scholarly journals Modular orbits at higher genus

2020 ◽  
Vol 2020 (2) ◽  
Author(s):  
Daniel Robbins ◽  
Thomas Vandermeulen
Keyword(s):  
Author(s):  
Benson Farb ◽  
Dan Margalit

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.


2021 ◽  
Vol 9 ◽  
Author(s):  
Pierrick Bousseau ◽  
Honglu Fan ◽  
Shuai Guo ◽  
Longting Wu

Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda _g$ -insertion is related to Gromov-Witten theory of the total space of ${\mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${\mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={\mathbb P}^2$ , we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${\mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${\mathbb P}^2$ , we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${\mathbb P}^2$ in the Nekrasov-Shatashvili limit.


2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.


2020 ◽  
Vol 2020 (5) ◽  
Author(s):  
Lorenz Eberhardt
Keyword(s):  

2019 ◽  
Vol 17 (1) ◽  
pp. 1631-1651
Author(s):  
Ick Sun Eum ◽  
Ho Yun Jung

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.


2006 ◽  
Vol 167 (2) ◽  
pp. 223-294 ◽  
Author(s):  
Mark Haskins ◽  
Nikolaos Kapouleas

1990 ◽  
Vol 41 (2) ◽  
pp. 478-483 ◽  
Author(s):  
R. K. Kaul ◽  
R. P. Malik ◽  
N. Behera

2016 ◽  
Vol 12 (08) ◽  
pp. 2043-2060
Author(s):  
Dania Zantout

We define a global linear operator that projects holomorphic modular forms defined on the Siegel upper half space of genus [Formula: see text] to all the rational boundaries of lower degrees. This global operator reduces to Siegel's [Formula: see text] operator when considering only the maximal standard cusps of degree [Formula: see text]. One advantage of this generalization is that it allows us to give a general notion of cusp forms in genus [Formula: see text] and to bridge this new notion with the classical one found in the literature.


Paleobiology ◽  
2021 ◽  
pp. 1-23
Author(s):  
Pablo S. Milla Carmona ◽  
Dario G. Lazo ◽  
Ignacio M. Soto

Abstract Despite the paleontological relevance and paleobiological interest of trigoniid bivalves, our knowledge of their ontogeny—an aspect of crucial evolutionary importance—remains limited. Here, we assess the intra- and interspecific ontogenetic variations exhibited by the genus Steinmanella Crickmay (Myophorellidae: Steinmanellinae) during the early Valanginian–late Hauterivian of Argentina and explore some of their implications. The (ontogenetic) allometric trajectories of seven species recognized for this interval were estimated from longitudinal data using 3D geometric morphometrics, segmented regressions, and model selection tools, and then compared using trajectory analysis and allometric spaces. Our results show that within-species shell shape variation describes biphasic ontogenetic trajectories, decoupled from ontogenetic changes shown by sculpture, with a gradual decay in magnitude as ontogeny progresses. The modes of change characterizing each phase (crescentic growth and anteroposterior elongation, respectively) are conserved across species, thus representing a feature of Steinmanella ontogeny; its evolutionary origin is inferred to be a consequence of the rate modification and allometric repatterning of the ancestral ontogeny. Among species, trajectories are more variable during early ontogenetic stages, becoming increasingly conservative at later stages. Trajectories’ general orientation allows recognition of two stratigraphically consecutive groups of species, hinting at a potentially higher genus-level diversity in the studied interval. In terms of functional morphology, juveniles had a morphology more suited for active burrowing than adults, whose features are associated with a sedentary lifestyle. The characteristic disparity of trigoniids could be related to the existence of an ontogenetic period of greater shell malleability betrayed by the presence of crescentic shape change.


Sign in / Sign up

Export Citation Format

Share Document