Abelian surfaces with odd bilevel structure

Convolution and square in abelian groups II

A critical value on an abelian group G of odd order d is a value $λ$ such that the functional equation f$\star$f (2 t) = $λ$f (t)^2 on G has a nonzero solution f. We construct many critical values by using abelian varieties with complex multiplication.

关于2阶Jordán常数的2阶Cremona群在奇特征域上的研究

On p-Jordan constant of Cremona group of rank 2 in odd characteristic

研究2阶Cremona群在奇特征域上正常阿贝尔子群的索引界限，揭示其结构特性。

Canonical maps of surfaces defined by Abelian covers

In this paper, we classified the surfaces whose canonical maps are abelian covers over $\mathbb{P}^2$. Moveover, we construct a new Campedelli surface with fundamental group $\mathbb{Z}_2^{\oplus 3}$ and give defining equations for Perssson's surface and Tan's surfaces with odd canonical degrees explicitly.

Absolutely simple polarized abelian varieties of odd Sophie Germain prime dimension over finite fields with maximal automorphism groups

For each Sophie Germain prime $g \geq 5,$ we construct an absolutely simple polarized abelian variety of dimension $g$ over a finite field, whose automorphism group is a cyclic group of order $4g+2$. We also provide a description on the asymptotic behavior of prime numbers that are related to our construction.

Varieties of sums of powers and moduli spaces of (1,7)-polarized abelian surfaces

We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle.

枚举$\u2119^1$上的奇次超椭圆曲线与阿贝尔曲面

Enumerating odd-degree hyperelliptic curves and abelian surfaces over $\mathbb{P}^1$

针对Venkatesh提出的数论渐近计数中低阶项拓扑本质的问题，研究者通过考察有限域上代数栈惯性栈的算术方面部分作答，并得到$\u2119_q(t)$上拟可容许奇次超椭圆曲线的新精确枚举结果。

阿贝尔簇的德拉姆-贝蒂结构的重量

Weight of the De Rham-Betti Structures of Abelian Varieties

本文证明了任何定义在有理闭包上的阿贝尔簇的德拉姆-贝蒂群必然包含乘法群作为同源群，从而排除了阿贝尔簇在奇数次上同调群中非零德拉姆-贝蒂类的存在性。

阿贝尔簇的挠率与扭形

Torsion and twists of abelian varieties

该研究探讨具好约化阿贝尔簇扭形的挠子群，证明相关二次扩张分歧素数定理，并指出满足Heegner假设的特定虚二次域上有理椭圆曲线无奇阶挠率增长。

Gradients of odd theta functions

We show that a generic principally polarized abelian variety (ppav) is uniquely determined by its theta hyperplanes. These are the non-projectivized version of those studied by Caporaso and Sernesi (see math.AG/0204164), which in a sense are a generalization to ppavs of bitangents of plane quartics, and of hyperplanes tangent to a canonical curves of genus $g$ in $g-1$ points. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli …

On CM abelian varieties over imaginary quadratic fields

In this paper, we associate canonically to every imaginary quadratic field $K=\Bbb Q(\sqrt{-D})$ one or two isogenous classes of CM abelian varieties over $K$, depending on whether $D$ is odd or even ($D \ne 4$). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to $\Bbb Q$. When $D$ is odd or divisible by 8, they are the `canonical' ones first studied by Gross and Rohrlich. We prove that these abelian varieties have the striking property that the vanishing order of their $L$-function at the center …

Moduli of abelian surfaces, symmetric theta structures and theta characteristics

We study the birational geometry of some moduli spaces of abelian varieties with extra structure: in particular, with a symmetric theta structure and an odd theta characteristic. For a $(d_1,d_2)$-polarized abelian surface, we show how the parities of the $d_i$ influence the relation between canonical level structures and symmetric theta structures. For certain values of $d_1$ and $d_2$, a theta characteristic is needed in order to define Theta-null maps. We use these Theta-null maps and preceding work of other authors on the representations of the Heisenberg group to study the birational geom…

The Tate conjecture for K3 surfaces in odd characteristic

We show that the classical Kuga-Satake construction gives rise, away from characteristic 2, to an open immersion from the moduli of primitively polarized K3 surfaces (of any fixed degree) to a certain regular integral model for a Shimura variety of orthogonal type. This allows us to attach to every polarized K3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety. In turn, this reduces the Tate conjecture for K3 surfaces over finitely generated fields of odd characteristic to a version of…

奇特征下超奇异阿贝尔簇的Oort猜想

Oort's conjecture on supersingular abelian varieties in odd characteristic

针对奇特征p>2，构造含a不变量g-2开稠密集的闭子簇，为Oort猜想提供独立证明，补充了Viehmann对a=1轨迹的研究。

$\mathfrak{sl}(2)$-type singular fibres of the symplectic and odd orthogonal Hitchin system

We define and parametrise so-called $\mathfrak{sl}(2)$-type fibres of the $\mathsf{Sp}(2n,\mathbb{C})$- and $\mathsf{SO}(2n+1,\mathbb{C})$-Hitchin system. These are (singular) Hitchin fibres, where the spectral curve induces a two-sheeted covering of a second Riemann surface $Y$. This identifies the $\mathfrak{sl}(2)$-type Hitchin fibres with fibres of an $\mathsf{SL}(2,\mathbb{C})$- respectively $\mathsf{PSL}(2,\mathbb{C})$-Hitchin map on $Y$. We give a stratification of these singular spaces by semi-abelian spectral data, study their irreducible components and obtain a global description of …

The trace formula and the existence of PEL type Abelian varieties modulo p

We show, using the trace formula, that any Newton stratum of a Shimura variety of PEL-type of types (A) and (C) is non-empty at the primes of good reduction. Furthermore we prove conditionally the non-emptiness for Shimura data associated to odd Spin groups. Our results are conditional on Rapoport-Langlands conjecture and Arthur's conjectures on the discrete spectrum. Both these results have been announced by Arthur and Kisin in significant cases.