Transferring imaginaries How to eliminate imaginaries in p-adic fields Silvain Rideau (joint work with E. Hrushovski and B. Martin) Paris 11, École Normale Supérieure April 3, 2013 1 / 35
Contents Imaginaries Valued fields Imaginary Transfer Unary types The p -adic imaginaries 2 / 35
Codes and Quotients Definition (Code) In some structure M , a set X definable (with parameters) is said to be Definition (Representable quotient) Let M be some structure, D be a definable set and E be a definable M if there exists a definable function f with domain D such that 3 / 35 coded by some tuple a if there is a formula φ [ x , y ] such that ⇒ a ′ = a . φ [ M , a ′ ] = X ( M ) ⇐ equivalence relation on D . The quotient D / E is said to be representable in ⇒ f ( x ) = f ( y ) . xEy ⇐
Eliminating imaginaries Proposition Let M be some structure with at least two constants, the following are equivalent: (i) Any subset of M definable (with parameters) is coded, (ii) Every quotient definable in M is representable. A theory is said to eliminate imaginaries if every model of T verifies any of the two statements in the previous proposition. Example 4 / 35 ▸ A non-example : infinite sets, ▸ An example : algebraically closed fields.
Shelah’s construction Definition projection. Proposition Proposition 5 / 35 Let M be a L -structure, we define a new language L eq and a L eq -structure M eq as follows: ▸ For any definable equivalence relation E on a product of L -sorts ∏ i S i , we add to L a sort S E and a function f E ∶ ∏ i S i → S E , ▸ In M eq , S E is interpreted as ∏ i S i ( M )/ E ( M ) and f E as the canonical Let T be a complete theory. The language L eq and the theory T eq = Th ( M eq ) does not depend on the choice of M ⊧ T . Let T be a complete theory. The theory T eq eliminates imaginaries.
Shelah’s construction Definition projection. Proposition Let T be a complete theory. The following are equivalent: (i) T eliminates imaginaries, 5 / 35 Let M be a L -structure, we define a new language L eq and a L eq -structure M eq as follows: ▸ For any definable equivalence relation E on a product of L -sorts ∏ i S i , we add to L a sort S E and a function f E ∶ ∏ i S i → S E , ▸ In M eq , S E is interpreted as ∏ i S i ( M )/ E ( M ) and f E as the canonical (ii) For all, M ⊧ T and e ∈ M eq , there exists a tuple d ∈ M such that: d ∈ dcl eq ( e ) and e ∈ dcl eq ( d ) .
Finite sets Definition (Weak elimination of imaginaries) Example Infinite sets weakly eliminate imaginaries. Proposition Suppose T has weak elimination of imaginaries and every finite set in every model of T is coded, then T eliminates imaginaries. 6 / 35 A complete theory T weakly eliminates imaginaries if for all M ⊧ T and e ∈ M eq , there exists a tuple d ∈ M such that: d ∈ acl eq ( e ) and e ∈ dcl eq ( d ) .
Finite imaginaries Definition (EI/UFI) A complete theory T eliminates imaginaries up to uniform finite that: Proposition Suppose T has EI/UFI and any finite quotient definable (with parameters) in any model of T is representable, then T eliminates imaginaries. 7 / 35 imaginaries if for all M ⊧ T and e ∈ M eq , there exists a tuple d ∈ M such d ∈ dcl eq ( e ) and e ∈ acl eq ( d ) .
Contents Imaginaries Valued fields Imaginary Transfer Unary types The p -adic imaginaries 8 / 35
Some definitions Definition 9 / 35 Let K be a field, a valuation on K is a map v from K ⋆ to some abelian ordered group Γ that satisfies the following axioms: (i) v ( xy ) = v ( x ) + v ( y ) , (ii) v ( x + y ) ≥ min { v ( x ) , v ( y )} ▸ We usually add a point ∞ to Γ to denote v ( 0 ) , greater than any other point in Γ . ▸ The set O = { x ∈ K ∣ v ( x ) ≥ 0 } is a ring, called the valuation ring of K . ▸ It has a unique maximal ideal M = { x ∈ K ∣ v ( x ) > 0 } . ▸ The residue field O / M will be denoted k . ▸ We will also be considering the group RV ∶ = K ⋆ /( 1 + M ) .
Some examples Q for the p -adic valuation. It is also a valued field, field (in some language to be specified). 10 / 35 ▸ Let p be a prime number, then we can define the p -adic valuation on Q by taking v p ( p n a / b ) = n whenever a ∧ b = a ∧ p = b ∧ p = 1, ▸ We will denote by Q p , the field of p -adic numbers, the completion of ▸ We will denote by ACVF the theory of algebraically closed valued
Imaginaries in valued fields Remark field nor in Q p However, in the case of ACVF , Haskell, Hrushovski and Macpherson have shown what imaginary sorts it suffjces to add. 11 / 35 In the language of rings enriched with a predicate for v ( x ) ≤ v ( y ) , the quotient Γ = K ⋆ / O ⋆ is not representable in any algebraically closed valued
The geometric sorts Definition (The sorts S n ) Definition (The sorts T n ) 12 / 35 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 . ▸ We can give an alternative definition of these sorts, for example 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 the ring language 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 (The sorts S n ) Definition (The sorts T n ) Theorem (Haskell, Hrushovski and Macpherson, 2006) Question new imaginaries in this theory? 2. Can these imaginairies be eliminated uniformly in p . 12 / 35 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 L G -theory ACVF eliminates imaginaries. 1. Are all imaginaries in Q p coded in the geometric sorts or are there
Contents Imaginaries Valued fields Imaginary Transfer Unary types The p -adic imaginaries 13 / 35
A first example : real-closed fields Example (Square roots) Let K be a real closed field and K considered as ring language structures). K alg , alg , 14 / 35 alg be its algebraic closure (both fields are ▸ Let a ∈ K , the function f ∶ x ↦ √ x − a can be defined in K but not in ▸ However, the 1-to-2 correspondance F = {( x , y ) ∣ y 2 = x − a } is quantifier free definable both in K and K ▸ F is the Zariski closure of the graph of f and f ( x ) can be defined (in K ) as the greatest y such that ( x , y ) ∈ F , alg (which is K ). ▸ In fact, f can be coded by the code of F in K
The general setting Question Similarly for acl, tp and TP. M eq . M , M . Let us fix some notations: Under what hypotheses can we deduce that T eliminates imaginaries? 15 / 35 ▸ Let ̃ L ⊆ L be two languages, ▸ Let ̃ T be a ̃ L theory that eliminates quantifiers and imaginaries, ▸ Let T be a L -theory such that ̃ T ∀ ⊆ T . Let ̃ M ⊧ ̃ T and M ⊧ T such that M ⊆ ̃ ▸ Let A ⊆ ̃ L ( A ) for the (quantifier-free) ̃ L -definable M , we will write dcl ̃ closure in ̃ L ( A ) for the L eq -definable closure in ▸ Let A ⊆ M eq , we will write dcl eq
alg . The specific cases a non principal ultrafilter on the set of primes, with constants added for some (2-generated) subfield F verifying certain properties. Remark By Ax-Kochen-Ersov, the theories of [PL] are the completions of the theory of equicharacteristic zero Henselian valued fields with a pseudo-finite residue field and a Z -group as valuation group. 16 / 35 ▸ The theory ̃ T will be either ACVF 0 , 0 or ACVF 0 , p , in L G . ▸ The theory T will be either : [ p C] The L G -theory of L a finite extension of Q p , with a constant added for a generator of L ∩ Q [PL] The L G -theory of ∏ L p / U where L p is a finite extension of Q p and U is
Dominant sorts Definition Example M eq , For any choice of theory T , we will suppose that a set of dominant sorts dominant sorts in any model of T . 17 / 35 In a theory, a set of sorts S will be called dominant if for any other sort S of the language, there is a surjective ∅ -definable function f ∶ ∏ i S i → S where the S i are in S . ▸ The set consisting of all the sorts is dominant. ▸ The set of “real” sorts (i.e. the original sorts from M ) are dominant in ▸ In a valued field in the geometric language, the sort K is dominant. has been chosen, and we will write dom ( M ) for the union of the
Algebraic boundedness Hypothesis (i) Proof. [ p C] Follows immediately from the fact that for all models M and [PL] A lot more technical. 18 / 35 For all M 1 ≼ M and c ∈ dom ( M ) , dcl eq L ( M 1 c ) ∩ M ⊆ acl ̃ L ( M 1 c ) . A ⊆ K ( M ) , acl ̃ L ( A ) ≼ M .
19 / 35 Proposition work. Proof. Hypothesis (ii) implies that finite sets are coded in T . Hypothesis (ii) L ( M ) Coping with dcl ̃ L ( M ) , there exists a tuple e ′ ∈ M such that for all σ ∈ Aut (̃ M ) For all e ∈ dcl ̃ with σ ( M ) = M , σ fixes e if and only if it fixes e ′ . It suffjces to consider e ∈ S n ( dcl ̃ L ( M )) . Such a lattice has a basis in some finite extension L ∣ K ( M ) . With the added constants, O ( L ) is generated over O ( M ) by has an element a whose minimal polynomial is over the prime field. Then the image of e ( L ) by the function ∑ x i a i ↦ ( x i ) will
Unary imaginaries Hypothesis (iii) Proof. We need a precise description of unary types. 20 / 35 Any L ( M ) -definable unary set X ⊆ dom ( M ) 1 is coded.
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