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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


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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

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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

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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).

  • Rexp (o-minimal, model complete, decidable if Schanuel’s

Conjecture is true).

  • Cexp (undecidable, interprets Peano’s Arithmetic).

Involutions on Zilber fields Vincenzo Mantova

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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 z1, . . . , zn ∈ C linearly independent over Q, tr.deg.Q(z1, . . . , zn, ez1, . . . , ezn) ≥ n. If Schanuel’s Conjecture holds at least for z1, . . . , zn ∈ R, then the first order theory of Rexp is decidable [1]. On the other hand, Cexp defines (Z, +, ·), hence it is always

  • undecidable. First order theory may not be sufficient.

Involutions on Zilber fields Vincenzo Mantova

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Conjectural, but categorical axioms for Cexp in Lω1,ω(Q)

Zilber looked for (uncountably) categorical axioms in Lω1,ω(Q).

Properties of Cexp: (ACF0 ) C is an algebraically closed field of characteristic 0. (E) exp is a homomorphism exp : (C, +) → (C×, ·). (LOG) exp is surjective. (STD) ker(exp) = 2πiZ (needs Lω1,ω). Conjectures on Cexp: (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 Cexp: (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

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Zilber’s categoricity result

Theorem (Zilber, 2005 [2]) The axioms are uncountably categorical. We call “Zilber field”, or BE, the unique model of cardinality 2ℵ0. The conjecture becomes the following. Conjecture (Zilber, 2005 [2]) Cexp is isomorphic to BE.

Involutions on Zilber fields Vincenzo Mantova

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Automorphisms

Definition An involution of KE is an automorphism σ : KE → KE s.t. σ2 = Id. Cexp has one involution, complex conjugation.

  • It is the unique known automorphism of Cexp.
  • exp is continuous in the induced topology.
  • exp is the unique continuous exponential (up to constants).

If BE ∼ = Cexp, BE would have an involution as well. Theorem (M., 2011)

1 There is an involution σ on BE (such that Bσ ∼

= R).

2 There are 22ℵ0 non-conjugate involutions on BE.

Involutions on Zilber fields Vincenzo Mantova

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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↾Bσ is not increasing.

This is also in contrast with the fact that on Cexp 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.

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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(iR) ⊂ S1(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).

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Summary

Zilber produced a sentence ψ in Lω1,ω(Q) which is uncountably categorical, and conjecturally an axiomatization of Cexp. Its unique model in cardinality 2ℵ0 is called BE. Looking for an analogue of complex conjugation, we found that

  • There are 22ℵ0 involutions on BE.
  • One of them is such that Bσ ∼

= R.

  • However, E is not continuous w.r.t. them.

Thanks for your attention!

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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