Download Arithmetic Noncommutative Geometry by Matilde Marcolli PDF

Number Theory

By Matilde Marcolli

ISBN-10: 0821838334

ISBN-13: 9780821838334

Marcolli works from her invited lectures brought at a number of universities to handle questions and reinterpret effects and structures from quantity concept and arithmetric algebraic geometry, in general is that they are utilized to the geometry and mathematics of modular curves and to the fibers of archimedean areas of mathematics surfaces and forms. one of many effects is to refine the boundary constitution of sure periods of areas, resembling moduli areas (like modular curves) or arithmetric kinds accomplished by way of compatible fibers at infinity through including barriers that aren't noticeable inside algebraic geometry. Marcolli defines the noncommutative areas and spectral triples, then describes noncommutable modular curves, quantum statistical mechanics and Galois conception, and noncommutative geometry at arithmetric infinity.

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K−1 k0 k1 . . kn . . The equivalence relation of passing to the quotient by the group action is implemented by the invertible (double sided) shift: 1 1 1 −[1/ω + ] 1 , − , ·s . 1 0 ω+ ω + ω − − [1/ω + ] In particular, the closed geodesics in XG correspond to the case where the endpoints ω ± are the attractive and repelling fixed points of a hyperbolic element in the group. This corresponds to the case where (ω, s) is a periodic point for the shift T . The Selberg zeta function is a suitable “generating function” for the closed geodesics in XG .

Kn (α)), qn (α) := Qn (k1 (α), . . , kn (α)) so that pn (α)/qn (α) is the sequence of convergents to α. We also set gn (α) := pn−1 (α) pn (α) qn−1 (α) qn (α) ∈ GL(2, Z). Written in terms of the continued fraction expansion, the shift T is given by T : [k0 , k1 , k2 , . ] → [k1 , k2 , k3 , . ]. 24) can be used to extend the notion of modular symbols to geodesics with irrational ends ([70]). Such geodesics correspond to infinite geodesics on the modular curve XG , which exhibit a variety of interesting possible behaviors, from closed geodesics to geodesics that approximate some limiting cycle, to geodesics that wind around different homology class exhibiting a typically chaotic behavior.

5. Two theorems on limiting modular symbols. The result on the T -invariant measure allows us to study the general behavior of limiting modular symbols. A special role is played by limiting modular symbols {{∗, β}} where β is a quadratic irrationality in R Q. 6. Let g ∈ G be hyperbolic, with eigenvalue Λg corresponding to the attracting fixed point αg+ . Let Λ(g) := | log Λg |, and let be the period of the continued fraction expansion of β = αg+ . Then {{∗, β}}G = = 1 λ(β) k=1 {0, g(0)}G Λ(g) {gk−1 (β) · 0, gk−1 (β) · i∞}G .

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