Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient $PGL(2,\bold{Z})\setminus\bold{P}^1(\bold{R})$ should be considered as non-commutative modular curve, and show that the modular complex can be seen as a sequence of $K_0$-groups of the related crossed-product $C^*$-algebras.
This paper is an expanded version of the previous "On the distribution of continued fractions and modular symbols". The main new features are Section 4 on non-commutative geometry and the modular complex and Section 1.2.2 on the Mixmaster Universe.
We study the Kronecker symbol $\left(\frac st\right)$ for the sequence of the convergents $s/t$ of a purely periodic continued fraction expansion. Whereas the corresponding sequence of Jacobi symbols is always periodic, it turns out that the sequence of Kronecker symbols may be aperiodic. Our main result describes the period length in the periodic case in terms of the period length of the sequence of Jacobi symbols and gives a necessary and sufficient condition for the occurrence of the aperiodic case.
In this paper, we prove that almost every translation of $\mathbb{T}^2$ admits a symbolic coding which has linear complexity $2n+1$. The partitions are constructed with Rauzy fractals associated with sequences of substitutions, which are produced by a particular extended continued fraction algorithm in projective dimension $2$. More generally, in dimension $d\geq 1$, we study extended measured continued fraction algorithms, which associate to each direction a subshift generated by substitutions, called $S$-adic subshift. We give some conditions which imply the existence, for almost every direc…
The main goal of this paper is to construct non-commutative Hilbert modular symbols. However, we also construct commutative Hilbert modular symbols. Both the commutative and the non-commutative Hilbert modular symbols are generalizations of Manin's classical and non-commutative modular symbols. We prove that many cases of (non-)commutative Hilbert modular symbols are periods in the sense on Kontsevich-Zagier. Hecke operators act naturally on them. Manin defines the non-commutative modilar symbol in terms of iterated path integrals. In order to define non-commutative Hilbert modular symbols, we…
We define and study generalized Dedekind symbols with values in non--necessarily commutative groups, generalizing constructions of Sh. Fukuhara in [Fu1], [Fu2]. Basic examples of such symbols are obtained by replacing period integrals of modular forms (cf. [Ma1], [Ma2], [Kn1], [Kn2], [ChZ]) by iterated period integrals introduced and studied in [Ma3], [Ma4].
We define iteration over a two dimensional manifold as analog of iteration over a path defined by Chen. We give several applications. Some of them include constructions of non-abelian modular symbol for $SL(3,\Z)$ and for $SL_{2/K}$, where $K$ is a real quadratic field. We construct the latter as generating series of iterated integrals over a two-dimensional manifold of Hilbert modular forms. We give a motivic interpretation of this non-abelian modular symbol. Other applications give motivic interpretations of certain iterated completed Dedekind zeta functions.
It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.
We develop a theory of $p$-adic continued fractions for a quaternion algebra $B$ over $\mathbb Q$ ramified at a rational prime $p$. Many properties holding in the commutative case can be proven also in this setting. In particular, we focus our attention on the characterization of elements having a finite continued fraction expansion. By means of a suitable notion of quaternionic height, we prove a criterion for finiteness. Furthermore, we draw some consequences about the solutions of a family of quadratic polynomial equations with coefficients in $B$.
This paper consists of variations upon the theme of limiting modular symbols. Topics covered are: an expression of limiting modular symbols as Birkhoff averages on level sets of the Lyapunov exponent of the shift of the continued fraction, a vanishing theorem depending on the spectral properties of a generalized Gauss-Kuzmin operator, the construction of certain non-trivial homology classes associated to non-closed geodesics on modular curves, certain Selberg zeta functions and C^* algebras related to shift invariant sets.
This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2,Z) with respect to the standard fundamental domain F = {z=x+iy: -1/2 <= x <= 1/2 and |z|>=1} of PSL(2,Z). The cutting sequence for a vertical geodesic {θ+it: t > 0} is related to a one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space {L,…
Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued fractions, and generalize them to evaluate infinite families (naturally generalizing theirs). Our work complements their beautiful approach, since we use symbolic rather than numeric computations, and we instruct the computer to not only discover such evaluations, but at the same time prove them rigorously.
The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of quaternion algebra once a pair of (positive) integers $x,y$ with $x^2+y^2\equiv-1\mod n$ is given. In this paper, we introduce a new method to find a representation of odd integer $w$ given $x,y$ satisfying the above requirement. This method can avoid the complicated non-commutative structure in quaternion algebra, which is similar to the one we use to obtain …
In a previous paper, I have defined non--commutative generalized Dedekind symbols for classical $PSL(2,Z)$--cusp forms using iterated period polynomials. Here I generalize this construction to forms of real weights using their iterated period functions introduced and studied in a recent article by R.~Bruggeman and Y.~Choie.
We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in $(0,1]$ ordered by denominators. We show convergence to a stable law in a general setting, by proving an estimate with power-saving error term for the associated characteristic function. This extends results of Baladi and Vallée on Gaussian behaviour for costs of moderate growth. We apply our result to obtain the limiting distribution of values of several key examples of quantum modular forms. We show that central values of th…
The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. In all the note a general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers. T…
Extreme value theorem for geodesic flow on the quotient of the theta group
HUN-REN Alfréd Rényi Institute of Mathematics（HUN-REN 阿尔弗雷德·雷尼数学研究所）；Department of Mathematical Sciences, Seoul National University（首尔大学数学科学系）；Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）
We consider new series expansions for variants of the so-termed ordinary geometric square series generating functions originally defined in the recent article titled "Square Series Generating Function Transformations" (arXiv: 1609.02803). Whereas the original square series transformations article adapts known generating function transformations to construct integral representations for these square series functions enumerating the square powers of $q^{n^2}$ for some fixed non-zero $q$ with $|q| < 1$, we study the expansions of these special series through power series generated by Jacobi-type …
The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map x-->{1/x} eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of noncocompact triangle groups of quadratic invariant trace field, we constru…
The divided cell algorithm was introduced by Delone in 1947 to calculate the inhomogeneous minima of binary quadratic forms and developed further by E. S. Barnes and H. P. F. Swinnerton-Dyer in the 1950s. We show how advances of the past fifty years in both symbolic computation and our understanding of homogeneous spectra can be combined to make divided cells more useful for organizing information about inhomogeneous approximation problems. A crucial part of our analysis relies on work of Jane Pitman, who related the divided cell algorithm to the regular continued fraction algorithm. In partic…
We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in \mathbb{C}$ are non-zero and defined such that $|q| < 1$ and $|b/a| < |z| < 1$. If we set the parameters $(a, b) := (q, q^2)$ in these generalized series expansions, then we have a corresponding J-fraction enumerating the sequence of terms $(1-q) / (1-q^{n+1})$ over all integers $n \geq 0$. Thus we are able to define new $q$-series expansions which correspond to the…
