Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTeichmueller curves in genus three and just likely intersections in $G_m^n x G_a^n$
182
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmueller curves. For the stratum consisting of holomorphic one-forms in genus three with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmueller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmueller curves in terms of cross-ratios of six points on a projective line. These constraints are akin to those that appear in Zilber and Pinks conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of $G_m^n times G_a^n$ (rather than the more standard $G_m^n$). The ambient algebraic group lies outside the scope of Zilbers Conjecture but we are nonetheless able to prove a sufficiently strong height bound. For the generic stratum in genus three, we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of $G_m^9$. These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.
rate research
Read More
We construct a Teichmueller curve uniformized by the Fuchsian triangle group (m,n,infty) for every m<n. Our construction includes the Teichmueller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find Billiard tables that generate these Teichmueller curves. We interprete some of the so-called Lyapunov exponents of the Kontsevich--Zorich cocycle as normalized degrees of some natural line bundles on a Teichmueller curves. We determine the Lyapunov exponents for the Teichmueller curves we construct.
This paper is devoted to the classification of the infinite families of Teichmuller curves generated by Prym eigenforms of genus 3 having a single zero. These curves were discovered by McMullen. The main invariants of our classification is the discriminant D of the corresponding quadratic order, and the generators of this order. It turns out that for D sufficiently large, there are two Teichmueller curves when D=1 modulo 8, only one Teichmueller curve when D=0,4 modulo 8, and no Teichmueller curves when D=5 modulo 8. For small values of D, where this classification is not necessarily true, the number of Teichmueller curves can be determined directly. The ingredients of our proof are first, a description of these curves in terms of prototypes and models, and then a careful analysis of the combinatorial connectedness in the spirit of McMullen. As a consequence, we obtain a description of cusps of Teichmueller curves given by Prym eigenforms. We would like also to emphasis that even though we have the same statement compared to, when D=1 modulo 8, the reason for this disconnectedness is different. The classification of these Teichmueller curves plays a key role in our investigation of the dynamics of SL(2,R) on the intersection of the Prym eigenform locus with the stratum H(2,2), which is the object of a forthcoming paper.
Let $M_{g, n}$ (respectively, $overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n geqslant 0$ if $g geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g geqslant 1$. Finding the precise range of pairs $(g, n)$, where $M_{g, n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $overline{M_{g, n}}$ defined over an arbitrary field $F$ of characteristic $ eq 2$. We show that all $F$-forms of $overline{M_{g, n}}$ are stably rational for $g = 1$ and $3 leqslant n leqslant 4$, $g = 2$ and $2 leqslant n leqslant 3$, $g = 3$ and $1 leqslant n leqslant 14$, $g = 4$ and $1 leqslant n leqslant 9$, $g = 5$ and $1 leqslant n leqslant 12$.
Reducing a 6d fivebrane theory on a 3-manifold $Y$ gives a $q$-series 3-manifold invariant $widehat{Z}(Y)$. We analyse the large-$N$ behaviour of $F_K=widehat{Z}(M_K)$, where $M_K$ is the complement of a knot $K$ in the 3-sphere, and explore the relationship between an $a$-deformed ($a=q^N$) version of $F_{K}$ and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of $F_K$ in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for $a$-deformed $F_K$ for $(2,2p+1)$-torus knots. They suggest a further $t$-deformation based on superpolynomials, which can be used to obtain a $t$-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how $F_K$ transforms under natural geometric operations on $K$ indicates relations to quantum modularity in a new setting.
We investigate the geometry of etale $4:1$ coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work cite{bo} the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on $mathbb{P}^1$ defining the genus 2 curve. The main result of the paper is the fact that, in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
