Model theory of a quantum 2-torus Real Multiplication Program - PowerPoint PPT Presentation

Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Model theory of a quantum 2-torus Real Multiplication Program Intuitive Masanori Itai Descriptions joint work with Boris Zilber Details Summary and more Dept of Math Sci, Tokai University Aug 29, 2012 at Yamanakako

References Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real IZxx Masanori Itai, Boris Zilber, Model Theory of a quantum Multiplication Program 2-torus , submitted Intuitive Descriptions M02 Y. Manin, Real Multiplication and Noncommutative Details geometry , arXiv, 2002 Summary and more Z09 B. Zilber, Structural Approximation , preprint, 2009 Z10 B. Zilber, Zariski Geometries , London Math. Soc. Lect Note Ser. 360, Cambridge, 2010

Hilbert 12th Problem Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Let K denote either Multiplication Program 1 Q , or Intuitive Descriptions 2 an imaginary quadratic extension of Q , or Details 3 a real quadratic extenstion of Q . Summary and more Problem (Hilbert 12th) Describe K ab , the maximal abelian extension of K .

Kronecker-Weber (KW) theorem Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Multiplication Program Intuitive Descriptions Q ab = Q ( all roots of unity ) Details Summary and more

Complex multiplication (CM) case Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber √ Real Let K = Q ( − d ) . Then Multiplication Program Intuitive K ab = K ( t ( E K , tors ) , j ( E K )) Descriptions Details E K is the elliptic curve with complex multiplication by O K , Summary and more t is a canonical coordinate of E K / Aut E K ≃ P 1 , j ( E K ) is the absolute invariant.

Real multiplication (RM) case Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Multiplication √ Program Let K = Q ( d ) . Then Intuitive Descriptions K ab = K ( Stark’s numbers ) Details Summary and more (Stark’s conjectures, not yet proven)

Manin’s RM Program Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Multiplication Program Intuitive Use two-dimensional quantum tori corresponding to real Descriptions quadratic irrationalities as a replacement of elliptic curves with Details CM. Summary and more

Model Theorests may make some contributions Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Multiplication Program Intuitive Descriptions Construct quantum tori by model theoretic tools so that we can Details study their algebro-geometric structures. Summary and more

Quantum Tori Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Multiplication Quantum tori are geometric objects associated with Program Intuitive non-commutative algebras A q of unitary operations with q Descriptions generating multiplicative groups. Details Summary and more When q is a root of unity, we have a quantum torus which is a Zariski structure (Zilber’s result).

Example 1: Noncummutative Geometry, Connes Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Consider the algebra generated by P , Q satisfying the Real Heisenberg commutation relation Multiplication Program QP − PQ = i � , Intuitive Descriptions Details where � = h / 2 π and h is Planck’s constant. This algebra is Summary and usually represented by actions on various Hilbert spaces and more its generalizations (known also as rigged Hilbert spaces). This results in calculations in terms of inner products, eigenvectors and eigenvalues of certain operators expressed in terms of P and Q . See the page 39 of Connes’ book.

Example 2: Manin’s quantum plane Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Manin’s quantum plane is the following skew polynomial ring in two indeterminates; Real Multiplication Program O q ( k 2 ) = k ⟨ x , y | xy = qyx ⟩ Intuitive Descriptions Details where k is a field and q is a constant. Generalizing this Summary and definition to algebraic tori we obtain the notion of quantum more torus of rank n as the k -algebra O q (( k × ) n ) with generators x ± 1 , · · · , x ± n with the relation x i x j = qx j x i .

Main theorems of [IZxx] Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Two main theorems proved in [IZxx] are; Multiplication Program 1 The theory of quantum line-bundles is superstable. Intuitive Descriptions 2 With the pairing function, within ( Γ , · , 1 , q ) we can define Details ( Γ , ⊕ , ⊗ , 1 , q ) and ( Γ , ⊕ , ⊗ , 1 , q ) ≃ ( Z , + , · , 0 , 1) . Hence the Summary and theory of the quantum 2-torus (U , V , F ∗ , Γ ) with the pairing more function is undecidable and unstable. In this talk I give a brief overview of [IZxx].

Quantum tori over C Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Multiplication Program Intuitive Descriptions First we give the description of a quantum torus defined over Details the complex numbers C . Summary and more

Quantum tori over C , cnt’d Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Consider a C -algebra A 2 Multiplication q generated by operators Program U , U − 1 , V , V − 1 satisfying Intuitive Descriptions Details VU = qUV Summary and more where q = e 2 π ih with h ∈ R . Let Γ q = q Z be a multiplicative subgroup of C ∗ .

Model theory of a quantum The quantum 2-torus T 2 q ( C ) associated with the algebra A 2 2-torus q Masanori Itai and the group Γ q is the 3-sorted structure (U φ , V φ , C ∗ ) with the joint work with Boris Zilber actions U and V satisfying ( γ ∈ Γ ) Real Multiplication U : u( γ u , v ) �→ γ u u( γ u , v ) Program (1) u( γ u , v ) �→ v u( q − 1 γ u , v ) V : Intuitive Descriptions Details and U : v( γ v , u ) �→ u v( q γ v , u ) Summary and (2) more V : v( γ v , u ) �→ γ v v( γ v , u ) where u( γ u , v ) ∈ U φ , v( γ v , u ) ∈ V φ and a function ⟨· | ·⟩ called the pairing ⟨· | ·⟩ : V φ × U φ → Γ

Intuitive Ideas Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber The intuitive ideas of U , V and operations U and V . Real Both U and V are two dimensional objects. Multiplication Program Both U and V are bases for an ambient module which we Intuitive do not give any formal description in the theory. Descriptions Details The operator U moves each element (vector) of U on its Summary and fibre, say vertically . On the other hand the operator V more moves each element of U to another element of U , say horizontally . The operator V does the same actions on U and V . The pairing function works as an inner product.

Γ -bundles Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Let φ : C ∗ / Γ → C ∗ be a (non-definable) “choice function”. Real Multiplication Put Φ = ran( φ ) . Program We work with Φ 2 . Intuitive Descriptions Consider ( u , v ) ∈ C ∗ × C ∗ . Details Summary and Let more { γ 1 · u( γ 2 u , v ) : ⟨ u , v ⟩ ∈ Φ 2 , γ 1 .γ 2 ∈ Γ } U φ : = (3) { γ 1 · v( γ 2 v , u ) : ⟨ u , v ⟩ ∈ Φ 2 , γ 1 .γ 2 ∈ Γ } V φ : = U φ , V φ are called Γ -bundles.

Γ -bundle over (u , v) Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber C ∗ U ( q − 1 u , v ) C ∗ U ( u , v ) C ∗ U ( qu , v ) C ∗ U ( q n u , v ) Real Multiplication q 2 u( q − 1 u , v ) q 2 u( u , v ) q 2 u( qu , v ) q 2 u( q n u , v ) Program s s s s Intuitive Descriptions · · · · · · · · · q u( q − 1 u , v ) q u( q n u , v ) q u( u , v ) q u( qu , v ) Details s s s s Summary and more u( q − 1 u , v ) u( q n u , v ) u( u , v ) u( qu , v ) s s s s C ∗ × C ∗ / Γ Figure: Γ -bundle over ( u , v ) inside an ambient C -module

Line-bundles Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Consider the following definable set C ∗ U φ . Real Multiplication C ∗ U φ { x · u( γ u , v ) : ⟨ u , v ⟩ ∈ Φ 2 , x ∈ C ∗ , γ ∈ Γ } Program : = (4) Intuitive Descriptions Notice that we have Details Summary and C ∗ U φ ≃ ( C × U φ ) / E (5) more where E is an equivalence relation identifying γ ∈ Γ as an element of C ∗ . We also consider the similar definable set C ∗ V φ . We call C ∗ U φ and C ∗ V φ , line-bundles over C ∗ .

Pairing function -1 Model theory of a quantum 2-torus Masanori Itai joint work with Boris Zilber Real Multiplication Program Consider a function ⟨· | ·⟩ called the pairing function which plays Intuitive as an inner product of two Γ -bundles U φ and V φ : Descriptions Details ( ) ( ) Summary and ⟨· | ·⟩ : V φ × U φ ∪ U φ × V φ → Γ . (6) more