Transferring imaginaries How to eliminate imaginaries in p-adic - PowerPoint PPT Presentation

Transferring imaginaries How to eliminate imaginaries in p-adic fields Silvain Rideau joint work with E. Hrushovski and B. Martin in “Definable equivalence relations and zeta functions of groups” with an appendix by R. Cluckers Orsay Paris-Sud 11, École Normale Supérieure May 12, 2014 1 / 18

Some notations 2 / 18 Let ( K , v ) be a valued field. ▸ We will denote by O = { x ∈ K ∣ v ( x ) ≥ 0 } the valuation ring; ▸ It has a unique maximal ideal M = { x ∈ K ∣ v ( x ) > 0 } ; ▸ The residue field O / M will be denoted k ; ▸ The value group will be denoted by Γ ; ▸ Let also RV ∶ = K ⋆ /( 1 + M ) ⊇ k ⋆ .

First model theory results Theorem (A. Robinson, 1956) quantifiers. Theorem (Macintyre, 1976) 3 / 18 Let L div = { K ; 0 , 1 , + , − , ⋅ , ∣} where x ∣ y is interpreted by v ( x ) ≤ v ( y ) . The L div -theory ACVF of algebraically closed valued fields eliminates Let L P = L div ∪ { P n ∣ n ∈ N > 0 } where x ∈ P n if and only if ∃ y , y n = x . The L P -theory of Q p eliminates quantifiers.

Imaginaries Let T be a theory add a point for each imaginary. Remark The theory ACF of algebraically closed fields in the language Theorem (Poizat, 1983) elimination of imaginaries. Positive answers to these two questions are equivalent and is called such that automorphisms fix c if and only if they stabilize X set-wise? function f — a representation — such that 4 / 18 ▸ For all definable equivalence relation E , does there exist a definable ∀ x , y , xEy ⇐ ⇒ f ( x ) = f ( y ) . ▸ For all definable (with parameters) set X , is there a tuple c — a code — L rg = { K ; 0 , 1 , + , − , ⋅ } eliminates imaginaries. To any L -structure M we can associate the L eq -structure M eq where we

Imaginaries in valued fields Remark algebraically closed valued field nor in Q p . However, in the case of ACVF — the theory of algebraically closed valued fields — Haskell, Hrushovski and Macpherson have shown what imaginary sorts it suffjces to add. 5 / 18 In the language L div , the quotient Γ = K ⋆ / O ⋆ is not representable in

The geometric sorts Definition We can give an alternative definition of these sorts, for example Definition 6 / 18 ▸ The elements of S n are the free O -module in K n of rank n . ▸ The elements of T n are of the form a + M s where s ∈ S n and a ∈ s . S n ≃ GL n ( K )/ GL n ( O ) . The geometric language L G is composed of the sorts K , S n and T n for all n , with L rg on K and functions ρ n ∶ GL n ( K ) → S n and τ n ∶ S n × K n → T n . ▸ S 1 can be identified with Γ and ρ 1 with v; ▸ T 1 can be identified with RV; ▸ The set of balls (open and closed, possibly with infinite radius) B can be identified with a subset of K ∪ S 2 ∪ T 2 .

The geometric sorts Definition Definition Theorem (Haskell, Hrushovski and Macpherson, 2006) 6 / 18 ▸ The elements of S n are the free O -module in K n of rank n . ▸ The elements of T n are of the form a + M s where s ∈ S n and a ∈ s . The geometric language L G is composed of the sorts K , S n and T n for all n , with L rg on K and functions ρ n ∶ GL n ( K ) → S n and τ n ∶ S n × K n → T n . ▸ The L G -theory ACVF G eliminates imaginaries. ▸ In particular, the imaginaries in ACVF G 0 , p (respectively those in ACVF G p , p ) can be eliminated uniformly in p .

The geometric sorts Definition Definition Question new imaginaries in this theory? 2. Can these imaginaries be eliminated uniformly in p ? 6 / 18 ▸ The elements of S n are the free O -module in K n of rank n . ▸ The elements of T n are of the form a + M s where s ∈ S n and a ∈ s . The geometric language L G is composed of the sorts K , S n and T n for all n , with L rg on K and functions ρ n ∶ GL n ( K ) → S n and τ n ∶ S n × K n → T n . 1. Are all imaginaries in Q p coded in the geometric sorts or are there

The general setting M . Let us fix some notations: Similarly for acl, tp and TP (the space of types). M eq . M , In the paper, we give a more general setting, but here we will only 7 / 18 consider substructures of ACVF. ▸ Let T ⊇ ACVF G ∀ be an L G -theory. Let ̃ M ⊧ ACVF G and M ⊧ T such that M ⊆ ̃ ▸ Let A ⊆ ̃ M ( A ) for the L G -definable closure in ̃ M , we will write dcl ̃ M ( A ) for the L eq -definable closure in ▸ Let A ⊆ M eq , we will write dcl eq

The specific cases of interest constants added: ultraproducts. Ax-Kochen-Eršov principle any model of PLF is equivalent to one of these principal ultrafilter on the set of primes is a model of PLF. In fact, By the Remark The theory T will be either : 8 / 18 with a pseudo-finite residue field, a Z -group as valuation group and 2 alg ; [ p CF] The L G -theory of K a finite extension of Q p , with a constant added for a generator of K ∩ Q alg over Q p ∩ Q [PLF] The L G -theory of equicharacteristic zero Henselian valued fields ▸ A uniformizer, i.e. π ∈ K with minimal positive valuation; ▸ An unramified Galois-unifomizer. i.e an element c ∈ K such that res ( c ) generates k ⋆ /( ⋂ n P n ( k ⋆ )) . Every ∏ K p / U where K p is a finite extension of Q p and U is a non

A first example: extracting square roots in Q 3 alg . 9 / 18 defined by: ▸ Let a ∈ Q 3 and f ∶ P 2 ( Q ⋆ 3 ) + a → Q 3 , where P 2 is the set of squares, f ( x ) 2 = x − a and ac ( f ( x )) = 1 . alg ⊧ ACVF 0 , 3 . ▸ This function can be defined in Q 3 but not in Q 3 ▸ However, the 1-to-2 correspondence F = {( x , y ) ∣ y 2 = x − a } is quantifier free definable both in Q 3 and Q 3 ▸ F is the Zariski closure of the graph of f and f ( x ) can be defined (in Q 3 ) as the y such that ( x , y ) ∈ F and ac ( y ) = 1. M ( Q 3 ) = Q 3 . alg and this code is in dcl ̃ ▸ F is coded in Q 3 ▸ The graph of f is coded by the code of F .

An abstract criterion Theorem Then T eliminates imaginaries. 10 / 18 Assume the following holds: (i) Any L ( M ) -definable unary set X ⊆ K ( M ) is coded; (ii) For all M 1 ≼ M and c ∈ K ( M ) , dcl eq M ( M 1 c ) ∩ M ⊆ acl ̃ M ( M 1 c ) ; M ( M ) , there exists a tuple e ′ ∈ M such that for all (iii) For all e ∈ dcl ̃ σ ∈ Aut (̃ M ) with σ ( M ) = M , σ fixes e if and only if it fixes e ′ ; M ( A ) ∩ M and c ∈ K ( M ) , there exists an (iv) For any A = acl eq Aut (̃ M (̃ M / A ) -invariant type ̃ M ) such that ̃ p ∣ M is consistent p ∈ TP ̃ with tp L ( c / A ) ; M ( A ) ∩ M and c ∈ K ( M ) , acl eq M ( Ac ) ∩ M = dcl eq M ( Ac ) ∩ M . (v) For all A = acl eq

Another abstract criterion Theorem Then T eliminates imaginaries. 11 / 18 Assume the following holds: (i) Any L ( M ) -definable unary set X ⊆ K ( M ) is coded; (ii) For all M 1 ≼ M and c ∈ K ( M ) , dcl eq M ( M 1 c ) ∩ M ⊆ acl ̃ M ( M 1 c ) ; M ( M ) , there exists a tuple e ′ ∈ M such that for all (iii) For all e ∈ dcl ̃ σ ∈ Aut (̃ M ) with σ ( M ) = M , σ fixes e if and only if it fixes e ′ ; M ( A ) ∩ M and c ∈ K ( M ) , there exists an (iv) For any A = acl eq Aut (̃ M (̃ M / A ) -invariant type ̃ M ) such that ̃ p ∣ M is consistent p ∈ TP ̃ with tp L ( c / A ) ; M ( A ) there exists e ′ ∈ M such that (v’) For all A ⊆ M and any e ∈ acl eq M ( Ae ′ ) and e ′ ∈ dcl eq M ( Ae ) . e ∈ dcl eq

p -adic imaginaries Theorem imaginaries. Proof. It follows from the first EI criterion. 12 / 18 Let K be a finite extension of Q p , then the theory of K in the language L G with a constant added for a generator of K ∩ Q alg over Q p ∩ Q alg eliminates

Ultraproducts Theorem and an unramified Galois-uniformizer, eliminate imaginaries. Proof. It follows from the second EI criterion. Remark 13 / 18 Let K = ∏ K p / U be an ultraproduct of finite extensions K p of Q p . The theory of K in the language L G , with constants added for a uniformizer The sorts T n are useless in those two cases.

Uniformity Corollary unramified m 0 -Galois uniformizer and function 14 / 18 Definition Let L ⋆ G be L G with two constants in K added. An unramified m -Galois uniformizer is a point c ∈ K such that res ( c ) generates k ⋆ / P m ( k ⋆ ) . For any equivalence relation E p on a set D p definable in K p uniformly in p , there exists m 0 and an L ⋆ G -formula φ ( x , y ) such that for all p , φ defines a p × S m ( K p ) f p ∶ D → K l where K p is made into a L ⋆ G -structure by choosing a uniformizer and an K p ⊧ ∀ x , y , xE p y ⇐ ⇒ f p ( x ) = f p ( y ) .

Definable families of equivalence relations Definition an equivalence relation on R and 15 / 18 Fix p a prime and let K p be a finite extension of Q p . A family ( R l ) l ∈ N r ⊆ K n p is said to be uniformly definable if there is an L G formula φ ( x , y ) such that for all l ∈ N r , φ ( K p , l ) = R l . We say that E ⊆ R 2 is a definable family of equivalence relations on R if E is ∀ x , y ∈ R , xEy ⇒ ∃ l ∈ N r , x , y ∈ R l . In particular, for all l ∈ N r , E induces an equivalence relation E l on R l .

Definable families of equivalence relations Definition 15 / 18 For all prime p , let K p be a finite extension of Q p . A family ( R p , l ) l ∈ N r ⊆ K n p is said to be definable uniformly in p if there is an L G formula φ ( x , y ) such that for all prime p and l ∈ N r , φ ( K p , l ) = R p , l . We say that E p ⊆ R 2 p is a family of equivalence relations on R p definable uniformly in p if E p is an equivalence relation on R p and ∀ p ∀ x , y ∈ R p , xE p y ⇒ ∃ l ∈ N r , x , y ∈ R p , l . In particular, for all l ∈ N r , E p induces an equivalence relation E p , l on R p , l .