C. Consani, C. Faber
We prove that the moduli space of stable n-pointed curves of genus one and the projector associated to the alternating representation of the symmetric group on n letters define (for n>1) the Chow motive corresponding to cusp forms of weight n+1 for SL(2,Z). This provides an alternative (in level one) to the construction of Scholl.
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The moduli space of twisted stable maps into the stack $B(\Z/m\Z)^2$ carries a natural $S_n$-action and so its cohomology may be decomposed into irreducible $S_n$-representations. Working over $\Spec \Z[1/m]$ we show that the alternating part of the cohomology of one of its connected components is exactly the cohomology associated to cusp forms for $Γ(m)$. In particular this offers an alternative to Scholl's construction of the Chow motive associated to such cusp forms. This answers in the affirmative a question of Manin on whether one can replace the Kuga-Sato varieties used by Scholl with so…
Chow motives of genus one fibrations
本文证明亏格1纤维化曲面与其雅可比纤维化的周氏动机同构，并由此将Bloch-Kas-Lieberman关于几何亏格0非一般型光滑射影曲面的木村有限维性结果推广到任意特征。
We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties $S_K$ corresponding to the group $\mbox{GSp}_{4,F}$ over a totally real field $F$, along with the relative Chow motives $^λ\mathcal{V}$ of abelian type over $S_K$ obtained from irreducible representations $V_λ$ of $\mbox{GSp}_{4,F}$. We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight $λ$ which characterises the absence of the middle weights 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, th…
In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.
We give a natural construction of unramified over Z framed mixed Tate motives, whose periods are the multiple zeta values. Namely, for each convergent multiple zeta-value we define two boundary divisors A and B in the moduli space M_{0,n+3} of stable curves of genus zero. The corresponding multiple zeta-motive is the n-th cohomology of the pair (M_{0,n+3} -A,B).
In this paper, we give a natural construction of mixed Tate motives whose periods are a class of iterated integrals which include the multiple polylogarithm functions. Given such an iterated integral, we construct two divisors $A$ and $B$ in the moduli spaces $\mathcal{\bar{M}}_{0,n}$ of $n$-pointed stable curves of genus 0, and prove that the cohomology of the pair $(\mathcal{\bar{M}}_{0,n}-A,B-B\cap A)$ is a framed mixed Tate motive whose period is that integral. It generalizes the results of A. Goncharov and Yu. Manin for multiple zeta values. Then we apply our construction to the dilogarit…
The Kudla-Millson lift of Siegel cusp forms
研究亏格2 Siegel尖点形式的Kudla-Millson提升的 injectivity，证明在偶格分裂出两个双曲平面且秩足够大时该提升是 injective 的，并引入向量值不定Siegel theta函数推广了Borcherds的结果。
The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s,F) is a product of L-series of a pair of elliptic cusp forms. Two key examples are the forms F defined by the symmetric cube of a non-CM elliptic curve E over the rationals, and those defined by anticyclotomic twists (of even non-zero weight) of the base change of such an E to an imaginary qu…
Let $C$ be an algebraic curve of genus $g \geq 2$ and $M_L$ be the moduli space of rank 2 stable vector bundles on $C$ whose determinants are isomorphic to a fixed line bundle $L$ of degree 1 on $C.$ S. del Bano studied motives of moduli spaces of rank 2 vector bundles on $C$ and computed the motive of $M_L.$ In this note, we prove that his result gives an interesting decomposition of the motive of $M_L.$ This motivic decomposition is compatible with a conjecture of M. S. Narasimhan which predicts semi-orthogonal decomposition of derived category of the moduli space.
We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative geometry. The same technique applies to enumeration of real plane cuspidal curves: We show that, for any fixed $r\ge1$ and $d\ge2r+3$, there exists a generic real $2r$-dimensional linear family of plane curves of degree $d$ in which the number of real $r$-cuspidal curves is asymptotically comparable with the total number of complex $r$-cuspidal curves in the family…
In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $\Q(\sqrt{5})$. In those examples, we identify Hilbert-Siegel eigenforms that are possible lifts from Hilbert eigenforms.
Given a Prym-Teichmüller curve in $\mathcal{M}_3$, this note provides an invariant that sorts the cusp prototypes of Lanneau and Nguyen by component. This can be seen as an analogue of McMullen's genus $2$ spin invariant, although the source of this invariant is different. Moreover, we describe the Galois action on the cusps of these Teichmüller curves, extending the results of Bouw and Möller in genus $2$. We use this to show that the components of the genus $3$ Prym-Teichmüller curves are homeomorphic.
We consider genus 1 enumerative invariants arising from the Smyth-Viscardi moduli space of stable maps from curves with nodes and cusps. We prove that these invariants are equal to the Vakil-Zinger reduced invariants for the quintic threefold, providing a modular interpretation of the latter.
Injectivity of the genus 1 Kudla-Millson lift on locally symmetric spaces
本文证明了当偶不定格分裂出双曲平面与缩放双曲平面时，亏格1的Kudla-Millson提升是单射的，该结果推广了已知结论并给出了正交型局部对称空间上的几何应用。
We give a practical formula for counting irreducible nodal genus-three plane curves that a fixed generic complex structure on the normalization. As an intermediate step, we enumerate rational plane curves that have a $(3,4)$-cusp.
Let $C$ be a smooth projective curve of genus $g\geq 2$ over $\mathbb C$. Fix $n\geq 1$, $d\in {\mathbb Z}$. A pair $(E,ϕ)$ over $C$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $C$ and a section $ϕ\in H^0(E)$. There is a concept of stability for pairs which depends on a real parameter $τ$. Let ${\mathfrak M}_τ(n,d)$ be the moduli space of $τ$-polystable pairs of rank $n$ and degree $d$ over $C$. Here we prove that for a generic curve $C$, the moduli space ${\mathfrak M}_τ(n,d)$ satisfies the Hodge Conjecture for $n \leq 4$. For obtaining this, we prove first that…
Consider the family S of irreducible plane curves of degree n with d nodes and k cusps as singularities. Let W be an irreducible component of S. We consider the natural rational map from W to the moduli space of curves of genus g=(n-1)(n-2)/2-d-k. We define the "number of moduli of W" as the dimension of the image of W with respect to this map. If W has the expected dimension equal to 3n+g-1-k, then the number of moduli of W is at most equal to the min(3g-3, 3g-3+ρ-k), dove ρis the Brill-Neother number of the linear series of degree n and dimension 2 on a smooth curve of genus g. We say that W…
On motives of parabolic Higgs bundles and parabolic connections
本文研究紧 Riemann 曲面上抛物型 Higgs 丛与抛物型联络的模空间，证明一般权重下二者在 Grothendieck 动机类、$E$-多项式及 Voevodsky、Chow 动机上等价，并推广至固定行列式情形。
We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichmüller cusp forms on \overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases.
Mixed Hodge Structures on Generalized Theta Divisors and Graph Motives
本文研究一般化θ除子上的混合Hodge结构，并探讨其与图动机的可能联系。
We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A_3 of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of d…
Motives meet SymPy: studying $λ$-ring expressions in Python
本文开发了基于 SymPy 的 Python 包 motives，用于在 λ-环中符号操作和化简 Chow 动机表达式，并借此验证了 Mozgovoy 关于扭曲 Higgs 丛模空间动机的猜想在秩 2、3 及亏格不超过 18 时成立。
We introduce the BCOV formula for the lattice polarized K3 surfaces. We find that it yields cusp forms expressed by certain eta products for many families of rank 19 lattice polarized K3 surfaces over $\mathbb{P}^{1}$. Moreover, for Clingher-Doran's family of $U\oplus E_{8}(-1)\oplus E_{7}(-1)$-polarized K3 surfaces, we obtain the Igusa cusp forms $χ_{10}$ and $χ_{12}$ from the formula. Inspired by the arithmetic properties of mirror maps studied by Lian-Yau, we also derive the K3 differential operators for all the genus zero groups of type $Γ_{0}(n)_{+}$.
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.
This paper proves the Beilinson-Soul{é} vanishing conjecture for motives attached to the moduli spaces of curves of genus 0 with n marked points. As part of the proof, it is also proved that these motives are mixed Tate. As a consequence of Levine's work, one obtains then well defined categories of mixed Tate motives over the moduli spaces of curves . It is shown that morphisms between moduli spaces forgetting marked points and embedding as boundary components induce functors between those categories and how tangential bases points fit in these functorialities. Tannakian formalism attaches gro…
Noncongruence cusp forms, Veech groups and enumeration
本文证明某些 Veech 群存在权 2 模形式，其尖点傅里叶系数计数分支 Galois 覆盖，并构造非同余尖点形式计数亏格 2 覆盖，类比经典模曲线构造。
We show that, for each $n>0$, there is a family of elliptic surfaces which are covered by the square of a curve of genus $2n+1$, and whose Hodge structures have an action by ${\mathbb Q}(\sqrt{-n})$. By considering the case $n=3$, we show that one particular family of K3 surfaces are covered by the square of genus $7$. Using this, we construct a correspondence between the square of a curve of genus $7$ and a general K3 surface in ${\mathbb P}^4$ with $15$ ordinary double points up to isogeny. This gives an explicit proof of the Kuga-Satake-Deligne correspondence for these K3 surfaces and any K…
We prove arithmetic Riemann-Roch and Hilbert-Samuel type formulae for pointed stable curves. We give applications to volumes of lattices of integral cusps forms for pointed stable curves of genus 0.
We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various "codimension 1" behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety…
We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in the complex projective plane. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular point, having connected link, and the curve is of genus 0. Generalizations apply in the case of multiple singular points.
We show that for $α\in (2/3, 7/10)$, the log canonical model $\bar M_4(α)$ of the pair $(\bar M_4, αδ)$ is isomorphic to the moduli space $\bar M_4^{hs}$ of h-semistable curves, and that there is a birational morphism $Ξ: \bar M_4^{hs} \to \bar M_4(2/3)$ that contracts the locus of curves $C_1\cup_p C_2$ consisting of genus two curves meeting in a node $p$ such that $p$ is a Weierstrass point of $C_1$ or $C_2$. To obtain this morphism, we construct a compact moduli space $\bar M_{2,1}^{hs}$ of pointed genus two curves that have nodes, ordinary cusps and tacnodes as singularity, and prove that …
In this paper we show that Atkin and Swinnerton-Dyer type of congruences hold for weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are nontrivial even for congruence subgroups. On the way we provide an explicit interpretation of the de Rham cohomology groups associated to modular forms in terms of "differentials of the second kind". As an example, we consider the space of cusp forms of weight 3 on a certain genus zero quotient of Fermat curve X^N+Y^N=Z^N. We show that the Galois representat…
Let S be the variety of irreducible sextics with six cusps as singularities. Let W be one of irreducible components of W. Denoting by M_4 the space of moduli of smooth curves of genus 4, the moduli map of W is the rational map from W to M_4 sending the general point of W, corresponding to a plane curve D, to the point of M_4 parametrizing the normalization curve of D. The number of moduli of W is, by definition the dimension of the image of W with respect to the moduli map. We know that this number is at most equal to seven. In this paper we prove that both irreducible components of S have num…
We determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL_n over Q of any given infinitesimal character, for essentially all n <= 8. For this, we compute the dimensions of spaces of level 1 automorphic forms for certain semisimple Z-forms of the compact groups SO_7, SO_8, SO_9 (and G_2) and determine Arthur's endoscopic partition of these spaces in all cases. We also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of GL_n with trivial infinitesimal c…
Stable cuspidal curves and the integral Chow ring of $\overline{\mathscr{M}}_{2,1}$
本研究引入n标记至多尖点稳定曲线的模栈$\tilde{\text{mod}}_{g,n}$，以此确定了$\bar{\text{mod}}_{2,1}$的整周环，并附带得到$\bar{\text{mod}}_{1,2}$的整周环结果。
Punctual Quot schemes and Cohen--Lenstra series of the cusp singularity
本文研究尖点曲线上点的 Quot 概形 motive 生成级数，证明其有理性并在 d≤3 时显式计算，进而通过 Quot 概形与 Cohen–Lenstra 级数的关系支持 Huang 的有理性猜想。
Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields
本文研究全实域上亏格2超椭圆曲线的模2 Galois表示，在其像为$S_5$且在$\mathrm{GSp}_4(\mathbb{F}_2)\cong S_6$下传递时，证明它对应平行权2的Hilbert–Siegel Hecke特征尖点形式。
A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formula to identify eigenforms for real multiplication on genus two Jacobians. We also present evidence that Teichmüller curves admit a rich arithmetic geometry by exhibiting examples with small primes of bad reduction and notable divisors supported at their cusps.
Semistable Reduction of Plane Quartics
研究平面四次曲线稳定模型计算问题，建立抽象稳定模型与GIT - 稳定平面模型联系，证明GIT - 稳定平面模型存在条件，明确稳定模型与主导态射性质，为计算稳定模型提供几何框架。
Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces
该研究证明亏格≥2的光滑射影曲线上，秩与次数互素的李代数胚联络模空间的动机类仅依赖线丛次数，与李代数胚结构及线丛本身无关，且等于对应扭曲希格斯丛模空间的动机类，并推广至E多项式。
The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $Γ$ is located between $Γ_2(4,8)$ and $Γ_2(2,4)$ in such a way the map factors on the related level moduli space $A_Γ$, the new map being injective on $A_Γ$. Satake's compactification $\text{Proj}A(Γ)$ and the desingularization $\text{Proj}S(Γ)$ are also due to be investigated, since the map does not extend to the boundary of the compactification; to aim at this, an algebraic descript…
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…
The finite symplectic group Sp(2g) over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this representation, for various (small) values of the genus and the level, into irreducible representations. As a consequence we obtain uniqueness results for certain modular forms related to the superstring measure, a better understanding of certain modular forms in genus three studied by D'Hoker and Phong as well as a new construction of Miyawaki's cusp form of…
Log MMP Constraints on Curves with Cuspidal Singularities
该研究运用对数极小模型程序证明陈维民提出的单尖有理曲线自交数上界，推广到任意尖点有理曲线并与辛界比较，还证明不存在特定次数、亏格及尖点的平面曲线，回答了相关问题。
Prym-Teichmüller curves $W_D(4)$ constitute the main examples of known primitive Teichmüller curves in the moduli space $\mathcal{M}_3$. We determine, for each non-square discriminant $D>1$, the number and type of orbifold points in $W_D(4)$. These results, together with the formulas of Lanneau-Nguyen and Möller for the number of cusps and the Euler characteristic, complete the topological characterisation of Prym-Teichmüller curves in genus 3. Crucial for the determination of the orbifold points is the analysis of families of genus 3 cyclic covers of degree $4$ and $6$, branched over four poi…
Unirational quasi-hyperelliptic surfaces in characteristic five
本文研究特征5下的单有理拟超椭圆曲面，对其奇异纤维分类，给出算术亏格和典范除子自交数公式，并作为推论通过确定奇异纤维组合、定义方程和截面分类有理拟超椭圆曲面。
Hodge-Elliptic genera, K3 surfaces and Enumerative Geometry
本文将K3曲面的Elliptic亏格推广为Hodge-Elliptic亏格，并据此给出K3 × E精细化BPS不变量的显式乃至闭式计算方法。
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae are derived from lattice point count involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann surfaces. A similar formula for the Masur-Veech volume (though without exp…
We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. {\it cusps}; and to do so, we stratify cusps according to value semigroup. We show that {\it generalized Severi varieties} of maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$ with images of fixed degree and arithmetic genus are often {\it reducible} whenever $n \geq 3$. We also prove that the Severi variety of degree-$d$ maps with a hyperelliptic cusp of delta-invariant $g \ll d$ is of codimension at least $(n-1)g$ inside the space of degree-$d$ holomorphic maps $\mathbb{P}^1 \rightarro…
Large-$n$ asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces
研究大$n$极限下Weil-Petersson随机曲面的谱理论，证明小Laplace特征值个数与$n$线性相关且最优，并给出模空间Weil-Petersson体积的新渐近公式。
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