Svetlana Katok

HEEGNER POINTS, CYCLES AND MAASS FORMS

We derive for Hecke-Maass cusp forms on the full modular group a relation between the sum of the form at Heegner points (and integrals over Heegner cycles) and the product of two Fourier coefficients of a corresponding form of half-integral weight. Specializing to certain cycles we obtain the nonnegativity of theL-function of such a form at the center of the critical strip. These results generalize similar formulae known for holomorphic forms.

Symbolic dynamics for the modular surface and beyond

In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.

Coding of closed geodesics after Gauss and morse

Closed geodesics associated to conjugacy classes of hyperbolic matrices in SL(2, ℤ) can be coded in two different ways. The geometric code, with respect to a given fundamental region, is obtained by a construction universal for all Fuchsian groups, while the arithmetic code, given by ‘—’ continued fractions, comes from the Gauss reduction theory and is specific for SL(2, ℤ). In this paper we give a complete description of all closed geodesics for which the two codes coincide.

Rigidity of measurable structure for \(\mathbb Z^d\)-actions by automorphisms of a torus

Abstract. We show that for certain classes of actions of

Higher cohomology for Abelian groups of toral automorphisms

\mathbb Z^d,\ d\ge 2

Closed geodesics, periods and arithmetic of modular forms

, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we construct various examples of

Applications of (a,b)-continued fraction transformations

\mathbb Z^d

Higher cohomology for abelian groups of toral automorphisms II: the partially hyperbolic case, and corrigendum

-actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomorphic. We show that the structure of the centralizer for these actions may or may not serve as a distinguishing measure-theoretic invariant.

Spanning sets for automorphic forms and dynamics of the frame flow on complex hyperbolic spaces

We give a complete description of smooth untwisted cohomology with coefficients in ℝ l for ℤ k -actions by hyperbolic automorphisms of a torus. For 1 ≤ n ≤ k − 1 the nth cohomology trivializes, i.e. every cocycle is cohomologous to a constant cocycle via a smooth coboundary. For n = k a counterpart of the classical Livshitz Theorem holds: the cohomology class of a smooth k -cocycle is determined by periodic data.

Livshitz theorem for the unitary frame flow

is the upper half-plane and F is a discrete cocompact subgroup of SL(2,N) acting by fractional linear transformations. We demonstrate that in this case the periods over closed geodesics play a role somewhat similar to that of Fourier coefficients of modular forms on SL(2,Z) and its congruence subgroups. More specifically, those periods uniquely determine a modular form (Theorem 2). This result is valid for cusp forms on any Fuchsian group of the first kind with or without cusps and is closely related to the study of relative Poincar6 series associated to closed geodesics. For each integer k > 2 and each closed geodesic [7o] we define special cusp forms of weight 2k, called relative Poincar6 series Ok, t~ol and prove that they generate the whole space S2k(F ) of cusp forms (Theorem 1). In w 3 we give an expression for periods of a relative Poincar6 series over closed geodesics in purely geometrical terms through the intersection of the corresponding geodesics (Theorem 3). An application of Theorems 1 and 3 to arithmetic subgroups of SL(2,1R) gives two natural rational structures o n S2k(l N) (Theorem 4). The relative Poincar6 series have been studied for general F by Petersson [18, 19] and Hejhal [5] (g>2). Wolpert [23] gives a basis of S4(F ) for g>2. For SL(2, TZ) the relative Poincar6 series have been studied by Zagier [25], Kohnen [8], Kohnen and Zagier [9], and Kramer [13]. A related problem of constructing cusp forms of weight two associated to closed geodesics has been treated by Kudla and Millson in [143. In connection with the problem of