The ground states, when evaluated at points in the arithmetic subalgebra, give generators of maximal abelian extension of Q. In the cases N =±1, the associated algebra can be interpreted as a limit algebra of the GL2 system.
Number fields and geometry
Recall that any real number can be represented as a continued fraction expansion of the form. This is equivalent to approximating the image of the path CO,α in the modular curve by geodesics whose endpoints approach α.
C ∗ -dynamical systems and KMS states
Recall that in the finite-dimensional setting our dynamics at the observatories was given by the Hamiltonian H. The definition of the Gibbs state in the finite-dimensional setting was based on the use of the trace: tr(e−βH).
Bost-Connes type systems
The arithmetic subalgebra of the Bost-Connes system, denoted A1,Q, is the algebra over Q generated by the elements e(λ) for λ∈Q/Z and µn. First we take the algebra of the space coordinates of all scaled 1-d Q lattices, which is C(ˆZ). In the Q grid image, the time evolution of the Bost-Connes system is given on C(L1/R+) by .
On the C∗(Q/Z) part of the algebra, the restriction of the condition is given by the function onQ/Z. We first extend the original definition of the GL2 system by [CM06b] to other subgroups of GL2(Q) that include the case of SL2(Z) and GL2(Z). We show that the structure of KMS states depends on the Cuntz-Krieger-.
Toeplitz-type system generated by isometries implementing the continued fraction algorithm and in the generalization of the GL2 system. The same mapping can be applied to the arithmetic algebra of the bulk system (which as in the original GL2 case is an algebra of unbounded multipliers consisting of modular functions and Hecke operators). The sub-algebra generated by the images is the arithmetic algebra of the limit system.
The modified GL 2 -system
Our goal here is to adapt this construction to obtain a specialization of the GL2- system in the limit P1(R) and a further specialization for real quadratic fields. Our starting point will be the same GL2 system algebra of [CM06b], so we begin by briefly reviewing that construction in order to use it in our setting. It is convenient, for the setting we consider below, to extend the construction of the GL2 system mentioned above to the case where we replace GL+2(Q) acting on the upper half-plane H by GL2(Q) acting on H± (upper and lower halves) and we consider a subgroup with finite fixed indexGof GL2(Z), where the latter replaces Γ = SL2(Z) = GL+2(Q)∩GL2(ˆZ) in the construction of GL2 - the system.
The Hecke operators are built into the algebra of the GL2 system in this form, through the dependence on the variable g ∈ Γ\GL2(Q)/Γ, see the discussion in [CM08], Proposition 3.87. 2(Z),G,P of the GL2 algebra, where an additional variable is introduced which is responsible for the selection of the finite index subgroup G⊂ GL2(Z) by the composite spacesPα =GL2(Z)αG/G , forα ∈GL2(Q), which for α= 1 includes the composite space P=GL2(Z)/G. The operation of the algebraZΞ of Hecke operators is given by the usual multiplication of adjoints.
This description of the Hecke algebra is obtained directly from the following properties of right cosets and double cosets (Chapter 4 of [Kri90]). We proceed in exactly the same way as in the case of the GL2 system of [CM06b] to construct an arithmetic algebra associated with AΓN,G,PN. This construction and its properties are fully analogous to the original case of the GL2 system described in section 2.3.2 and we refer the reader to [CM06b], [CM08] for details.
Boundary GL 2 -system
We include the side spaces in the construction of the decoupling algebra in the following way. The commutation with the action of Hecke operators can be checked as in the case of the shift TN in Lemma 3.2.3. In the algebra of the Γ = GL2 system on the bulk space, we consider functions f(g, ρ, z) that are invariant under the action of Γ×Γ mapping. and similar for the H×P case).
Next, we perform the operation of the shift operator T in the form of a semigroup cross product algebra. Note that in the construction of the algebra we implicitly use the fact that the action of the semigroup RedN and the action of the Hecke operators (which is embedded in the convolution product B∂,N) commute as in Lemma 3.2.6. If π : A → B(H) is represented as bounded operators on Hilbert space and S` act as isometries on H, compatible with the above relations, then the semigroup C∗-algebra of the cross product (which will also be denoted by A oS) is is the C∗-complement in B(H) of the above algebraic cross product.
The algebra A∂,G,PN can be identified with the semigroup crossed product B∂,N orRedN of the algebra B∂,N of Definition 3.2.7 and the semigroup of. Let E = {Eα} be a collection of subsets Eα ⊂ [0,1] that is invariant under the action of the shift TN of the N-continued fractional expansion. As in [CM06b], we consider the ground states at zero temperature as the weak limit of the Gibbs states for β.
Averaging on geodesics and boundary values
On the level sets of the Lyapunov exponent, the limit is again given by an average of modular symbols associated with the successive terms of the continued fraction expansion. More precisely, as in the previous section, let T denote the shift map of the continued fraction expansion. The Lyapunov exponent of the shift map is given by the limit (when it exists).
However, we obtain uncountable values of limit modular symbols, for example, for all quadratic irrationalities. A more complete analysis of the value of limiting modular symbols was then carried out in [KS07b], where it was shown to be in fact a limiting modular symbol. Let L ⊂[0,1] denote the subset of points for which the Lyapun exponent (3.34) of the proper fraction displacement exists.
Lyapunov and Birkhoff spectra for the displacement of the continuing fracture extension are analyzed in [PW99], [Fan+09]. In a similar way, one can consider the multifractal spectrum associated with the level sets of the limiting modular symbol, as analyzed in [KS07b]. The first step in extending the work of the previous chapter to a higher weight setting is to define the limiting modular symbols for weight greater than 2.
Shokurov modular symbols of higher weight
This has the important property that the functional invariants are given by JG, where JG is a meromorphic function on XG given by the composition of the morphism. The Kuga modular manifold is obtained by taking the Kuga manifold, which can be constructed from any non-singular projective surface over a modular curve, of the modular elliptic surface. Note that this does not give an action directly on H1(XG,Π,(R1Φ∗Q)w), but rather on representations of homology classes as modular symbols. 4.2) Again, due to the additivity property, it is sufficient to consider modular symbols of the form, for α∈Q.
To study the weight-2 limiting modular symbols, we consider a modular curve of the form XG=PGL2(Z)\(H×P) where P=PGL2(Z)/Multiply the associated shift map. The level sets are given by the Lyapunov spectrum of the shift map on the unit interval. It is easy to check that the displacement of the continued fractional expansion is gauge-preserving with respect to the Gaussian gauge.
The operation of the shift operator on modular symbols of higher weight is now described by a relation. Note that again this action is not on H1(XG,Π,(R1Φ∗Q)w) but on representations of homology classes as modular symbols. It was introduced by Kessenbómer and Stratmann in [KS07b] to obtain a more complete description of standard modular symbols and their level set structure.
Twisted continued fraction coding and shift space
However, it is also known that there exists an exceptional set of measure 0 and Hausdorff dimension 1 where λ(β) does not exist ([PW99] Theorem 3). Finally, in the special case that β is a quadratic irrationality (and therefore has a periodic continuous fraction expansion), it is shown ([Mar03] Lemma 2.2) that the limiting modular symbol is given by . In this case, it is also known that the limit λ(β) converges to a positive finite number, so that, in particular, the limiting modular symbol does not vanish.
However, instead of continuing with this setup, let's move to a related setup where we encode each geodesic in the hyperbolic plane using cells of the Farey tessellation. Consider the Farey tessellation of H formed by PSL2(Z) translations of the triangle with vertices at 0,1, and i∞. Traveling along a geodesic in a positive direction, each tile is intersected so that one vertex of the triangle is on one side and two vertices of the triangle are on the other side.
If the single vertex is on the left, we say the visit to the tile is of type L, and if the single vertex is on the right, we say it is of type R. Formulating the limiting modular symbols in terms of the shift space rather than directly in terms of points in R is useful because the shift space (ΣG, σ) is known to be finitely irreducible (Prop 3.1 [KS07b]). Similar to equation 4.8, we have the relation describing the operation of gk(x) on higher weight modular symbols.
Limiting modular symbol for the shift space
This means that there exists a finite set W ⊂Σ∗G, where Σ∗G is the set of finite admissible words in the alphabet Z××EG such that for each a, b∈Z××EG there exists w∈W such that awb∈Σ ∗Mr. We proceed to obtain a similar result as in Equation 4.12, but now with modular symbols of greater weight. The proof follows the strategy described in [KS07b], but here we follow the additional (n, m) coordinate data of the modular symbol of higher weight.
We start by showing it. 4.14) exists if and only if there is a sequence(tn)n∈N tending to infinity such that. Then, we will show that the limit 4.15 does not depend on the particular sequence (tn)n∈N chosen. The main idea is to write the geodesic passing through e1(i∞) and e1(x), in terms of the geodesic associated with continued fractional approximants of x.
Note that the geodesic path oriented from ξn+1 to yn+1 is homologous to the geodesic path from ξn+1 to yn+1. From property 4.2 of modular symbols, and acting directly on gn at the points 0 and i∞ with linear fractional transformations, we obtain. Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory”.
