Arvid Perego
We give a geometrical criterion to determine when a quaternion algebra over the function field of a stable elliptic surface X is an Azumaya algebra over X.
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527 elliptic fibrations on Enriques surfaces
本文证明每个 Enriques 曲面在适当多重性计数下都恰好有 527 个椭圆纤维化同构类，并通过构造模空间研究遗忘映射的分歧行为，将其与 Morrison-Kawamata 锥猜想中的双曲体积联系起来。
Elliptic Exceptional Belyi Coverings
本文研究亏格1紧黎曼曲面上由分歧方案唯一确定的椭圆例外Belyi覆盖，并通过Maple算法给出次数不超过12的完整计算结果表。
The quantum integrable hierarchy for the Gromov-Witten theory of elliptic curves
本文提出通过椭圆曲线的Gromov-Witten理论构造量子可积层次，并首次展示包含费米子场的协同场论中的非平凡量子可积层次实例。
We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space, one for the ramification divisor and another for the rational curve divisor. In the course of the proof, we further develop the theory of integral affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types ${\bf A_n}$ ($n\ge1$), ${\bf C_n}$ ($n\ge0$) and …
Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.
Faber's socle intersection numbers via Gromov--Witten theory of elliptic curve
本文通过改进Oberdieck-Pixton的工作，重新证明了Faber在模空间tautological环中关于socle交数的公式，并揭示了新的tautological关系。
We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension are disjoint.
In this paper, we give an analogue of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic. As an application, we obtain an analogue of Jorgensen's inequality in 2-dimensional Mobius group of the above case.
Moduli of Higgs bundles over the two punctured elliptic curve
该研究通过构建双覆盖映射，分析了椭圆曲线上有两个极点的Higgs丛模空间的奇点结构及Hitchin映射的奇纤维特性。
Brauer groups of conic bundles over elliptic curves
本文研究数域上椭圆曲线的正则圆锥丛的Brauer群，在奇异纤维位于2-可除点且分歧域同构的条件下给出显式计算，并应用于一类Châtelet型曲面，进而分析实位点产生的Brauer-Manin障碍对弱逼近的影响。
A non-linear differential equation for the periods of elliptic surfaces
本文推导了椭圆面周期的非线性偏微分方程，并证明了其周期映射的通用微分Torelli定理。
Non-split Cartan curves and a characterization of fake elliptic curves
该文针对虚二次域上的无潜在CM且L-函数为平方的阿贝尔曲面，结合非分裂Cartan模曲线的二次点研究，给出了其具有四元数乘法的刻画条件，部分情形为无条件结论。
We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck's dessins d'enfants. In the particular case of the index 36 subgroups, the corresponding Calabi-Yau threefolds are identified, in analogy with the index 24 cases being associated with K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over P^1 as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative.
A K3 surface is a quaternionic analogue of an elliptic curve from a view point of moduli of vector bundles. We can prove the algebraicity of certain Hodge cycles and a rigidity of curve of genus eleven and gives two kind of descriptions of Fano threefolds as applications. In the final section we discuss a simplified construction of moduli spaces.
Triangulations of non-archimedean curves, semi-stable reduction, and ramification
本文利用 Berkovich 曲线的最小三角剖分研究非阿基米德曲线的半稳定约化，证明最小三角剖分重数的最小公倍数整除 Galois 扩张次数，并在驯顺情形下给出 Saito 定理的新证明及椭圆曲线和野分歧情形的推广分析。
We prove a higher dimensional generalization of Gross and Zagier's theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields $K$ and $K^\prime$, when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross-Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in $B_{p, \infty}$, the definite quaternion a…
The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their $p$-torsion Galois representations is one-to-one over function fields of small genus complex curves for sufficiently large $p$ relative to the genus.
Nakajima-extremal Artin--Schreier covers of ordinary elliptic curves in characteristic $2$
本文在特征2中构造亏格q+1的普通双椭圆曲线，其自同构群达到极值4q，实现Giulietti-Korchmáros分类中的情形(ib)，并给出该情形的完全刻画与参数化。
Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over Q without complex multiplication and χis a ring class character such that L(E/K,χ,1) is not 0 we prove a Gross-Zagier type formula relating Darmon points to a suitably defined algebraic part of L(E/K,χ,1); this generalizes results of Bertolini, Darmon and Dasgupta to the case of division quaternion algebr…
The genus of division algebras over discrete valued fields
本文研究离散赋值域上除代数的亏格，建立了与剩余代数亏格及分歧数据的关系，证明四元数情形在多种域上的平凡性，并联系整体域上曲线函数域的亏格问题与雅可比簇的 $2$-挠，给出了椭圆曲线函数域上亏格大小的更优界及计算方法。
For the study of the Mordell-Weil group of an elliptic curve ${\bf E}$ over a complex function field of a projective curve $B$, the first author introduced the use of differential-geometric methods arising from Kähler metrics on $\mathcal H \times \mathbb C$ invariant under the action of the semi-direct product ${\rm SL}(2,\mathbb R) \ltimes \mathbb R^2$. To a properly chosen geometric model $π: \mathcal E \to B$ of ${\bf E}$ as an elliptic surface and a non-torsion holomorphic section $σ: B \to \mathcal E$ there is an associated ``verticality'' $η_σ$ of $σ$ related to the locally defined Bett…
Moduli of surfaces fibered in log Calabi-Yau pairs
本文研究容许 log Calabi-Yau 纤维化的曲面对的模空间，利用稳定约化结果给出 KSBA 紧致化边界的显式描述，并应用于双度除子、K3 曲面和椭圆曲面等情形。
Number of Kummer structures and Moduli spaces of generalized Kummer surfaces
本文研究广义Kummer曲面的Kummer结构数量与模空间分类，发现代数情形下数量与Shimura曲线上3阶椭圆点相关，非代数情形仅一个结构，并证明相关四元数序由判别式决定。
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