Johannes Sprang
In this work we use elementary methods to discuss the question of the minimal number of points with bad reduction over the projective line for elliptic curves E/k(T) which are non-constant resp. have non-constant j-invariant.
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Modular curves and bad reduction
研究证明在不同假设下，所有模曲线上的点对应的椭圆曲线在特定素数处有不良约减，例如定义在有理数域上的椭圆曲线在特定素数处有不良约减。
We investigate the modularity of three non-rigid Calabi-Yau threefolds with bad reduction at 11 which arise as fibre products of rational elliptic surfaces. For this purpose, we apply a method by Serre to compare two-dimensional 2-adic Galois representations with uneven trace. Hereby, we associate two of the threefolds (or, more precisely, a two-dimensional piece of their middle cohomology group) to newforms of weight 4 and level 22 and 55, respectively. This enables us to compute the zeta-functions of these varieties up to the Euler factors of the bad primes.
We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. We study minimality and semi-stability, considering what conditions imply minimality (Theorem 4.4) and whether semi-stable models and presentations are minimal, proving results in the degree two case (Theorems 4.6, 4.7). We prove the singular reduction of a semi-stable pre…
We construct $13$ projective $\mathbb{Q}$-factorial Fano toric varieties and show that for any minimal rational elliptic surface $X$ there is one such toric variety $Z_X$ and a divisor class $δ_X\in {\rm Cl}(Z_X)$ such that the number of $(-1)$-curves of $X$ equals the dimension of the Riemann-Roch space of $δ_X$. As an application we give the number of $(-1)$-curves of any such elliptic fibration of Halphen index $2$.
Lame curves are a particular class of elliptic curves (with a torsion point attached to them) which naturally arise when studying Lame operators with finite monodromy. They can be realized as covers of the projective line unramified outside three points and can be defined over number fields. This paper investigates their p-adic properties. The main ingredient is formal/rigid geometry and in particular the use of p-adic theta functions. As a consequence, we can completely enumerate Lamé curves with bad reduction (by giving, among the others, the p-adic valuation of their j-invariant) and descri…
Let $X$ and $Y$ be smooth projective varieties over $\C$. We say that $X$ and $Y$ are \emph{D-equivalent} (or, $X$ is a \emph{Fourier--Mukai partner} of $Y$) if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories. The aim of this short note is to find an example of mutually D-equivalent but not isomorphic relatively minimal elliptic surfaces.
We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators…
We explicitly construct Brill--Noether general $K3$ surfaces of genus $4,6$ and $8$ having the maximal number of elliptic pencils of degrees $3, 4$ and $5$, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill--Noether general $K3$ surfaces of genus $4$ and $6$ without stable Lazarsfeld--Mukai bundles of minimal $c_2$.
The classification of reflexive modules of rank one over rational and minimally elliptic singularities
本文分类了有理和极小椭圆奇点上的秩一自反模，确定了特殊与平坦模，证明Behnke猜想，并推广McKay对应及相关结果。
Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties
本文研究Calabi-Yau簇椭圆亏格的模微分方程，证明严格六维情形最小阶为四，并为六维超Kähler簇构造7阶方程，刻画非一般解的结构。
We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic fourfold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely $3780$ elliptic curves of minimal degree with fixed (general) $j$-invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-Kähler varieties with fixed $j$-invariant in terms of Gromov--Witten invariants. In $K3[2]$-type this leads to explicit f…
Bad Reduction of Genus Two Curves with complex multiplication
本文研究具有双二次CM域复乘的亏格二曲线，通过正交Shimura簇上的CM环显式描述其雅可比簇，推导稳定坏约化素数的上界并证明其存在性。
Explicit bounds on common projective torsion points of elliptic curves
本文在给定素数处约化类型的温和假设下，基于Raynaud对Manin-Mumford猜想的工作并结合对数代数几何技术，给出了椭圆曲线公共射影挠点数量的显式有效界。
We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second homology. In the case we consider we get the minimal possible genus allowed by the adjunction inequality.
We give an elementary proof of the fact that any elliptic curve $E$ over an algebraically closed non-archimedean field $K$ with residue characteristic $\neq{2,3}$ and with $v(j(E))<0$ admits a tropicalization that contains a cycle of length $-v(j(E))$. We first define an adapted form of minimal models over non-discrete valuation rings and we recover several well-known theorems from the discrete case. Using these, we create an explicit family of marked elliptic curves $(E,P)$, where $E$ has multiplicative reduction and $P$ is an inflection point that reduces to the singular point on the reducti…
In this article, we study the minimal degree [K(T):K] of a p-subgroup T <= E(\overline{K})_tors for an elliptic curve E/K defined over a number field K. Our results depend on the shape of the image of the p-adic Galois representation ρ_{E,p^infty}:Gal_K-->GL(2,Z_p). However, we are able to show that there are certain uniform bounds for the minimal degree of definition of T. When the results are applied to K=Q and p=2, we obtain a divisibility condition on the minimal degree of definition of any subgroup of E[2^n] that is best possible.
Let f : X -> S be any elliptic fibration. If X has dimension 3 and is not uniruled, then X has a minimal model (with terminal singularities) [Mori]. In earlier work we have shown that there exists a birationally equivalent elliptic fibration p: Y -> T such that Y is minimal and a multiple of K_Y can be expressed as the pullback of a divisor from T. Moreover T has at worst quotient singularities; it is not difficult to find examples where T is actually singular. In this paper we describe the singularities of this parameter surface T in the case of no multiple fibers. Although T is not uniquely …
In this paper we determine a minimal set of generators for the Cox rings of extremal rational elliptic surfaces. Moreover, we develop a technique for computing the ideal of relations between them which allows, in all but three cases, to provide a presentation of the Cox ring.
We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such surfaces that admit non-trivial automorphisms that act trivially on the Picard group. As an application, we classify classical Enriques surfaces in characteristic $2$ that admit non-trivial numerically trivial automorphisms.
Infinitesimal Torelli for elliptic surfaces revisited
该研究针对不同j-不变量条件下的无多重纤维极小椭圆曲面，给出无穷小Torelli定理的新证明，并补充了若干新反例，丰富了代数几何中该定理的研究结果。
Non-minimal Elliptic Threefolds at Infinite Distance I: Log Calabi-Yau Resolutions
本文研究椭圆 Calabi-Yau 三维流形在无穷距离极限下的退化，通过基座爆破解消非极小米型奇点，得到对数 Calabi-Yau 空间并分析其几何与 F-理论规范代数。
This work was initially motivated by Miranda's work on elliptic Weierstrass threefolds. Miranda [Mi] describes a smooth equidimensional (flat) model for any elliptic Weierstrass threefold; such models occur naturally in the study of moduli spaces. In this paper we use minimal model theory to link birational maps of log surfaces (log contractions) to equidimensional fibrations of elliptic threefolds. As a corollary we show that an elliptic fibration of positive Kodaira dimension has a minimal model with an equidimensional birationally equivalent elliptic fibration satisfying certain good proper…
An arithmetic intersection for squares of elliptic curves with complex multiplication
本研究针对雅可比簇同构于复乘椭圆曲线平方的亏格2曲线，证明其稳定模型坏约化素数满足$p \leq d/4$，给出基于精细化Humbert不变量的素数计算方法，结合已有公式算出素数在稳定模型判别式中的指数，并完成d<100的显式计算与验证。
We define the notion of a $G$-structure for elliptic curves, where $G$ is a finite 2-generated group. When $G$ is abelian, a $G$-structure is the same as a classical congruence level structure. There is a natural action of $\text{SL}_2(\mathbb{Z})$ on these level structures. If $Γ$ is a stabilizer of this action, then the quotient of the upper half plane by $Γ$ parametrizes isomorphism classes of elliptic curves equipped with $G$-structures. When $G$ is "sufficiently" nonabelian, the stabilizers $Γ$ are noncongruence. As a result we realize noncongruence modular curves as moduli spaces of elli…
We list the elliptic curves defined over $Q(\sqrt 5)$ with good reduction away from 2. There are 368 isomorphism classes. We give a global minimal model for each class.
For each of n=1,2,3 we find the minimal height h^(P) of a nontorsion point P of an elliptic curve E over C(T) of discriminant degree d=12n (equivalently, of arithmetic genus n), and exhibit all (E,P) attaining this minimum. The minimal h^(P) was known to equal 1/30 for n=1 (Oguiso-Shioda) and 11/420 for n=2 (Nishiyama), but the formulas for the general (E,P) were not known, nor was the fact that these are also the minima for an elliptic curve of discriminant degree 12n over a function field of any genus. For n=3 both the minimal height (23/840) and the explicit curves are new. These (E,P) also…
The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2
该研究针对特征2的整体域情形，提出超Weierstrass曲线的新理论，证明椭圆曲线2-Selmer群平均大小的上界，补全了任意固定整体域上椭圆曲线平均秩有限的证明。
We apply the modular approach to computing the topological string partition function on non-compact elliptically fibered Calabi-Yau 3-folds with higher Kodaira singularities in the fiber. The approach consists in making an ansatz for the partition function at given base degree, exact in all fiber classes to arbitrary order and to all genus, in terms of a rational function of weak Jacobi forms. Our results yield, at given base degree, the elliptic genus of the corresponding non-critical 6d string, and thus the associated BPS invariants of the 6d theory. The required elliptic indices are determi…
Belyi's theorem asserts that a smooth projective curve $X$ defined over a number field can be realized as a cover of the projective line unramified outside three points. In this short paper we investigate the bejaviour of the minimal degree of such a cover. More precisely, we start by defining the absolute Belyi degree of X, which only depends on the $\bar{\bold Q}$-isomorphism class of $X$. We then give a lower bound of this invariant, only depending on the stable primes of bad reduction (as defined in the paper) and we show that this bound is sharp. In the second part of the paper, we introd…
We classify elliptic fibrations birational to a nonsingular, minimal cubic surface over a field of characteristic zero. Our proof is adapted to provide computational techniques for the analysis of such fibrations, and we describe an implementation of this analysis in computer algebra.
Congruences of models of elliptic curves
本文证明了在离散赋值环上，当无穷小纤维层级足够大时，椭圆曲线的最小 Weierstrass 模型和最小正则模型的特异纤维可由该无穷小纤维及伽罗瓦群作用唯一确定。
Non-minimal Elliptic Threefolds at Infinite Distance II: Asymptotic Physics
本文通过F理论六维紧化中的非极小奇点，分类无穷距离极限，将其中心纤维解释为带缺陷的去紧致化端点，支持涌现弦猜想。
We give an asymptotic formula for the number of elliptic curves over $\mathbb{Q}$ with bounded Faltings height. Silverman has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in $\mathbb{R}^2$.
We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an "isogeny estimate", providing an explicit bound on the degree of a minimal isogeny between two isogenous elliptic curves. We also give several corollaries of these two results.
Recently Witten proposed to consider elliptic genus in $N=2$ superconformal field theory to understand the relation between $N=2$ minimal models and Landau-Ginzburg theories. In this paper we first discuss the basic properties satisfied by elliptic genera in $N=2$ theories. These properties are confirmed by some fundamental class of examples. Then we introduce a generic procedure to compute the elliptic genera of a particular class of orbifold theories, {\it i.e.\/} the ones orbifoldized by $e^{2πiJ_0}$ in the Neveu-Schwarz sector. This enables us to calculate the elliptic genera for Landau-Gi…
We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve if one imposes appropriate boundary conditions. Algebraic curves of the type thus obtained appear as irreducible components of spectral curves of minimal tori with planar ends in R^3. It appears that these curves coincide with the spectral curves of certain elliptic KP solitons as studied by Krichever.
We consider the stable ruled surface $S_1$ over an elliptic curve. There is a unique foliation on $S_1$ transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.
Quantum Elliptic Cohomology From Four-Dimensional Minimal Supersymmetric Gauge Theories
本文通过四维N=1超对称规范理论的涡旋配分函数提出量子椭圆上同调概念，利用等变局域化计算U(1)理论，并探讨反常与Thom层及虚拟结构层的联系。
Noncommutative geometry of elliptic surfaces
本文用非交换环面和周期连分数的单参数族重新表述射影直线上的椭圆曲面，利用该对应关系研究了此类曲面的Picard数、秩和极小模型，并计算了纤维具有复乘的椭圆曲面的Picard数。
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