Ronno Das
In this paper we compute the distributions of various markings on smooth cubic surfaces defined over the finite field $\mathbb{F}_q$, for example the distribution of pairs of points, `tritangents' or `double sixes'. We also compute the (rational) cohomology of certain associated bundles and covers over complex numbers.
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Cubic arithmetic-geometric mean and isogenies of Hesse elliptic curves
本文在特征非三的任意域上构造Hesse椭圆曲线的三次同源，对应三次算术几何平均的一步，并应用于特征三中普通椭圆曲线的Serre-Tate典范提升，同时探讨与超几何函数的联系。
In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic forms), rigid Calabi-Yau threefolds and their arithmetic (Diophantine geometry) are intimately related to Hessian polyhedra and their invariants. We construct a Galois representation whose image is a proper subgroup of $W(E_6)$, the Weyl group of the exceptional Lie algebra $E_6$. We give a conjecture about the identification of two different kinds of $L$-function…
Arithmetic Period Map and Complex Multiplication for Cubic Fourfolds
本文构造了三次四次曲面的算术周期映射，并发展了其复乘法理论，证明了CM类型的三次四次曲面定义在阿贝尔扩张上。
It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the prime 2, an old question of Deligne and a recent question of Kudla and Rapoport. We further classify the Mumford-Tate groups of the abelian varieties which arise, and give additional arithmetic applications.
We study dynamical systems induced by birational automorphisms on smooth cubic surfaces defined over a number field $K$. In particular we are interested in the product of non-commuting birational Geiser involutions of the cubic surface. We present results describing the sets of $K$ and $\bar{K}$-periodic points of the system, and give a necessary and sufficient condition for a dynamical local-global property called strong residual periodicity. Finally, we give a dynamical result relating to the Mordell--Weil problem on cubic surfaces.
Rank 2 indecomposable arithmetically Cohen-Macaulay bundles E on a nonsingular cubic surface X in P^3 are classified, by means of the possible forms taken by the minimal graded free resolution of E over P^3. The admissible values of the Chern classes of E are listed and the vanishing locus of a general section of E is studied. Properties of E such as (semi) stability and simplicity are investigated; the number of relevant families is computed together with their dimension.
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over $\mathbb{C}$ there are $27$ lines, and over $\mathbb{R}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $α$ in $L^*/(L^*)^2$. There is an equality in the Grothendieck-Witt group $\operatorname{GW}(k)$ of $k$ $$\sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle α\rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, $$ where $\o…
We establish Manin's conjecture for a cubic surface split over Q and whose singularity type is 2A_2+A_1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three variables in arithmetic progressions. This result is due to Friedlander and Iwaniec (and was later improved by Heath-Brown) and draws on the work of Deligne.
The geometry of secondary terms in arithmetic statistics
本文证明了函数域三次扩张计数中次级项的存在，利用几何参数化方法对光滑曲线计数，推广了有理数域上的结果。
We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.
In this expository note we discuss some arithmetic aspects of the mirror symmetry for plane cubic curves. We also explain how the Picard-Fuchs equation can be used to reveal part of these arithmetic properties. The application of Picard-Fuchs equations in studying the genus zero Gromov-Witten invariants of more general Calabi-Yau varieties and the Weil-Petersson geometry on their moduli spaces will also be discussed.
Prym varieties and cubic threefolds over $\mathbb{Z}$
本文发展了特征2域上的Prym簇与三次三维簇理论，证明了光滑三次三维簇在任意域上的非有理性及Torelli定理，并解决了Deligne关于算术Torelli映射的猜想。
The moduli space of stable real cubic surfaces is the quotient of real hyperbolic four-space by a discrete, nonarithmetic group. The volume of the moduli space is 37π^2/1080 in the metric of constant curvature -1. Each of the five connected components of the moduli space can be described as the quotient of real hyperbolic four-space by a specific arithmetic group. We compute the volumes of these components.
The moduli space of cubic surfaces in complex projective space is known to be isomorphic to the quotient of the complex 4-ball by a certain arithmetic group. We apply Borcherds' techniques to construct automorphic forms for this group and show that these provide an embedding of the moduli space in 9-dimensional projective space. We also show that our automorphic forms directly encode the geometry of cubic surfaces, by showing that each of Cayley's invariants (certain cross-ratios) is simply a quotient of two of our automorphic forms.
Quadrics in arithmetic statistics
本文将圆法与 Bhargava 计数技术结合，提出一种处理由二次不变量定义的仿射空间子簇轨道计数问题的通用方法，并通过计算 2-Selmer 群平均大小验证其有效性。
Generalizing a recent construction of Yang and Yu, we study to what extent one can intersect Hassett's Noether-Lefschetz divisors $\mathcal{C}_d$ in the moduli space of cubic fourfolds $\mathcal{C}$. In particular, we exhibit arithmetic conditions on 20 indexes $d_1,\dots, d_{20}$ that assure that the divisors $\mathcal{C}_{d_1},\dots,\mathcal{C}_{d_{20}}$ all intersect one another. This allows us to produce examples of rational cubic fourfolds with an associated K3 surface with rank 20 Néron-Severi group, i.e. a singular K3 surface.
Exceptional covers of the projective line with genus-one Galois closure: arithmetic forms over finite fields
本文对几何伽罗瓦闭包亏格1的可分不可约驯服例外射影直线映射分类，给出有限域上算术固定域形式的编码准则，获秩2支撑公式等结果，明确特征与障碍的相关性质。
A special cubic fourfold is a smooth hypersurface of degree three and dimension four that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether-Lefschetz divisors C_d in the moduli space C of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the "low-weight cusp form trick" of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of C_d. For example, if d = 6n + 2, then we show that C_d is of general type …
In this paper we show that the moduli space of nodal cubic surfaces is isomorphic to a quotient of a 4-dimensional complex ball by an arithmetic subgroup of the unitary group. This complex ball uniformization uses the periods of certain K3 surfaces which are naturally associated to cubic surfaces. A similar uniformization is given for the covers of the moduli space corresponding to geometric markings of the Picard group or to the choice of a line on the surface. We also give a detailed description of the boundary components corresponding to singular surfaces.
Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for stable cubic surfaces: the moduli space is biholomorphic to a quotient of the compex 4-ball by an explict arithmetic group generated by complex reflections. This identification gives interesting structural information on the moduli space and allows one to locate the points in complex hyperbolic 4-space corresponding to cubic surfaces with symmetry, e.g., the Ferm…
Rational lines on smooth cubic surfaces
本文证明了光滑三次曲面上直线的计数几何受基域算术性质支配，并基于基域的Galois理论为Segre在1949年确定的每种可能直线数目给出了实现的充分判据。
We consider a family of abelian surfaces over $\mathbb{Q}$ arising as Prym varieties of double covers of genus-$1$ curves by genus-$3$ curves. These abelian surfaces carry a polarization of type $(1,2)$ and we show that the average size of the Selmer group of this polarization equals $3$. Moreover we show that the average size of the $2$-Selmer group of the abelian surfaces in the same family is bounded above by $5$. This implies an upper bound on the average rank of these Prym varieties, and gives evidence for the heuristics of Poonen and Rains for a family of abelian varieties which are not …
We contribute to the arithmetic/topology dictionary by relating asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over $\Cb$ in the example of configuration spaces of $n$ points in smooth varieties. To do this, we import the method of homological stability from the realm of topology into the theory of étale cohomology; in particular we give the first examples of stability of étale cohomology groups as Galois representations where the Galois actions are not already explicitly known. We then establish subexponential bounds …
Arithmetic aspects of the Burkhardt quartic threefold
本文研究 Burkhardt 四次三维簇的算术性质，证明其在特征非 3 域上有理、计算有限域 zeta 函数，并显式给出 genus 2 泛曲线模型及 j-平面与 3 阶子群、Hesse 铅笔的关系。
Motivic obstruction to rationality of a general cubic hypersurface in $\mathbb P^5$
研究$\mathbb{P}^5$中一般三次超曲面合理性问题，引入整本质不可分解动机，证明相关动机性质及提升定理，为将三次四维形猜想归结为正特征下算术现象提供可能。
This is a sequel to the paper [Cas]. Here, we extend the methods of Farb-Wolfson using the theory of FI_G-modules to obtain stability of equivariant Galois representations of the etale cohomology of orbit configuration spaces. We establish subexponential bounds on the growth of unstable cohomology, and then use the Grothendieck-Lefschetz trace formula to obtain results on arithmetic statistics for orbit configuration spaces over finite fields. In particular, we show that the average value, across polynomials over F_q, of certain Gauss sums over their roots, stabilizes as the degree goes to inf…
Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the action of the Hecke operators. The goal of that paper was to test the modularity of elliptic curves over F. In the present paper, we complement and extend this prior work in two ways. First, we tabulate more elliptic curves than were found in our prior work by using various heuristics ("old a…
Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics
本文完善了Thorne的猜想，证明了单连Dynkin图导出曲线族的光滑曲线雅可比簇的2-Selmer元素可由相应表示的整轨道参数化，并利用数的几何技术推导出统一的算术统计结果，恢复并推广了前人工作。
Monodromy in the space of symmetric cubic surfaces with a line
研究对称三次曲面中直线的monodromy群，发现其Galois群为克莱因四元群，揭示了对称性对计数问题的影响。
The moduli stack $\mathcal A_2$ of principally polarized abelian surfaces comes equipped with the universal abelian surface $π: \mathcal X_2 \to \mathcal A_2$. The fiber of $π$ over a point corresponding to an abelian surface $A$ in $\mathcal A_2$ is $A$ itself. We determine the $\ell$-adic cohomology of $\mathcal X_2$ as a Galois representation. Similarly, we consider the bundles $\mathcal X_2^n \to \mathcal A_2$ and $\mathcal X_2^{\operatorname{Sym}(n)} \to \mathcal A_2$ for all $n \geq 1$, where the fiber over a point corresponding to an abelian surface $A$ is $A^n$ and $\operatorname{Sym}^…
We construct the moduli space of cubic surfaces which do not admit a Sylvester form as an arithmetic quotient, and determine the graded ring of modular forms of even weights.
We continue the study of a general class of spaces of 0-cycles on a manifold defined and begun by Farb-Wolfson-Wood. Using work of Gadish on linear subspace arrangements, we obtain representation stability for the cohomology of the ordered version of these spaces. We establish subexponential bounds on the growth of unstable cohomology, and the Grothendieck-Lefschetz trace formula then allows us to translate these topological stability phenomena to stabilization of statistics for spaces of 0-cycles over finite fields. In particular, we show that the average value of certain arithmetic quantitie…
Cubic Fourfolds with an Order-$7$ Automorphism
本文研究含7阶自同构的光滑三次四维簇，确定其全部完全自同构群，并刻画两类模空间、边界奇异簇及单值群可公度类。
The moduli space of cubic surfaces and its orbifold fundamental group
本文基于彼得森图的阿廷群得到三次曲面模空间orbifold基本群的新表示，其商群对应$W(E_6)$与$\text{PU}(1,4)$中的算术群，还给出亏格0带任意穿孔映射类群的$\text{Z}$中心扩张。
Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4 and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in PO(4,1). We also derive several new and several old results on the topology of M_0^R. Let M_s^R be the moduli space of real cubic surfaces that are stable in the sense o…
On the arithmetic of rational hypersurfaces in toric varieties
本文研究环簇中特定多重次数超曲面的算术性质，证明其为k-有理且双有理等价于射影空间，推导其有理点分布结果，d=1时对应偶维费马三次曲面。
An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo $p$. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of $\mathbb F_3$. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over $\mathbb F_q$ which attains Perret's and Weil's upper boun…
This article focuses on the study of toric algebraic statistical models which correspond to toric Del Pezzo surfaces with Du Val singularities. A closed-form for the Maximum Likelihood Estimate of algebraic statistical models which correspond to cubic and quartic toric Del Pezzo surfaces with Du Val singular points is given. Also, we calculate the ML degrees of some toric Del Pezzo surfaces of degree less than or equal to six, which equals the degree of the surface in all the case but one, namely the quintic with two points of type $\mathbb{A}_1$.
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