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.