Orbitwise countings in H(2) and quasimodular forms

Lelièvre, Samuel; Royer, Emmanuel

doi:10.1155/imrn/2006/42151

Cited by 20 publications

(27 citation statements)

References 10 publications

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“…. Volumes of orbit closures and their applications to dynamics are studied in Eskin and Okounkov [24], Eskin, Okounkov and Pandharipandhe [25], Eskin, Masur and Schmoll [22], Eskin, Masur and Zorich [23], Hubert and Lelièvre [35], Lelièvre and Royer [47], and Lelièvre [46].…”

Section: Notes and Referencesmentioning

confidence: 99%

“…Theorem 1.4 calculating .W d 2 / was established in [35] when d is prime and was conjectured for arbitrary d . In [47] Lelièvre and Royer established Theorem 1.4 independently by counting square-tiled surfaces using the theory of quasimodular forms.…”

Section: Notes and Referencesmentioning

confidence: 99%

“…The curve W 25 has six two-cylinder cusps; it meets C .2; 3; 2/ twice and the other four curves C P corresponding to nonterminal prototypes once each. W 25 also has two one-cylinder cusps (see Section 13, McMullen [54], or Lelièvre and Royer [47] for formulas for the number of one-cylinder cusps) corresponding to two intersections with S p maps each curve C P onto a cusp of y X D , and this induces a bijection between the connected components of S P C P and the set of cusps C y X D .…”

Section: Proposition 83 the Normalization Of The Varietymentioning

confidence: 99%

“…X D which is covered by the involution .z; w/ 7 ! .w; z/ of ‫ވ‬ ‫.ވ‬ This involution replaces an Abelian surface A together with a choice of real multiplication, .r / r 2 : and Remark These formulas were established independently by Lelièvre and Royer [47]. A billiards path on P .D/ is a path which is geodesic on the interior of P .D/ and bounces off the walls as a physical billiards ball would (angle of incidence equals angle of reflection).…”

Section: Introductionmentioning

confidence: 99%

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Euler characteristics of Teichmüller curves in genus two

Bainbridge

2007

Geom. Topol.

119

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We calculate the Euler characteristics of all of the Teichmüller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves are naturally embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmüller curve is proportional to the Euler characteristic of the Hilbert modular surface on which it lies.The idea is to use techniques from algebraic geometry to calculate the fundamental classes of these Teichmüller curves in certain compactifications of the Hilbert modular surfaces. This is done by defining meromorphic sections of line bundles over Hilbert modular surfaces which vanish along these Teichmüller curves.We apply these results to calculate the Siegel-Veech constants for counting closed billiards paths in certain L-shaped polygons. We also calculate the Lyapunov exponents of the Kontsevich-Zorich cocycle for any ergodic, SL 2 .‫-/ޒ‬invariant measure on the moduli space of Abelian differentials in genus two (previously calculated in unpublished work of Kontsevich and Zorich). The Jacobian Jac.X / has real multiplication by O D , the unique real quadratic order of discriminant D .There is an Abelian differential ! on X which is an eigenform for real multiplication and has a double zero. where the top map is an embedding, and the vertical map is a two-to-one map sending an Abelian surface A to the unique Riemann surface X 2 M 2 such that Jac.X / Š A.The curve W D is not a Shimura curve on X D ; however, it is a Teichmüller curve, a curve which is isometrically immersed in the moduli space M 2 with respect to the Teichmüller metric. In fact, the curves W D with D nonsquare are all but one of the primitive Teichmüller curves in M 2 , where a Teichmüller curve is said to be primitive if it does not arise from a Teichmüller curve of lower genus by a certain branched covering construction. The curves W D arise from the study of billiards in certain L-shaped polygons, and the study of these curves has applications to the dynamics of billiards in these polygons. Euler characteristics

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Section: Notes and Referencesmentioning

confidence: 99%

Section: Notes and Referencesmentioning

confidence: 99%

Section: Proposition 83 the Normalization Of The Varietymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Euler characteristics of Teichmüller curves in genus two

Bainbridge

2007

Geom. Topol.

119

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“…Various arithmetic and geometric aspects of quasimodular forms have been investigated actively in recent years (see, for example, [2,6,8,10,12]). Given integers m and λ with m ≥ 0, a holomorphic function f on the Poincaré upper halfplane H is a quasimodular form for Γ of weight λ and depth at most m if there are holomorphic functions f 0 , f 1 , .…”

Section: Introductionmentioning

confidence: 99%

Quasimodular Forms and Cohomology

Lee¹

2011

Bull. Aust. Math. Soc.

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We construct linear maps from the spaces of quasimodular forms for a discrete subgroup Γ of SL(2, R) to some cohomology spaces of the group Γ and prove that these maps are equivariant with respect to appropriate Hecke operator actions. The results are obtained by using the fact that there is a correspondence between quasimodular forms and certain finite sequences of modular forms.2010 Mathematics subject classification: primary 11F11, secondary 11F12, 11F25, 11F75.

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