The first-order theory of finitely generated fields Bjorn Poonen The first-order theory of finitely generated Statements Distinguishing fields F.g. fields fields 3 questions Rumely’s work Preparations Quadratic forms Building anisotropic Bjorn Poonen forms Pfister forms Kronecker dimension Constant field University of California at Berkeley Algebraic dependence Scanlon’s theorem Bi-interpretability (on work of Robinson, Ershov, Rumely, Pop, myself, and Scanlon) Proof Final comments Non-uniformity Antalya Algebra Days X Uniform bi-interpretability June 1, 2008
Distinguishing fields The first-order theory of finitely generated fields Bjorn Poonen Statements Distinguishing fields F.g. fields 3 questions Example Rumely’s work Preparations The first-order sentence Quadratic forms Building anisotropic forms ( ∃ x )( ∃ y ) x 2 + y 2 = − 1 Pfister forms Kronecker dimension Constant field Algebraic dependence holds for every finite field, and hence for every field of Scanlon’s theorem positive characterstic. But it is false for Q . Bi-interpretability Proof Final comments To what extent can we distinguish fields by the truth values Non-uniformity of first-order sentences? Uniform bi-interpretability
The first-order Some fields cannot be distinguished by first-order sentences. theory of finitely generated fields Example Bjorn Poonen Any first-order sentence true for C is true also for any Statements algebraically closed field of characteristic 0. Distinguishing fields F.g. fields 3 questions Rumely’s work Corollary (Lefschetz principle) Preparations Quadratic forms Many theorems of algebraic geometry proved over C using Building anisotropic forms analytic methods automatically transfer to any algebraically Pfister forms Kronecker dimension closed field of characteristic 0 . Constant field Algebraic dependence Scanlon’s theorem Example Bi-interpretability Proof Elementary model theory shows that any first-order sentence Final comments Non-uniformity true for C holds also for any algebraically closed field of Uniform bi-interpretability sufficiently large positive characteristic (depending on the sentence). To hope to be able to distinguish fields, we must restrict the class of fields considered.
Finitely generated fields The first-order theory of finitely generated fields Bjorn Poonen Every field K has a minimal subfield, isomorphic to either Q Statements Distinguishing fields or F p for some prime p . F.g. fields 3 questions Rumely’s work Definition Preparations Quadratic forms Call K finitely generated (f.g.) if it is finitely generated as a Building anisotropic forms field extension of its minimal subfield. Pfister forms Kronecker dimension Constant field Algebraic dependence F.g. fields arise naturally as the function fields of Scanlon’s theorem varieties over number fields and finite fields. Bi-interpretability Proof The theory of transcendence bases shows that every f.g. Final comments Non-uniformity field is a finite extension of Q ( t 1 , . . . , t n ) or Uniform bi-interpretability F p ( t 1 , . . . , t n ), where n is the absolute transcendence degree.
Three questions about the richness of the The first-order theory of finitely generated fields arithmetic of f.g. fields Bjorn Poonen Statements Question (Sabbagh 1980s, Pop 2002) Distinguishing fields F.g. fields Given non-isomorphic f.g. fields K and L, is there a sentence 3 questions Rumely’s work that is true for K and false for L? Preparations Quadratic forms (Previously Duret had asked for the analogue for f.g. Building anisotropic forms extensions of algebraically closed fields, and he and later Pfister forms Kronecker dimension Constant field Pierce proved some cases of this.) Algebraic dependence Scanlon’s theorem Question (P. 2007) Bi-interpretability Proof Given a f.g. field K, is there a sentence that is true for K Final comments Non-uniformity and false for all f.g. fields not isomorphic to K? Uniform bi-interpretability Question (P. 2007) Is every reasonable class of infinite f.g. fields cut out by a single sentence?
Three questions about the richness of the The first-order theory of finitely generated fields arithmetic of f.g. fields Bjorn Poonen Statements Question (Sabbagh 1980s, Pop 2002, proved by Scanlon) Distinguishing fields F.g. fields Given non-isomorphic f.g. fields K and L, is there a sentence 3 questions Rumely’s work that is true for K and false for L? Preparations Quadratic forms (Previously Duret had asked for the analogue for f.g. Building anisotropic forms extensions of algebraically closed fields, and he and later Pfister forms Kronecker dimension Constant field Pierce proved some cases of this.) Algebraic dependence Scanlon’s theorem Question (P. 2007, proved by Scanlon) Bi-interpretability Proof Given a f.g. field K, is there a sentence that is true for K Final comments Non-uniformity and false for all f.g. fields not isomorphic to K? Uniform bi-interpretability Conjecture (P. 2007, still open) Every reasonable class of infinite f.g. fields is cut out by a single sentence.
Reasonable classes of f.g. fields The first-order theory of finitely generated fields Bjorn Poonen Definition (P. 2007, idea due to Hrushovski) Choose a natural bijection between a recursive A ⊂ N and Statements Distinguishing fields { ( r , f 1 , . . . , f m ) : r ∈ N , f 1 , . . . , f m ∈ Z [ x 1 , . . . , x r ] } . F.g. fields 3 questions Construction of Z [ x 1 , . . . , x r ] / ( f 1 , . . . , f m ) yields a map Rumely’s work Preparations Quadratic forms A → { f.g. Z -algebras } . Building anisotropic forms Pfister forms Kronecker dimension The set of a ∈ A such that the corresponding Z -algebra is a Constant field Algebraic dependence domain is a recursive subset B ⊂ A . Construction of the Scanlon’s theorem fraction field yields a map Bi-interpretability Proof Final comments B → { isomorphism classes of f.g. fields } . Non-uniformity Uniform bi-interpretability For any class of f.g. fields, define the set of b ∈ B for which the corresponding field belongs to the class. Call the class reasonable if this subset of B is a first-order definable subset of ( N , + , · ).
Rumely’s work on the global field case The first-order theory of finitely generated fields Bjorn Poonen Definition A global field is a field of one of the following types: Statements Distinguishing fields 1. number field: finite extension of Q F.g. fields 3 questions Rumely’s work 2. global function field: f.g. extension of transcendence Preparations degree 1 over some F p . Quadratic forms Building anisotropic forms Pfister forms Rumely gave a positive answer to all three questions Kronecker dimension Constant field restricted to global fields. Algebraic dependence Scanlon’s theorem A key step was to build on work of Robinson and Bi-interpretability Ershov to give uniform definitions of the family of Proof Final comments valuation subrings. Non-uniformity Uniform Then, for example, he could distinguish number fields bi-interpretability from global function fields by using a first-order sentence saying “The intersection of all valuation rings in K is not a field.”
Quadratic forms The first-order theory of finitely generated fields Definition Bjorn Poonen A quadratic form over a field k in x 1 , . . . , x n is a Statements homogeneous polynomial of degree 2 in k [ x 1 , . . . , x n ]. Distinguishing fields F.g. fields 3 questions Rumely’s work Definition Preparations Quadratic forms A quadratic form q ( x 1 , . . . , x n ) is isotropic if there exist Building anisotropic forms a 1 , . . . , a n ∈ k not all 0 such that Pfister forms Kronecker dimension Constant field Algebraic dependence q ( a 1 , . . . , a n ) = 0 . Scanlon’s theorem Bi-interpretability It is called anisotropic otherwise. Proof Final comments Non-uniformity Example Uniform bi-interpretability x 2 − 3 y 2 over Q is anisotropic. Example x 2 1 + x 2 2 + x 2 over Q is anisotropic. 3
Quadratic forms The first-order theory of finitely generated fields Definition Bjorn Poonen A quadratic form over a field k in x 1 , . . . , x n is a Statements homogeneous polynomial of degree 2 in k [ x 1 , . . . , x n ]. Distinguishing fields F.g. fields 3 questions Rumely’s work Definition Preparations Quadratic forms A quadratic form q ( x 1 , . . . , x n ) is isotropic if there exist Building anisotropic forms a 1 , . . . , a n ∈ k not all 0 such that Pfister forms Kronecker dimension Constant field Algebraic dependence q ( a 1 , . . . , a n ) = 0 . Scanlon’s theorem Bi-interpretability It is called anisotropic otherwise. Proof Final comments Non-uniformity Example Uniform bi-interpretability x 2 − 3 y 2 over Q is anisotropic. Example x 2 1 + x 2 2 + x 2 3 + x 2 over Q is anisotropic. 4
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