of characteristic zero, is similar to Scanlon that for DCF0. Introduction
Additive If (K, σ) |= ACFAp is a difference closed field of groups in difference and characteristic p and τ : K → K is the Frobenius map differential p fields x 7→ x , then for any pair of integers (n, m) the field n m n m Drinfeld Fix(σ τ ) := {x ∈ K | σ τ (x) = x} is definable. modules It can happen that a definable additive group is not a finite dimensional vector space over any definable field. [This is the case with G := {x ∈ K | σ(x) = xp + x}.] Chatzidakis has constructed examples of one-based non-stably embedded additive groups.
Thomas Scanlon University of California, Berkeley Additive groups Positive characteristic additive groups
Additive groups The description of the definable groups in ACFA0, the theory
Thomas of difference closed fields of characteristic zero, is similar to Scanlon that for DCF0, but there are more complicated definable Introduction additive groups in ACFAp with p > 0. Additive groups in If (K, σ) |= ACFAp is a difference closed field of difference and differential characteristic p and τ : K → K is the Frobenius map fields x 7→ xp, then for any pair of integers (n, m) the field Drinfeld n m n m modules Fix(σ τ ) := {x ∈ K | σ τ (x) = x} is definable. It can happen that a definable additive group is not a finite dimensional vector space over any definable field. [This is the case with G := {x ∈ K | σ(x) = xp + x}.] Chatzidakis has constructed examples of one-based non-stably embedded additive groups.
Thomas Scanlon University of California, Berkeley Additive groups Positive characteristic additive groups
Additive groups The description of the definable groups in ACFA0, the theory
Thomas of difference closed fields of characteristic zero, is similar to Scanlon that for DCF0, but there are more complicated definable Introduction additive groups in ACFAp with p > 0. Additive groups in If (K, σ) |= ACFAp is a difference closed field of difference and differential characteristic p and τ : K → K is the Frobenius map fields x 7→ xp, then for any pair of integers (n, m) the field Drinfeld n m n m modules Fix(σ τ ) := {x ∈ K | σ τ (x) = x} is definable. It can happen that a definable additive group is not a finite dimensional vector space over any definable field. [This is the case with G := {x ∈ K | σ(x) = xp + x}.] Chatzidakis has constructed examples of one-based non-stably embedded additive groups.
Thomas Scanlon University of California, Berkeley Additive groups Positive characteristic additive groups
Additive groups The description of the definable groups in ACFA0, the theory
Thomas of difference closed fields of characteristic zero, is similar to Scanlon that for DCF0, but there are more complicated definable Introduction additive groups in ACFAp with p > 0. Additive groups in If (K, σ) |= ACFAp is a difference closed field of difference and differential characteristic p and τ : K → K is the Frobenius map fields x 7→ xp, then for any pair of integers (n, m) the field Drinfeld n m n m modules Fix(σ τ ) := {x ∈ K | σ τ (x) = x} is definable. It can happen that a definable additive group is not a finite dimensional vector space over any definable field. [This is the case with G := {x ∈ K | σ(x) = xp + x}.] Chatzidakis has constructed examples of one-based non-stably embedded additive groups.
Thomas Scanlon University of California, Berkeley Additive groups Positive characteristic additive groups
Additive groups The description of the definable groups in ACFA0, the theory
Thomas of difference closed fields of characteristic zero, is similar to Scanlon that for DCF0, but there are more complicated definable Introduction additive groups in ACFAp with p > 0. Additive groups in If (K, σ) |= ACFAp is a difference closed field of difference and differential characteristic p and τ : K → K is the Frobenius map fields x 7→ xp, then for any pair of integers (n, m) the field Drinfeld n m n m modules Fix(σ τ ) := {x ∈ K | σ τ (x) = x} is definable. It can happen that a definable additive group is not a finite dimensional vector space over any definable field. [This is the case with G := {x ∈ K | σ(x) = xp + x}.] Chatzidakis has constructed examples of one-based non-stably embedded additive groups.
Thomas Scanlon University of California, Berkeley Additive groups Separably closed fields
Additive groups
Thomas Scanlon
Introduction
Additive groups in Definition difference and differential A field K is separably closed if for any polynomial f ∈ K[x] fields with f 0(x) 6≡ 0 there is a root a ∈ K to f (a) = 0. Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Separably closed fields
Additive groups
Thomas Scanlon Definition Introduction A field K is separably closed if for any polynomial f ∈ K[x] Additive 0 groups in with f (x) 6≡ 0 there is a root a ∈ K to f (a) = 0. difference and differential fields
Drinfeld Theorem modules The theory SCFp,e of separably closed fields of characteristic p p and Erˇsovinvariant e := logp[K : K ] ∈ N ∪ {∞} is complete and stable.
Thomas Scanlon University of California, Berkeley Additive groups Separably closed fields
Additive groups
Thomas Scanlon Definition A field K is separably closed if for any polynomial f ∈ K[x] Introduction 0 Additive with f (x) 6≡ 0 there is a root a ∈ K to f (a) = 0. groups in difference and differential fields Theorem Drinfeld The theory SCFp,e of separably closed fields of characteristic p modules p and Erˇsovinvariant e := logp[K : K ] ∈ N ∪ {∞} is complete and stable.
SCFp,e (at least for 0 < e < ∞) is the positive characteristic analogue of DCF0
Thomas Scanlon University of California, Berkeley Additive groups Additive groups in SCFp,e
Additive groups
Thomas Scanlon • The definable groups in a separably closed field K of Erˇsov
Introduction invariant 1 ≤ e < ∞ are definably isomorphic to algebraic
Additive groups whilst the “interesting” groups are type-definable. groups in difference and differential fields
Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Additive groups in SCFp,e
Additive groups • The definable groups in a separably closed field K of Erˇsov Thomas Scanlon invariant 1 ≤ e < ∞ are definably isomorphic to algebraic groups whilst the “interesting” groups are type-definable. Introduction
Additive ∞ ∞ n groups in • The groups p A(K) := T p A(K) where A is an abelian difference and n=1 differential variety play an important rˆolein Hrushovski’s proof of the fields positive characterisitc Mordell-Lang theorem. Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Additive groups in SCFp,e
Additive • The definable groups in a separably closed field K of Erˇsov groups invariant 1 ≤ e < ∞ are definably isomorphic to algebraic Thomas Scanlon groups whilst the “interesting” groups are type-definable.
Introduction ∞ T∞ n Additive • The groups p A(K) := n=1 p A(K) where A is an abelian groups in difference and variety play an important rˆolein Hrushovski’s proof of the differential fields positive characterisitc Mordell-Lang theorem.
Drinfeld modules • Bouscaren and Delon completely classified the type definable groups having no additive part.
Thomas Scanlon University of California, Berkeley Additive groups Additive groups in SCFp,e
Additive • The definable groups in a separably closed field K of Erˇsov groups invariant 1 ≤ e < ∞ are definably isomorphic to algebraic Thomas Scanlon groups whilst the “interesting” groups are type-definable.
Introduction ∞ T∞ n Additive • The groups p A(K) := n=1 p A(K) where A is an abelian groups in difference and variety play an important rˆolein Hrushovski’s proof of the differential fields positive characterisitc Mordell-Lang theorem.
Drinfeld modules • Bouscaren and Delon completely classified the type definable groups having no additive part.
• In the same work, some pathological additive groups were identified, including a type definable group G which admits a definable isogeny to the additive group of the constants, p∞ T∞ pn K := n=1 K , but no isomorphism. Thomas Scanlon University of California, Berkeley Additive groups Additive groups in SCFp,e
Additive groups • The groups p∞A(K) := T∞ pnA(K) where A is an abelian Thomas n=1 Scanlon variety play an important rˆolein Hrushovski’s proof of the
Introduction positive characterisitc Mordell-Lang theorem.
Additive groups in difference and • Bouscaren and Delon completely classified the type definable differential fields groups having no additive part.
Drinfeld modules • In the same work, some pathological additive groups were identified, including a type definable group G which admits a definable isogeny to the additive group of the constants, p∞ T∞ pn K := n=1 K , but no isomorphism.
• Blossier has produced a menangerie of additive groups in SCFp,e including Thomas Scanlon University of California, Berkeley Additive groups Additive groups in SCFp,e
Additive groups
Thomas Scanlon • Bouscaren and Delon completely classified the type definable
Introduction groups having no additive part.
Additive groups in difference and • In the same work, some pathological additive groups were differential fields identified, including a type definable group G which admits a
Drinfeld definable isogeny to the additive group of the constants, modules p∞ T∞ pn K := n=1 K , but no isomorphism.
• Blossier has produced a menangerie of additive groups in SCFp,e including •a family of 2ℵ0 pairwise orthogonal additive groups of Lascar rank one,
Thomas Scanlon University of California, Berkeley Additive groups Additive groups in SCFp,e
Additive groups
Thomas Scanlon
Introduction • In the same work, some pathological additive groups were Additive identified, including a type definable group G which admits a groups in difference and definable isogeny to the additive group of the constants, differential ∞ n fields p T∞ p K := n=1 K , but no isomorphism. Drinfeld modules • Blossier has produced a menangerie of additive groups in SCFp,e including •a family of 2ℵ0 pairwise orthogonal additive groups of Lascar rank one, • additive groups of rank ω, and
Thomas Scanlon University of California, Berkeley Additive groups Additive groups in SCFp,e
Additive groups
Thomas Scanlon
Introduction
Additive groups in difference and differential • Blossier has produced a menangerie of additive groups in fields
Drinfeld SCFp,e including modules •a family of 2ℵ0 pairwise orthogonal additive groups of Lascar rank one, • additive groups of rank ω, and • ℵ0-categorical minimal additive groups.
Thomas Scanlon University of California, Berkeley Additive groups Additive polynomials
Additive groups
Thomas Scanlon Let R be a ring of characteristic p > 0. p Introduction Let τ : R → R be the p-power Frobenius x 7→ x . Additive groups in The (usually noncommutative) ring R{τ} of polynomials difference and differential is τ is the ring generated by R and a symbol for τ subject fields to the commutation conditions τλ = λpτ for λ ∈ R. Drinfeld modules There is a natural function R{τ} → R[X ] given by Pn i Pn pi i=0 λi τ → i=0 λi X which converts the additive P i operator λi τ into an additive polynomial whose action on R agrees with the additive operator.
Thomas Scanlon University of California, Berkeley Additive groups Additive polynomials
Additive groups
Thomas Scanlon Let R be a ring of characteristic p > 0. p Introduction Let τ : R → R be the p-power Frobenius x 7→ x . Additive groups in The (usually noncommutative) ring R{τ} of polynomials difference and differential is τ is the ring generated by R and a symbol for τ subject fields to the commutation conditions τλ = λpτ for λ ∈ R. Drinfeld modules There is a natural function R{τ} → R[X ] given by Pn i Pn pi i=0 λi τ → i=0 λi X which converts the additive P i operator λi τ into an additive polynomial whose action on R agrees with the additive operator.
Thomas Scanlon University of California, Berkeley Additive groups Additive polynomials
Additive groups
Thomas Scanlon Let R be a ring of characteristic p > 0. p Introduction Let τ : R → R be the p-power Frobenius x 7→ x . Additive groups in The (usually noncommutative) ring R{τ} of polynomials difference and differential is τ is the ring generated by R and a symbol for τ subject fields to the commutation conditions τλ = λpτ for λ ∈ R. Drinfeld modules There is a natural function R{τ} → R[X ] given by Pn i Pn pi i=0 λi τ → i=0 λi X which converts the additive P i operator λi τ into an additive polynomial whose action on R agrees with the additive operator.
Thomas Scanlon University of California, Berkeley Additive groups Additive polynomials
Additive groups
Thomas Scanlon Let R be a ring of characteristic p > 0. p Introduction Let τ : R → R be the p-power Frobenius x 7→ x . Additive groups in The (usually noncommutative) ring R{τ} of polynomials difference and differential is τ is the ring generated by R and a symbol for τ subject fields to the commutation conditions τλ = λpτ for λ ∈ R. Drinfeld modules There is a natural function R{τ} → R[X ] given by Pn i Pn pi i=0 λi τ → i=0 λi X which converts the additive P i operator λi τ into an additive polynomial whose action on R agrees with the additive operator.
Thomas Scanlon University of California, Berkeley Additive groups Introduction to Drinfeld modules
Additive groups
Thomas Scanlon
Introduction Definition Additive groups in A Drinfeld module over the field K is a ring homomorphism difference and differential ϕ : Fp[t] → K{τ} whose image is not contained in K. fields
Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Introduction to Drinfeld modules
Additive groups
Thomas Scanlon Definition Introduction A Drinfeld module over the field K is a ring homomorphism Additive groups in ϕ : p[t] → K{τ} whose image is not contained in K. difference and F differential fields • If R is any K-algebra, then the Drinfeld module ϕ gives R the Drinfeld modules structure of an Fp[t]-module via the action f · x := ϕ(f )(x).
Thomas Scanlon University of California, Berkeley Additive groups Introduction to Drinfeld modules
Additive groups Definition Thomas Scanlon A Drinfeld module over the field K is a ring homomorphism ϕ : p[t] → K{τ} whose image is not contained in K. Introduction F
Additive groups in • If R is any K-algebra, then the Drinfeld module ϕ gives R the difference and differential structure of an Fp[t]-module via the action f · x := ϕ(f )(x). fields
Drinfeld modules • The function which gives the constant term of a τ-operator is a ring homomorphism π : K{τ} → K. The composed by ι := π ◦ ϕ gives a ring homomorphism from Fp[t] to K. We say that ϕ has generic characteristic if ι is injective and that ϕ has special characteristic otherwise.
Thomas Scanlon University of California, Berkeley Additive groups Introduction to Drinfeld modules
Additive groups • If R is any K-algebra, then the Drinfeld module ϕ gives R the Thomas Scanlon structure of an Fp[t]-module via the action f · x := ϕ(f )(x).
Introduction • The function which gives the constant term of a τ-operator is Additive groups in a ring homomorphism π : K{τ} → K. The composed by difference and differential ι := π ◦ ϕ gives a ring homomorphism from p[t] to K. We say fields F that ϕ has generic characteristic if ι is injective and that ϕ has Drinfeld modules special characteristic otherwise.
• There is an extensive analytic theory of Drinfeld modules of generic characteristic. Special characteristic Drinfeld modules behave like abelian varieties in positive characteristic.
Thomas Scanlon University of California, Berkeley Additive groups Introduction to Drinfeld modules
Additive groups
Thomas Scanlon • The function which gives the constant term of a τ-operator is a ring homomorphism π : K{τ} → K. The composed by Introduction ι := π ◦ ϕ gives a ring homomorphism from p[t] to K. We say Additive F groups in that ϕ has generic characteristic if ι is injective and that ϕ has difference and differential special characteristic otherwise. fields
Drinfeld modules • There is an extensive analytic theory of Drinfeld modules of generic characteristic. Special characteristic Drinfeld modules behave like abelian varieties in positive characteristic.
• A Drinfeld module has generic characteristic just in case every operator ϕ(f ) is separable.
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld modules and ∞-definable groups in SCF
Additive groups
Thomas Scanlon If K is a separably closed field of characteristic p and Introduction ϕ : Fp[t] → K{τ} is a Drinfeld module, then the group ] T Additive ϕ (K) := a∈ [t] {0} ϕ(a)(K) is ∞-definable. groups in Fp r difference and ] differential If ϕ has generic characteristic, then ϕ (K) = K. fields ] Drinfeld If ϕ has special characteristic, then ϕ is thin. modules If ϕ has special characteristic, then either there is some λ ∈ K × for which λ−1ϕ(t)λ ∈ K p∞ {τ} or ϕ] is locally modular.
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld modules and ∞-definable groups in SCF
Additive groups
Thomas Scanlon If K is a separably closed field of characteristic p and Introduction ϕ : Fp[t] → K{τ} is a Drinfeld module, then the group ] T Additive ϕ (K) := a∈ [t] {0} ϕ(a)(K) is ∞-definable. groups in Fp r difference and ] differential If ϕ has generic characteristic, then ϕ (K) = K. fields ] Drinfeld If ϕ has special characteristic, then ϕ is thin. modules If ϕ has special characteristic, then either there is some λ ∈ K × for which λ−1ϕ(t)λ ∈ K p∞ {τ} or ϕ] is locally modular.
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld modules and ∞-definable groups in SCF
Additive groups
Thomas Scanlon If K is a separably closed field of characteristic p and Introduction ϕ : Fp[t] → K{τ} is a Drinfeld module, then the group ] T Additive ϕ (K) := a∈ [t] {0} ϕ(a)(K) is ∞-definable. groups in Fp r difference and ] differential If ϕ has generic characteristic, then ϕ (K) = K. fields ] Drinfeld If ϕ has special characteristic, then ϕ is thin. modules If ϕ has special characteristic, then either there is some λ ∈ K × for which λ−1ϕ(t)λ ∈ K p∞ {τ} or ϕ] is locally modular.
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld modules and ∞-definable groups in SCF
Additive groups
Thomas Scanlon If K is a separably closed field of characteristic p and Introduction ϕ : Fp[t] → K{τ} is a Drinfeld module, then the group ] T Additive ϕ (K) := a∈ [t] {0} ϕ(a)(K) is ∞-definable. groups in Fp r difference and ] differential If ϕ has generic characteristic, then ϕ (K) = K. fields ] Drinfeld If ϕ has special characteristic, then ϕ is thin. modules If ϕ has special characteristic, then either there is some λ ∈ K × for which λ−1ϕ(t)λ ∈ K p∞ {τ} or ϕ] is locally modular.
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld Mordell-Lang via ϕ]
Additive groups
Thomas Scanlon Theorem Introduction Let ϕ : [t] → K{τ} be a Drinfeld module of special Additive Fp groups in characteristic over the field K for which no conjugate is defined difference and differential over a finite field. If Γ ≤ (K, +) is a finitely generated fields [t]-module via the action of [t] provided by ϕ and Drinfeld Fp Fp n modules X ⊆ Ga is a closed subvariety of some power of the additive group over K, then X (K) ∩ Γn is a finite union of cosets of groups.
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld Mordell-Lang via ϕ]
Additive groups Theorem Thomas Scanlon Let ϕ : Fp[t] → K{τ} be a Drinfeld module of special Introduction characteristic over the field K for which no conjugate is defined Additive over a finite field. If Γ ≤ (K, +) is a finitely generated groups in difference and Fp[t]-module via the action of Fp[t] provided by ϕ and differential n fields X ⊆ Ga is a closed subvariety of some power of the additive n Drinfeld group over K, then X (K) ∩ Γ is a finite union of cosets of modules groups.
Proof. Follow Hrushovski’s proof of the function field Mordell-Lang theorem using the modularity of ϕ].
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld Manin-Mumford
Additive groups
Thomas Scanlon
Introduction Theorem
Additive Let ϕ : p[t] → K{τ} be a Drinfeld module of generic groups in F difference and characteristic. Let differential alg fields Ξ := ϕtor := {ξ ∈ K | (∃a ∈ Fp[t] r {0}) ϕ(a)(ξ) = 0} be n Drinfeld the torsion submodule. If X ⊆ a is a closed subvariety of modules G some power of the additive group, then X (K) ∩ Ξn is a finite union of translates of submodules of K algn.
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld Manin-Mumford
Additive groups Theorem
Thomas Let ϕ : p[t] → K{τ} be a Drinfeld module of generic Scanlon F characteristic. Let Introduction alg Ξ := ϕtor := {ξ ∈ K | (∃a ∈ Fp[t] r {0}) ϕ(a)(ξ) = 0} be Additive n groups in the torsion submodule. If X ⊆ Ga is a closed subvariety of difference and some power of the additive group, then X (K) ∩ Ξn is a finite differential n fields union of translates of submodules of K alg . Drinfeld modules Proof. Using the theory of reductions of Drinfeld modules, find (L, σ) |= ACFA extending (K, idK ) and a definable group Υ ≤ (L, +) such that: Υ has
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld Manin-Mumford
Additive groups Theorem
Thomas Let ϕ : p[t] → K{τ} be a Drinfeld module of generic Scanlon F characteristic. Let Introduction alg Ξ := ϕtor := {ξ ∈ K | (∃a ∈ Fp[t] r {0}) ϕ(a)(ξ) = 0} be Additive n groups in the torsion submodule. If X ⊆ Ga is a closed subvariety of difference and some power of the additive group, then X (K) ∩ Ξn is a finite differential n fields union of translates of submodules of K alg . Drinfeld modules Proof. Using the theory of reductions of Drinfeld modules, find (L, σ) |= ACFA extending (K, idK ) and a definable group Υ ≤ (L, +) such that: Υ has finite Lascar rank
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld Manin-Mumford
Additive groups Theorem
Thomas Let ϕ : p[t] → K{τ} be a Drinfeld module of generic Scanlon F characteristic. Let Introduction alg Ξ := ϕtor := {ξ ∈ K | (∃a ∈ Fp[t] r {0}) ϕ(a)(ξ) = 0} be Additive n groups in the torsion submodule. If X ⊆ Ga is a closed subvariety of difference and some power of the additive group, then X (K) ∩ Ξn is a finite differential n fields union of translates of submodules of K alg . Drinfeld modules Proof. Using the theory of reductions of Drinfeld modules, find (L, σ) |= ACFA extending (K, idK ) and a definable group Υ ≤ (L, +) such that: Υ has finite Lascar rank, (almost) contains Ξ
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld Manin-Mumford
Additive groups Theorem
Thomas Let ϕ : p[t] → K{τ} be a Drinfeld module of generic Scanlon F characteristic. Let Introduction alg Ξ := ϕtor := {ξ ∈ K | (∃a ∈ Fp[t] r {0}) ϕ(a)(ξ) = 0} be Additive n groups in the torsion submodule. If X ⊆ Ga is a closed subvariety of difference and some power of the additive group, then X (K) ∩ Ξn is a finite differential n fields union of translates of submodules of K alg . Drinfeld modules Proof. Using the theory of reductions of Drinfeld modules, find (L, σ) |= ACFA extending (K, idK ) and a definable group Υ ≤ (L, +) such that: Υ has finite Lascar rank, (almost) contains Ξ and is one-based.
Thomas Scanlon University of California, Berkeley Additive groups Drinfeld Manin-Mumford
Additive groups Theorem
Thomas Let ϕ : p[t] → K{τ} be a Drinfeld module of generic Scanlon F characteristic. Let Introduction alg Ξ := ϕtor := {ξ ∈ K | (∃a ∈ Fp[t] r {0}) ϕ(a)(ξ) = 0} be Additive n groups in the torsion submodule. If X ⊆ Ga is a closed subvariety of difference and some power of the additive group, then X (K) ∩ Ξn is a finite differential n fields union of translates of submodules of K alg . Drinfeld modules Proof. Using the theory of reductions of Drinfeld modules, find (L, σ) |= ACFA extending (K, idK ) and a definable group Υ ≤ (L, +) such that: Υ has finite Lascar rank, (almost) contains Ξ and is one-based. For the assertion about modules, use the analytic theory of Drinfeld modules. Thomas Scanlon University of California, Berkeley Additive groups Ghioca’s refined theorems on ϕ]
Additive groups Theorem
Thomas If ϕ : p[t] → K{τ} is a sufficiently general Drinfeld module of Scanlon F special characteristic, Γ ≤ K is a finitely generated Introduction n Fp[t]-module, and X ⊆ Ga is an irreducible variety for which Additive n groups in X (K) ∩ Γ is Zariski dense in X , then X is a translate of an difference and differential algebraic group which is fixed by ϕ(f ) for some nonconstant fields f ∈ Fp[t]. Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Ghioca’s refined theorems on ϕ]
Additive groups
Thomas Scanlon Theorem If ϕ : [t] → K{τ} is a sufficiently general Drinfeld module of Introduction Fp
Additive special characteristic, Γ ≤ K is a finitely generated groups in n difference and Fp[t]-module, and X ⊆ Ga is an irreducible variety for which differential n fields X (K) ∩ Γ is Zariski dense in X , then X is a translate of an
Drinfeld algebraic group which is fixed by ϕ(f ) for some nonconstant modules f ∈ Fp[t]. The proof of this theorem employs a theory of heights for Drinfeld modules to analyze the quasiendomorphism ring of the ∞-definable group ϕ].
Thomas Scanlon University of California, Berkeley Additive groups Ghioca’s refined theorems on ϕ]
Additive groups
Thomas Scanlon
Introduction Theorem Additive groups in If ϕ : Fp[t] → K{τ} is a Drinfeld module not conjugate to one difference and differential defined over a field of transcendence degree at most one, Γ ≤ K fields n is a finitely generated Fp[t]-module and X ⊆ Ga is a closed Drinfeld modules subvariety not containing a translate of a positive dimensional n n algebraic subgroup of Ga , then X (K) ∩ Γ is finite.
Thomas Scanlon University of California, Berkeley Additive groups Ghioca’s refined theorems on ϕ]
Additive groups
Thomas Scanlon Theorem Introduction If ϕ : Fp[t] → K{τ} is a Drinfeld module not conjugate to one Additive defined over a field of transcendence degree at most one, Γ ≤ K groups in difference and is a finitely generated [t]-module and X ⊆ n is a closed differential Fp Ga fields subvariety not containing a translate of a positive dimensional Drinfeld algebraic subgroup of n, then X (K) ∩ Γn is finite. modules Ga This theorem is proven via a specialization argument analogous to Hrushovski’s proof of the characterisitic zero function field Mordell-Lang theorem by reduction to positive characteristic.
Thomas Scanlon University of California, Berkeley Additive groups Ghioca’s refined theorems on ϕ]
Additive groups Theorem Thomas Scanlon If ϕ : Fp[t] → K{τ} is a Drinfeld module not conjugate to one defined over a field of transcendence degree at most one, Γ ≤ K Introduction n is a finitely generated p[t]-module and X ⊆ a is a closed Additive F G groups in subvariety not containing a translate of a positive dimensional difference and n n differential algebraic subgroup of Ga , then X (K) ∩ Γ is finite. fields
Drinfeld modules This theorem is proven via a specialization argument analogous to Hrushovski’s proof of the characterisitic zero function field Mordell-Lang theorem by reduction to positive characteristic. Since we do not have a good description of the quasiendomorphism ring of ϕ] in general, the proof breaks n down when one considers varieties X ⊆ Ga which do contain translates of algebraic groups.
Thomas Scanlon University of California, Berkeley Additive groups Geometric proofs of modularity
Additive groups
Thomas Scanlon
Introduction • The proofs of modularity of ϕ] and of the group Υ in the Additive groups in difference field proof of the Drinfeld Manin-Mumford theorem difference and differential rely upon a trichotomy theorem for minimal types in SCF and fields ACFA. Drinfeld modules
Thomas Scanlon University of California, Berkeley Additive groups Geometric proofs of modularity
Additive groups
Thomas ] Scanlon • The proofs of modularity of ϕ and of the group Υ in the
Introduction difference field proof of the Drinfeld Manin-Mumford theorem
Additive rely upon a trichotomy theorem for minimal types in SCF and groups in difference and ACFA. differential fields
Drinfeld • Using jet space techniques, Pink and Roessler, and later modules Pillay and Ziegler, proved the modularity of some of the auxilliary groups used in the proofs of the Manin-Mumford and function field Mordell-Lang conjectures.
Thomas Scanlon University of California, Berkeley Additive groups Geometric proofs of modularity
Additive groups
Thomas Scanlon • Using jet space techniques, Pink and Roessler, and later Introduction
Additive Pillay and Ziegler, proved the modularity of some of the groups in auxilliary groups used in the proofs of the Manin-Mumford and difference and differential function field Mordell-Lang conjectures. fields
Drinfeld modules • Adapting the jet space techniques to the additive group, Ealy showed that the induced structure on the torsion module of an abelian T -module (a higher dimensional analogue of a Drinfeld module) is modular.
Thomas Scanlon University of California, Berkeley Additive groups Geometric proofs of modularity
Additive groups
Thomas Scanlon
Introduction • Adapting the jet space techniques to the additive group, Ealy
Additive showed that the induced structure on the torsion module of an groups in difference and abelian T -module (a higher dimensional analogue of a Drinfeld differential fields module) is modular.
Drinfeld In fact, Ealy can show that the induced structure on the torsion modules module is modular without recourse to either a model theoretic trichotomy theorem or a jet space construction. Rather, all that is needed is the observation that the degree of a finite flat map cannot increase upon specialization and restriction to a subvariety.
Thomas Scanlon University of California, Berkeley Additive groups Is it adding or subtracting today?
Additive groups
Thomas Scanlon
Introduction
Additive groups in difference and differential fields
Drinfeld modules
If only HRH knew...
Thomas Scanlon University of California, Berkeley Additive groups