Christian Liedtke, Stefan Schroeer
We analyze the geometry of rational p-division points in degenerating families of elliptic curves in characteristic p. We classify the possible Kodaira symbols and determine for the Igusa moduli problem the reduction type of the universal curve. Special attention is paid to characteristic 2 and 3 where wild ramification and stacky phenomena show up.
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The numerical data of an embedded resolution determine the candidate poles of Igusa's p-adic zeta function. We determine in complete generality which real candidate poles are actual poles in the curve case.
Let S be a Dedekind scheme with field of functions K. We show that if X_K is a smooth connected proper curve of positive genus over K, then it admits a Néron model over S, i.e., a smooth separated model of finite type satisfying the usual Néron mapping property. It is given by the smooth locus of the minimal proper regular model of X_K over S, as in the case of elliptic curves. When S is excellent, a similar result holds for connected smooth affine curves different from the affine line, with locally finite type Néron models.
The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the denominators of the discriminants of the Igusa class polynomials all have the property 1) that they are bounded by d, the absolute value of the discriminant of K, and 2) that they divide d-x^2, for some integer x whose square is less than d. A slightly stronger condition is given in Section 3. Such primes are primes of bad reduction for the genus 2 curve and primes of …
It is known that K3 surfaces S whose Picard number rho (= rank of the Neron-Severi group of S) is at least 19 are parametrized by modular curves X, and these modular curves X include various Shimura modular curves associated with congruence subgroups of quaternion algebras over Q. In a family of such K3 surfaces, a surface has rho=20 if and only if it corresponds to a CM point on X. We use this to compute equations for Shimura curves, natural maps between them, and CM coordinates well beyond what could be done by working with the curves directly as we did in ``Shimura Curve Computations'' (199…
Let O be a maximal order in a totally indefinite quaternion algebra over a totally real number field. In this note we study the locus Q_O of quaternionic multiplication by O in the moduli space A_g of principally polarized abelian varieties of even dimension g with particular emphasis in the two-dimensional case. We describe Q_O as a union of Atkin-Lehner quotients of Shimura varieties and we compute the number of irreducible components of Q_O in terms of class numbers of CM-fields.
On Jacobians of geometrically reduced curves and their Néron models
本文研究任意域上几何约化曲线的雅可比簇结构，将完美域上的若干结构结果推广到非完美域，并据此证明 Bosch-Lütkebohmert-Raynaud 关于 Néron 模型存在性的两个猜想，还给出局部情形下半因子模型的存在性结果。
The auto-Igusa zeta function of a plane curve singularity is rational
研究证明平面曲线奇点芽的自 Igusa 函数具有有理性，并由此给出特征零代数闭域上平面曲线在一点光滑的新判据。
We consider the problem of constructing universal Néron models for families of curves with sections. By applying a construction of the author of universal compactified Jacobians over the moduli stack of reduced curves with markings and a result by J. Kass, we get a positive answer for smooth families of curves with planar singularities over Dedekind schemes.
We prove that Néron models of jacobians of generically-smooth nodal curves over bases of arbitrary dimension are quasi-compact (hence of finite type) whenever they exist. We give a simple application to the orders of torsion subgroups of jacobians over number fields.
We study a variant of the Neron models over curves which is recently found by the second named author in a more general situation using the theory of Hodge modules. We show that its identity component is a certain open subset of an iterated blow-up along smooth centers of the Zucker extension of the family of intermediate Jacobians and that the total space is a complex Lie group over the base curve and is Hausdorff as a topological space. In the unipotent monodromy case, the image of the map to the Clemens extension coincides with the Neron model defined by Green, Griffiths and Kerr. In the ca…
We prove that there exist some stacks, representable over the stack of stable curves, having the following universal property with respect to Néron models of Jacobians. For every one-parameter family of stable curves, with regular total space, the Néron model of the Jacobian of its generic fibre is isomorphic to the base change of the above stacks via the moduli map of the given family. We also obtain a stack compactification of the universal Picard scheme and hence a geometrically meaningful completion of the Néron model of the Jacobian for every family as above.
Modular curves and Néron models of generalized Jacobians
本文利用曲线在离散赋值环上的正则模型，描述广义 Jacobian 的 Néron 模型及其特殊纤维的特征群与分支群，推广 Raynaud 关于通常 Jacobian 的结果，并计算带尖点模数的模曲线情形。
In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to th…
Over the past two years we have improved several of the (Mordell-Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P_i in E(Q) (the previous record was 24, by R.Martin and W.McMillen 2000), and the first example of a curve over Q with Mordell-Weil group isomorphic with (Z/2Z) x Z^18 (the previous rank record for a curve with a 2-torsion point was 15, by Dujella 2002). In these lectures we give some of the background, theory, and computational tools that led to these new reco…
It was conjectured in \cite{Namikawa_ExtendedTorelli} that the Torelli map $M_g\to A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}_g^{\rm cent}$. A counterexample in genus $g=9$ was found in \cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $g\le8$, thus completely solving the problem in every genus.
Let S be a nonsingular projective K3 surface. Motivated by the study of the Gromov-Witten theory of the Hilbert scheme of points of S, we conjecture a formula for the Gromov-Witten theory (in all curve classes) of the Calabi-Yau 3-fold S x E where E is an elliptic curve. In the primitive case, our conjecture is expressed in terms of the Igusa cusp form chi_{10} and matches a prediction via heterotic duality by Katz, Klemm, and Vafa. In imprimitive cases, our conjecture suggests a new structure for the complete theory of descendent integration for K3 surfaces. Via the Gromov-Witten/Pairs corres…
We study endomorphisms of complete Noetherian local rings in the context of motivic integration. Using the notion of an auto-arc space, we introduce the (reduced) auto-Igusa zeta series at a point, which appears to measure the degree to which a variety is not smooth that point. We conjecture a closed formula in the case of curves with one singular point, and we provide explicit formulas for this series in the case of the cusp and the node. Using the work of Denef and Loeser, one can show that this series will often be rational. These ideas were obtained through extensive calculations in Sage. …
Let $X$ be a smooth geometrically connected projective curve of genus two over a complete non-archimedean field $K$. For discretely valued $K$, the first main theorem in \cite{liu} gives a set of criteria on the Igusa invariants of the curve that determine the minimal Berkovich skeleton of $X$ together with its edge lengths and vertex weights. In this paper we use the theory of Berkovich spaces to give a new proof of this theorem that works for arbitrary complete non-archimedean fields. We furthermore interpret the final result in terms of tropical moduli spaces and tropical Igusa invariants. …
Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).T_m, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and T_m is the Hirzebruch-Zagier divisors parameterizing products of elliptic curves with an m-isogeny between them. In this paper, we examine fields not covered by Yang's proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polyn…
Let $S$ be a K3 surface and let $E$ be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold $S \times E$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphi…
Motivic classes of fixed-generators Hilbert schemes of unibranch curve singularities and Igusa zeta functions
本文研究固定生成元数的单分支平面曲线 Hilbert 概形的动机类，给出公式、正性与拓扑不变性，并联系 Igusa zeta 函数。
The multiplicity-one theorem for the superspeciality of curves of genus two
本文证明了Rosenhain形式亏格2曲线超特殊性理想在每个超特殊点重数均为1，利用三变量Lauricella系统推广了Igusa关于椭圆曲线超奇异多项式可分性的经典结果。
Néron models of Jacobians over bases of arbitrary dimension
本文在任意维优良局部因子化基上，通过爆破结点模型构造雅可比簇的 Néron 模型，并给出其可分性的组合判据。
We relate a part of the abelian etale fundamental group of curves over local fields to the component group of the Neron model of the jacobian. We apply the result to the modular curve X_0(p)/Q_p to show that the unramified abelian covering X_1(p) over X_0(p) (Shimura covering) uses up all the possible ramification over the special fiber of X_0(p).
Let $X$ be a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, and let $K'/K$ be a tame extension. We study extensions of the $G = \Gal(K'/K)$-action on $ X_{K'} $ to certain regular models of $X_{K'}$ over $R'$, the integral closure of $R$ in $K'$. In particular, we consider the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model, and obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. We apply these results to study a natural filtr…
We define a new canonical height pairing on the rational points of elliptic curves over global function fields which takes values in the multiplicative group of a completion of the function field. This height serves as an analogue of both the classical Neron-Tate height and analytic p-adic heights. Our main result is that under some general hypotheses this pairing is non-degenerate.
Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\, Y/ \text{Br}_1\, Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron-Severi lattices. Over a field of characteristic $0$, we prove that the existe…
A PEL-type Igusa Stack and the $p$-adic Geometry of Shimura Varieties
本文通过构造一个称为 Igusa 叠的 v-叠，将无限 p-级 Shimura 簇的良好约化区域描述为与仿射格拉斯曼流形的 Schubert 胞腔在 Fargues-Fontaine 曲线上的纤维积，并推广到极小紧化，同时当 Schubert 胞腔替换为有界子叠时恢复积分模型。
The jacobian of the universal curve over $\mathcal{M}_{g,n}$ is an abelian scheme over $\mathcal{M}_{g,n}$. Our main result is the construction of an algebraic space $β\colon \tilde{\mathcal{M}}_{g,n} \rightarrow \bar{\mathcal{M}}_{g,n}$ over which this jacobian admits a Néron model, and such that $β$ is universal with respect to this property. We prove certain basic properties, for example that $β$ is separated, locally of finite presentation, and satisfies a certain restricted form of the valuative criterion for properness. In general, $β$ is not quasi-compact. We relate our construction to …
We construct explicitly moduli curves of polarized supersingular K3 surfaces in characteristic 2 with Artin invariant 2. As an application, we detect a "jump" phenomenon in a family of automorphism groups of supersingular K3 surfaces with a constant Néron-Severi lattice.
We classify webs of minimal degree rational curves on surfaces and give a criterion for webs being hexagonal. In addition, we classify Neron-Severi lattices of real weak del Pezzo surfaces. These two classifications are related to root subsystems of E8.
We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure on the associated minimal Berkovich skeleton. Our main tool is the description of genus two curves via hyperelliptic covers of P^1 with six branch points. Given the valuations of these six points and their differences, our algorithm provides an explicit harmonic 2-to-1 map to a metric tree on six leaves. Second, we use tropical modifications to produce a fait…
Let k be an algebraically closed field of characteristic p. Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a quotient of any X(p^e;N) by any subgroup of ((Z/p^e Z)^* x \SL_2(Z/NZ))/{+-1}. We prove that in any sequence of distinct modular curves over k, the k-gonality tends to infinity. This extends earlier work, in which the result was proved for particular sequences of modular curves, such as X_0(N) for p not dividing N. We give an application to the function field a…
We present a new method for constructing genus 2 curves over a finite field with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discrete-log based cryptosystems. Our algorithm provides an alternative to the traditional CM method for constructing genus 2 curves. For a quartic CM field K with primitive CM type, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem (CRT) and a bound on the denominators to construct the class polyn…
In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for two models of the same type to be k-isomorphic. As a consequence, we obtain an explicit formula for the number of k-isomorphism classes of curves of genus two over a finite field. Moreover, we prove that the field of moduli of any curve coincides with its field of definition, by exhibiting rational models of curves with any prescribed value of their Igusa i…
Specialization of Néron-Severi groups in positive characteristic
本文研究正特征下Néron-Severi群的特化问题，利用晶体上同调的变分Tate猜想等工具证明了几何Picard秩在大量闭点处与generic纤维一致，推广了特征零的相关结果，具有多方面应用。
Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields $k$ to the the specialization theorem for Néron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface $S$, for all vertical curves $S_x$ of a fibration $S \to U \subseteq \mathbf{P}^1_k$ with $x$ from the complement of a sparse subset of $|U|$, the Mordell-Weil rank of an abelian scheme o…
Explicit Uniform Lower Bounds for the Canonical Height on Elliptic Curves over Abelian Extensions
本文通过显式方法为椭圆曲线在阿贝尔扩张上的诺尔-塔特高度提供了一个不依赖基域判别式的下界，改进了 Baker 的定理证明。
We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curve $X$ over a finite field in term of euclidean relationships between the Neron Severi classes in $X\times X$ of the graphs of iterations of the Frobenius morphism. This allows us to recover Ihara's bound, which can be seen as a {\em second order} Weil upper bound, to establish a new {\em third order} Weil upper bound, and using {\tt magma} to produce numerical tables for {\em higher order} Weil upper bounds. We also give some interpretation for the defect…
If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is nonzero. This represents the analogue in the function field setting of a theorem for rational elliptic curves due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof e…
Regular models of modular curves in prime level over ${\mathbb Z}_p^{\mathrm{ur}}$
本文给出了与 ${\mathrm{GL}}_2 ({\mathbb F}_p )$ 的分裂和非分裂 Cartan 子群相关联的模曲线的正则模型，并计算了其雅可比的 Néron 模型在 $p$ 处纤维的连通分支群。
A uniform quantitative Manin-Mumford theorem for curves over function fields
本文提出一个关于函数域上曲线的统一定量Manin-Mumford定理，利用Zhang配对、算术Hodge指数定理和Elkies下界，证明了曲线在雅可比嵌入中的torsion点数量上限及Bogomolov型结果。
Let U be a smooth geometrically connected affine curve over $\mathbb{F}_p$ with compactification X. Following Dwork and Katz, a $p$-adic representation $ρ$ of $π_1(U)$ corresponds to an étale $F$-isocrystal. By work of Tsuzuki and Crew an $F$-isocrystal is overconvergent precisely when $ρ$ has finite monodromy at each $x \in X-U$. However, in practice most F-isocrystals arising geometrically are not overconvergent and have logarithmic growth at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log growth $F$-isocrystals …
Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of $E^g$. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null…
Positive Cones of Parabolic Grassmann Bundle over a curve
本文定义光滑射影簇上抛物向量丛的抛物格拉斯曼丛，推广抛物射影丛构造，并计算其Néron-Severi群及在光滑射影曲线上的nef、伪有效和Mori锥。
In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron--Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibr…
In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. …
We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the Néron-Severi group generated by these sections is not finitely generated. This also gives examples of K3 surfaces over the function field F of a complex curve with Zariski dense F-rational points, whose geometric model is Calabi-Yau.
We establish a $\mathbb{Z}[[t_1,\ldots, t_n]]$-linear derived equivalence between the relative Fukaya category of the 2-torus with $n$ distinct marked points and the derived category of perfect complexes on the $n$-Tate curve. Specialising to $t_1= \ldots =t_n=0$ gives a $\mathbb{Z}$-linear derived equivalence between the Fukaya category of the $n$-punctured torus and the derived category of perfect complexes on the standard (Néron) $n$-gon. We prove that this equivalence extends to a $\mathbb{Z}$-linear derived equivalence between the wrapped Fukaya category of the $n$-punctured torus and the…
Let $R$ be a complete discrete valuation ring with fraction field $K$ and with algebraically closed residue field. Let $X$ be a faithfully flat $R$-scheme of finite type of relative dimension 1 and $G$ be any affine $K$-group scheme of finite type. We prove that every $G$-torsor $Y$ over the generic fibre $X_η$ of $X$ can be extended to a torsor over ${X'}$ under the action of an affine and flat $K$-group scheme of finite type $G'$ where $X'$ is obtained by $X$ after a finite number of Néron blowing ups. Moreover if $G$ is finite and étale (resp. admits a finite and flat model) we find $X'$ su…
Enriched structures on stable curves over fields were defined by Maino in the late 1990s, and have played an important role in the study of limit linear series and degenerating jacobians. In this paper we solve three main problems: we give a definition of enriched structures on stable curves over arbitrary base schemes, and show that the resulting fine moduli problem is representable; we show that the resulting object has a universal property in terms of Néron models; and we construct a compactification of our stack of enriched structures.
We prove the discreteness of algebraic points (with respect to the Neron-Tate height) on a curve of genus greater than one embedded in his jacobian. This result was conjectured by Bogomolov. We also prove the positivity of the self intersection of the admissible dualizing sheaf.
We show that a particular subfunctor of the relative logarithmic Picard functor for families of aligned, log semistable curves over a regular base scheme and smooth over an open dense subscheme of the base is representable by a smooth algebraic space which is slightly more separated than the classical relative Picard functor. We show the existence of its maximal separated quotient and give a new functorial interpretation in terms of logarithmic geometry for the Néron model of the relative Picard functor of the smooth locus when we restrict to transversal pull backs from spectra of discrete val…
Elliptic curves over the rational numbers with semi-abelian reduction and two-division points
本研究分类了整数环上Néron模型半阿贝尔、p=2处良好约化且含非平凡二阶元的有理域椭圆曲线，还刻画了窄型二阶元及多二阶元的情形，并以亏格1带点曲线的Deligne–Mumford栈表述结果。
Positive Cones of the Projectivization of a parabolic vector bundle and Their Products over a Curve
本文计算了光滑复射影曲线上抛物向量束的射影化以及两个抛物射影束的纤维积的正锥，包括Néron-Severi群、nef锥和伪有效锥，并给出了抛物向量束半稳定性的充要条件。
We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form $y^2=f(x^2)$ defined over $\mathbb Q$ with height $h \leq 101$. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space $\mathcal M_2$ for which the field of moduli is a field of definiti…
Let $X$ be a smooth projective variety over the complex numbers. One knows by the Cone Theorem that the closed cone of curves of $X$ is rational polyhedral whenever $c_1(X)$ is ample. For varieties $X$ such that $c_1(X)$ is not ample, however, it is in general difficult to determine the structure of $\bar NE(X)$. The purpose of this paper is to study the cone of curves of abelian varieties. Specifically, the abelian varieties $X$ are determined such that the closed cone $\bar NE(X)$ is rational polyhedral. The result can also be formulated in terms of the nef cone of $X$ or in terms of the sem…
Let $K$ be a local field with algebraically closed residue field and $X_K$ a torsor under an elliptic curve $J_K$ over $K$. Let $X$ be a proper minimal regular model of $X_K$ over the ring of integers of $K$ and $J$ the identity component of the Néron model of $J_K$. We study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality isomorphism on generic fibres. We show that $q$ is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore we interpret our results in relation to Shafarevich's duality theory for…
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