Martin Moeller
We show that for each genus there are only finitely many algebraically primitive Teichmueller curves C, such that i) C lies in the hyperelliptic locus and ii) C is generated by an abelian differential with two zeros of order g-1. We prove moreover that for these Teichmueller curves the trace field of the affine group is not only totally real but cyclotomic.
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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 curv…
Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstrass points and the singularities. Our main tool is the Hodge-theoretic characterisation of Teichmueller curves. We deduce from it a finiteness result for the Mordell-Weil group of the family of Jacobians over a Teichmueller curve.
We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a Hodge-theoretic description of Shimura curves and of Teichmueller curves that reveals similarities and differences of the two classes of curves. The proof of the classification relies on the geometry of square-tiled coverings and on estimating the numerical invariants of these particular fibered surfaces. Finally we translate our main result into a classification of Teichmueller curves with totally de…
We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the computation of divisor classes of Hilbert Jacobi forms on the universal abelian surface over the pseudo-Hilbert modular surface.
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 constrai…
Teichmueller curves are geodesic discs in Teichmueller space that project to an algebraic curve in the moduli space $M_g$. We show that for all $g \geq 2$ Teichmueller curves map to the locus of real multiplication in the moduli space of abelian varieties. Remark that McMullen has shown that precisely for $g=2$ the locus of real multiplication is stable under the $\SL_2(\RR)$-action on the tautological bundle $Ω\M_g$. We also show that Teichmueller curves are defined over number fields and we provide a completely algebraic description of Teichmueller curves in terms of Higgs bundles. As a cons…
In the moduli space M_g of genus g Riemann surfaces, consider the locus RM_O of Riemann surfaces whose Jacobians have real multiplication by the order O in a totally real number field F of degree g. If g = 2 or 3, we compute the closure of RM_O in the Deligne-Mumford compactification of M_g and the closure of the locus of eigenforms over RM_O in the Deligne-Mumford compactification of the moduli space of holomorphic one-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of RM_O Boundary strata of RM_O are parameterized by configurations of el…
Teichmueller curves are geodesic discs in Teichmueller space that project to algebraic curves $C$ in the moduli space $M_g$. Some Teichmüller curves can be considered as components of Hurwitz spaces. We show that the absolute Galois group $G_\QQ$ acts faithfully on the set of these embedded curves. We also compare the action of $G_\QQ$ on $π_1(C)$ with the one on $π_1(M_g)$ and obtain a relation in the Grothendieck-Teichmueller group, seemingly independent of the known ones.
Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apery-like integrality statement for solutions of Pic…
We study a special Teichmueller curve in the moduli space of curves of genus 3 that is intersected by infinitely many other Teichmueller curves. The Veech group of the underlying translation surface is SL_2(Z). All occurring Teichmueller curves are induced by origamis, i.e. unramified coverings of the once punctured torus.
This paper focuses on the interplay between the intersection theory and the Teichmueller dynamics on the moduli space of curves. As applications, we study the cycle class of strata of the Hodge bundle, present an algebraic method to calculate the class of the divisor parameterizing abelian differentials with a non-simple zero, and verify a number of extremal effective divisors on the moduli space of pointed curves in low genus.
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmueller curves with various geometrically defined divisors on moduli spaces of curves.
We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9, -1), (6,3,-1), (3,3,3, -1) in genus three and (12), (9,3), (6,6), (6,3,3) and (3,3,3,3) in genus four. This result is part of a more general investigation of disjointness of Teichmueller curves with divisors of Brill-Noether type on the moduli space of curves. As a result we show that for many strata of quadratic differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for …
We study an example of a Teichmueller curve in the moduli space of algebraic curves of genus 2 coming from an origami S. It is particular in that its points admit the Klein four group as a subgroup of the automorphism group. We give an explicit description of its points in terms of affine plane curves, we show that the Teichmueller curve is a nonsingular, affine curve of genus 0 and we determine the number of cusps in the boundary of the moduli space.
We describe globally nilpotent differential operators of rank 2 defined over a number field whose monodromy group is a nonarithmetic Fuchsian group. We show that these differential operators have an S-integral solution. These differential operators are naturally associated with Teichmueller curves in genus 2. They are counterexamples to conjectures by Chudnovsky--Chudnovsky and Dwork. We also determine the field of moduli of primitive Teichmueller curves in genus 2, and an explicit equation in some cases.
The first part of this paper is a survey on Teichmueller curves and Veech groups, with emphasis on the special case of origamis where much stronger tools for the investigation are available than in the general case. In the second part we study a particular configuration of origami curves in genus 3: A "base" curve is intersected by infinitely many "transversal" curves. We determine their Veech groups and the closure of their locus in M_3, which turns out to be a three dimensional variety and the image of a certain Hurwitz space in M_3.
We prove the following statement, predicted by Clemens' conjecture: A generic quintic threefold contains only finitely many smooth rational curves of degree 12.
We give a simplified and more algebraic proof of the finiteness of the families of Calabi-Yau n-folds with non-vanishing of Yukawa-coupling over a fixed base curve and with fixed degeneration locus. We also give a generalization of this result. Our method is variation of Hodge structure and poly-stability of Higgs bundles.
We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then CH^{i}(X) is finitely generated for i<=4.
In this short note, making use of the recent theory of noncommutative mixed motives, we prove that the Voevodsky's mixed motive of a quadric fibration over a smooth curve is Kimura-finite.
The \emph{canonical degree} of a curve $C$ on a surface $X$ is $K_X\cdot C$. Our main result, is that on a surface of general type there are only finitely many curves with negative self--intersection and sufficiently large canonical degree. Our proof strongly relies on results by Miyaoka. We extend our result both to surfaces not of general type and to non--negative curves, and give applications, e.g. to finiteness of negative curves on a general blow--up of $\mathbb P^ 2$ at $n\geq 10$ general points (a result related to \emph{Nagata's Conjecture}). We finally discuss a conjecture by Vojta co…
Let $E\subseteq \mathbb{P}^2$ be a complex rational cuspidal curve and let $(X,D)\to (\mathbb{P}^2,E)$ be the minimal log resolution of singularities. We prove that $\bar E$ has at most six cusps and we establish an effective version of the Zaidenberg Finiteness Conjecture (1994) concerning Eisenbud-Neumann diagrams of $E$. This is done by analysing the Minimal Model Program run for the pair $(X,\frac{1}{2}D)$. Namely, we show that $\mathbb{P}^2\setminus E$ is $\mathbb{C}^{**}$-fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their se…
The paper contains a fundamental defect in its framework of using the gauge action to study the rigidity problem. As a result, the calculations leading to the main formula is also incorrect.
In this article we prove a finiteness result on the number of log minimal models for $3$-folds in char $p>5$. We then use this result to prove a version of Batyrev's conjecture on the structure of nef cone of curves on $3$-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic $0$. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_X$ of a smooth projective variety in arbitrary characteristic…
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of fixed genus $g\geq 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.
In this paper, we present a characterization for the Hausdorff distance between two given algebraic curves in the $n$-dimensional space (parametrically or implicitly defined) to be finite. The characterization is related with the asymptotic behavior of the two curves and it can be easily checked. More precisely, the Hausdorff distance between two curves $\cal C$ and $\overline{\cal C}$ is finite if and only if for each infinity branch of $\cal C$ there exists an infinity branch of $\overline{\cal C}$ such that the terms with positive exponent in the corresponding series are the same, and recip…
For every fixed genus $g\geq 1$, we consider all quadruples $Q=(w_0,w_1,w_2,d)\in\mathbb{Z}^4_{>0}$ with the property that any smooth degree-$d$ curve embedded in the weighted projective plane $\mathbb{P}^2(w_0,w_1,w_2)$ has genus $g$. We show there are infinitely many quadruples $Q$ satisfying this condition. For every such $Q$, we consider $Z_Q\subseteq M_g$ the locus in the moduli space of all smooth degree-$d$ curves embedded in $\mathbb{P}^2(w_0,w_1,w_2)$. We show that, as $Q$ varies over all these quadruples, there are only finitely many different loci $Z_Q\subseteq M_g$.
We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorize by the retraction through a locally finite subgraph of the curve.
Finiteness properties for Shimura curves and modified diagonal cycles
本文证明给定亏格界的 Shimura 曲线仅有有限多条，并猜想修正对角线环消失仅发生在有限多条曲线上，通过模形式数据与 Sage 计算给出分类结果和显式例子。
A unified finiteness theorem for curves
本文研究代数曲线上的算术性质，证明在可控的降解行为下，伽罗瓦不变点集在自同构作用下有限分割，扩展了Birch-Merriman、Siegel和Faltings的有限性定理。
Given a symplectic three-fold $(M,ω)$ we show that for a generic almost complex structure $J$ which is compatible with $ω$, there are finitely many $J$-holomorphic curves in $M$ of any genus $g\geq 0$ representing a homology class $β$ in $\H_2(M,\Z)$ with $c_1(M).β=0$, provided that the divisibility of $β$ is at most 4 (i.e. if $β=nα$ with $α\in H_2(M,\Z)$ and $n\in \Z$ then $n\leq 4$). Moreover, each such curve is embedded and 4-rigid.
A Unified Finiteness Theorem For Curves Over Function Fields
本文将数域情形的统一有限性结果推广到有限域上的函数域，证明光滑真曲线模型中给定次数且在基上平展的水平除子在自同构群及等平凡情形的Frobenius作用下仅有有限多个。
A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(N), then X is called a new modular curve. We prove that for each integer g at least 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of the new part …
In this paper we prove that the $\mathcal{E}^\dagger_K$-valued cohomology, introduced in [9] is finite dimensional for smooth curves over Laurent series fields $k((t))$ in positive characteristic, and forms an $\mathcal{E}^\dagger_K$-lattice inside `classical' $\mathcal{E}_K$-valued rigid cohomology. We do so by proving a suitable version of the p-adic local monodromy theory over $\mathcal{E}^\dagger_K$, and then using an étale pushforward for smooth curves to reduce to the case of $\mathbb{A}^1$. We then introduce $\mathcal{E}^\dagger_K$-valued cohomology with compact supports, and again prov…
We study the topology of a space parametrizing stable tropical curves of genus g with volume 1, showing that its reduced rational homology is canonically identified with both the top weight cohomology of M_g and also with the genus g part of the homology of Kontsevich's graph complex. Using a theorem of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least 7. This disproves a recent conjecture of Church, Farb, and Putman as well as an older, more general conjecture of Kontsevich. We also give …
It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL_2-type over \Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves.
On finiteness properties of separating semigroup of real curve
研究实代数曲线到射影直线的分离态射所定义的分离半群，证明对于给定亏格g，所有分离半群的集合是有限的。
Finiteness and rigidity of optimal destabilizing subvarieties for the J-equation
该研究证明J方程无光滑解时，紧致Kähler流形上最优去稳定化曲线与除子有限，三维流形上所有此类子簇无条件有限，还刻画其轨迹并证明最优闭链的刚性准则。
Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture
本文研究了极小族的有限性问题，证明了在固定点曲线和极小极种变种下，局部平凡族的同构类有限，并在假设条件成立时进一步证明了投影族的有限性。
Kolyvagin proved that the Tate-Shafarevich group of an elliptic curve over Q of analytic rank 0 or 1 is finite, and that its algebraic rank is equal to its analytic rank. A program of generalisation of this result to the case of some motives which are quotients of cohomology motives of Shimura varieties is offered. We prove some steps of this program, mainly for quotients of $H^7$ of Siegel sixfolds. For example, we "almost" find analogs of Kolyvagin's trace and reduction relations. Some steps of the present paper are new contribution, because they have no analogs in the case of elliptic curve…
Modules over a vertex operator algebra V give rise to sheaves of coinvariants on moduli of stable pointed curves. If V satisfies finiteness and semi-simplicity conditions, these sheaves are vector bundles. This relies on factorization, an isomorphism of spaces of coinvariants at a nodal curve with a finite sum of analogous spaces on the normalization of the curve. Here we introduce the notion of a factorization presentation, and using this, we show that finiteness conditions on V imply the sheaves of coinvariants are coherent on moduli spaces of pointed stable curves without any assumption of …
Finiteness and injectivity of Prym maps for cyclic coverings
本文研究了超椭圆曲线的循环覆盖普林映射的单射性，证明在特定次数条件下映射的单射性，并完成二阶曲线的普林映射研究。
We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the way, we prove a criterion for the finiteness of rational points of degree $5$ on a curve of large genus over a number field using the results of Abramovich--Harris and Faltings on subvarieties of Jacobians.
Finiteness of function field-valued points on exceptional Shimura varieties
研究证明了例外Shimura品种上函数域值点的有限性，并推广到更一般的Shimura品种和素数条件。
By using Christol and Mebkhout's algebrization and finiteness theorem, we prove that in the case of smooth curves, Berthelot's strongest conjecture on the stability of holonomicity is still valid without Frobenius structure but under some non-Liouville type hypotheses.
In the recent paper arXiv:1807.02721, B. Lawrence and A. Venkatesh develop a method of proving finiteness theorems in arithmetic geometry by studying the geometry of families over a base variety. Their results include a new proof of both the $S$-unit theorem and Faltings' theorem, obtained by constructing and studying suitable abelian-by-finite families over $\mathbb{P}^1\setminus\{0,1,\infty\}$ and over an arbitrary curve of genus $\geq 2$ respectively. In this paper, we apply this strategy to reprove Siegel's theorem: we construct an abelian-by-finite family on a punctured elliptic curve to …
The paper's main result is an effective uniform bound for the finiteness statement of the Shafarevich Conjecture over function fields. Several results on the projective geometry of curves are established in the course of the proof. These results should be of independent interest. As a corollary, a uniform effective bound for the Mordell Conjecture over function fields is derived via Parshin's trick.
Oort has conjectured that there do not exist Shimura curves contained generically in the Torelli locus of genus-$g$ curves when $g$ is large enough. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura curves parameterizing principally polarized $g$-dimensional abelian varieties isogenous to $g$-fold self-products of elliptic curves for $g>11$. We also prove that there do not exist Shimura curves contained generically in the Torelli locus of hyperelliptic curves of genus $g>7$. As a consequence, we obtain a finiteness result regarding smooth genus-$g$ curve…
We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is "elementary" in the sense that it does not assume the finiteness of any Shafarevich-Tate group. On the other hand, using Kolyvagin's construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show that there are infinitely many curves of every index over every number field.
We wish to give a short elementary proof of S. Saito's result that the Brauer-Manin obstruction for zero-cycles of degree 1 is the only one for curves, supposing the finiteness of the Tate-Shafarevich-group $\sha^1(A)$ of the Jacobian variety. In fact we show that we only need a conjecturally finite part of the Brauer-group for this obstruction to be the only one. We also comment on the situation in higher dimensions
Towards Lang--Vojta via Degeneration
本文通过研究曲线模空间的算术和几何性质，证明了关于数字段上变种S-积分点有限性和Zariski退化的结果，并提供了几何上不可约除子的例子，回答了关于正常射影品种的积分点有限性问题。
Triple intersections for algebraic curves on the real torus
本文受Corvaja和Zannier问题启发，研究实环面上三条实代数曲线投影的三重交点，对三条直线给出完整分类，对高次曲线用Siegel和Levin定理得有限性结果，并构造无穷多三重交点的显式例子。
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