By Matilde Marcolli

Marcolli works from her invited lectures added at numerous universities to deal with questions and reinterpret effects and structures from quantity idea and arithmetric algebraic geometry, usually is that they are utilized to the geometry and mathematics of modular curves and to the fibers of archimedean areas of mathematics surfaces and kinds. one of many effects is to refine the boundary constitution of convinced sessions of areas, reminiscent of moduli areas (like modular curves) or arithmetric kinds accomplished through appropriate fibers at infinity by means of including limitations that aren't obvious inside algebraic geometry. Marcolli defines the noncommutative areas and spectral triples, then describes noncommutable modular curves, quantum statistical mechanics and Galois concept, and noncommutative geometry at arithmetric infinity.

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**Example text**

21) is described by the C ∗ -algebra of the groupoid of the equivalence relation G([0, 1] × P, T ) = {((x, s), m − n, (y, t)) : T m (x, s) = T n (y, t)} with objects G 0 = {((x, s), 0, (x, s))}. 21). The corresponding groupoid C ∗ -algebra C ∗ (G(E, T )) encodes the dynamical properties of the map T on E. 21) on [0, 1] × P is related to the action of the geodesic flow on the horocycle foliation on the modular curves. 2. Arithmetic of modular curves and noncommutative boundary. 21) can be used to describe the geometry of the noncommutative boundary of modular curves.

The reason for this choice of functions is given by the following classical result of L´evy (1929). 1. For a function f as above, we have 1 f (q, q ) . q(q + q ) 0 Sums and integrals here converge absolutely and uniformly. 40) (f, α)dα = 42 2. NONCOMMUTATIVE MODULAR CURVES We can interpret the summing over pairs of successive denominators as a property that replaces modularity, when “pushed to the boundary”. Through this class of functions it is possible recast certain averages related to modular symbols on XG , completely in terms of function theory on the “boundary” space Γ\(P1 (R) × P).

An−1 (α), an (α) + xn (α) ]. The asymptotic behavior of the measures mn is a famous problem on the distribution of continued fractions formulated by Gauss, who conjectured that 1 ? log(1 + x). 35) was only proved by R. Kuzmin in 1928. Other proofs were then given by P. L´evy (1929), K. Babenko (1978) and D. Mayer (1991). The arguments used by Babenko and Mayer use the spectral theory on the Perron–Frobenius operator. 24). The Gauss problem can be formulated in terms of a recursive relation ∞ 1 1 m .