# Involutions on Zilber fields Vincenzo Mantova Scuola Normale - PowerPoint PPT Presentation

## Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Involutions on Zilber fields Vincenzo Mantova Scuola Normale Superiore di Pisa Logic Colloquium Barcelona July 12th, 2011 Involutions on Zilber fields

1. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Involutions on Zilber fields Vincenzo Mantova Scuola Normale Superiore di Pisa Logic Colloquium Barcelona – July 12th, 2011 Involutions on Zilber fields Vincenzo Mantova

2. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Outline 1 Exponential fields 2 Axiomatizations and Schanuel’s Conjecture 3 Automorphisms and topologies 4 Very few details Involutions on Zilber fields Vincenzo Mantova

3. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Exponential fields Definition An exponential field , or E-field , is a structure ( K , 0 , 1 , + , · , E ) where ( K , 0 , 1 , + , · ) is a field, and the following equation holds E ( x + y ) = E ( x ) · E ( y ) . • R exp ( o -minimal, model complete, decidable if Schanuel’s Conjecture is true). • C exp (undecidable, interprets Peano’s Arithmetic). Involutions on Zilber fields Vincenzo Mantova

4. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Schanuel’s Conjecture A special role in the model-theoretic study is played by a long standing conjecture in transcendental number theory. Conjecture (Schanuel) For any z 1 , . . . , z n ∈ C linearly independent over Q , tr . deg . Q ( z 1 , . . . , z n , e z 1 , . . . , e z n ) ≥ n . If Schanuel’s Conjecture holds at least for z 1 , . . . , z n ∈ R , then the first order theory of R exp is decidable [1]. On the other hand, C exp defines ( Z , + , · ) , hence it is always undecidable. First order theory may not be sufficient. Involutions on Zilber fields Vincenzo Mantova

5. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Conjectural, but categorical axioms for C exp in L ω 1 ,ω ( Q ) Zilber looked for (uncountably) categorical axioms in L ω 1 ,ω ( Q ) . Properties of C exp : (ACF 0 ) C is an algebraically closed field of characteristic 0. (E) exp is a homomorphism exp : ( C , +) → ( C × , · ) . (LOG) exp is surjective. (STD) ker ( exp ) = 2 π i Z (needs L ω 1 ,ω ). Conjectures on C exp : (SP) tr . deg . Q ( z , exp ( z )) ≥ lin . d . Q ( z ) (Schanuel’s Property). (SEC) every “rotund” variety contains a generic solution ( z , exp ( z )) . Another property of C exp : (CCP) every “rotund” variety of “depth 0” contains at most countably many generic solutions ( z , exp ( z )) (needs Q ). Involutions on Zilber fields Vincenzo Mantova

6. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Zilber’s categoricity result Theorem (Zilber, 2005 [2]) The axioms are uncountably categorical. We call “Zilber field”, or B E , the unique model of cardinality 2 ℵ 0 . The conjecture becomes the following. Conjecture (Zilber, 2005 [2]) C exp is isomorphic to B E . Involutions on Zilber fields Vincenzo Mantova

7. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Automorphisms Definition An involution of K E is an automorphism σ : K E → K E s.t. σ 2 = Id . C exp has one involution, complex conjugation. • It is the unique known automorphism of C exp . • exp is continuous in the induced topology. • exp is the unique continuous exponential (up to constants). If B E ∼ = C exp , B E would have an involution as well. Theorem (M., 2011) 1 There is an involution σ on B E (such that B σ ∼ = R ). 2 There are 2 2 ℵ 0 non-conjugate involutions on B E . Involutions on Zilber fields Vincenzo Mantova

8. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Problems in our proof Unfortunately, what we found is different from complex conjugation. • the solutions ( z , E ( z )) of rotund varieties are dense ; • hence, E is not continuous; • moreover, the restriction E ↾ B σ is not increasing. This is also in contrast with the fact that on C exp the solutions ( z , exp ( z )) of rotund varieties of “depth 0” are isolated. Remark. We are not refuting Zilber’s conjecture: other involutions can still be such that E is continuous. Involutions on Zilber fields Vincenzo Mantova

9. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography The construction We start from K and σ : K → K , and we build E . For instance, K = C and σ the complex conjugation. For any E , we know that σ ◦ E = E ◦ σ if and only if 1 E ( R ) ⊂ R > 0 ; 2 E ( i R ) ⊂ S 1 ( C ) . Hence, we build E on C by ‘back-and-forth’, while respecting the restrictions 1 , 2 . We can easily obtain an E satisfying all of the axioms except (CCP). In order to build E with (CCP), we add dense sets of solutions to rotund varieties (destroying continuity). Involutions on Zilber fields Vincenzo Mantova

10. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Summary Zilber produced a sentence ψ in L ω 1 ,ω ( Q ) which is uncountably categorical, and conjecturally an axiomatization of C exp . Its unique model in cardinality 2 ℵ 0 is called B E . Looking for an analogue of complex conjugation, we found that • There are 2 2 ℵ 0 involutions on B E . • One of them is such that B σ ∼ = R . • However, E is not continuous w.r.t. them. Thanks for your attention! Involutions on Zilber fields Vincenzo Mantova

11. Exponential fields Axiomatizations Automorphisms Details Summary Bibliography Bibliography I Angus Macintyre and A. J. Wilkie. On the decidability of the real exponential field. In Kreiseliana , pages 441–467. A K Peters, Wellesley, MA, 1996. Boris Zilber. Pseudo-exponentiation on algebraically closed fields of characteristic zero. Annals of Pure and Applied Logic , 132(1):67–95, 2005. doi:10.1016/j.apal.2004.07.001 . Involutions on Zilber fields Vincenzo Mantova

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