Modularity of abelian surfaces with Quaternionic Multiplication

Galois action on the $\bar\mathbb{Q}$-isogeny classes of abelian surfaces with quaternionic multiplication

An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a centr…

The field of moduli of quaternionic multiplication on abelian varieties

We consider principally polarized abelian varieties with quaternionic multiplication over number fields and we study the field of moduli of their endomorphisms in relation to the set of rational points on suitable Shimura varieties.

On abelian surfaces with potential quaternionic multiplication

An abelian surface A over a field K has potential quaternionic multiplication if the ring End_\bar K (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is sharp.

具有四元数乘法的阿贝尔曲面上的有理挠点

Rational torsion points on abelian surfaces with quaternionic multiplication

本文研究具有非分裂四元数乘法的阿贝尔曲面的有理挠点，证明其挠子群为12-挠且阶至多18，并在GL₂-型条件下给出完整分类。

Parametrization of abelian $K$-surfaces with quaternionic multiplication

We prove that the abelian $K$-surfaces whose endomorphism algebra is a quaternion algebra are parametrized, up to isogeny, by the $K$-rational points of the quotient of certain Shimura curves by the group of their Atkin-Lehner involutions.

Integral Points on Elliptic Curves and Modularity

In this paper we prove the finiteness of the set of S-integral points of a punctured rational elliptic curve without complex multiplication using the Chabauty-Kim method. This extends previous results of Kim in the complex multiplication case. The key input of our approach is the use of modularity techniques to prove the vanishing of certain Selmer groups involved in the Chabauty-Kim method.

三角线伽罗瓦表示的模性

Modularity of trianguline Galois representations

本文通过伪刚性空间上的三角线模理论，证明了特定三角线伽罗瓦表示的模性提升定理，并构造了修补的四元数本征簇。

具有定四元数乘法的 Jacobi 簇的单值群

Monodromy groups of Jacobians with definite quaternionic multiplication

本文利用带定四元数乘法的超椭圆 Jacobi 簇显式族，证明对任意偶数 $g\ge4$，$\mathbb{Q}$ 上存在无穷多个几何不同源的 $g$ 维阿贝尔簇，其连通单值域严格大于自同态定义域。

具有大次数复乘的K3曲面的显式模性

Explicit modularity of K3 surfaces with complex multiplication of large degree

本研究针对三个猜想具有大次数复乘的K3曲面，通过匹配其超越动机与显式代数Hecke拟特征、复乘阿贝尔三维簇，为Kuga-Satake对应下的相关结论提供了充分证据。

CM K3 曲面的动机模性

Motivic Modularity of CM K3 Surfaces

本文显式构造了定义在数域上具有复乘的K3曲面的代数Hecke特征，并特别给出了与具有复乘的简单阿贝尔曲面相关的Kummer曲面的代数Hecke特征的显式确定。

Examples of genuine QM abelian surfaces which are modular

Let $K$ be an imaginary quadratic field. Modular forms for GL(2) over $K$ are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over $K$ or an associated abelian surface with quaternionic multiplication over $K$. We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to genuine Bianchi newforms, which answers a question posed by J. Cremona as to whether this phenomenon can happen.

On the level p weight 2 case of Serre's conjecture

This brief note only contains a modest contribution: we just fix some inaccuracies in the proof of the prime level weight 2 case of Serre's conjecture given in Khare's preprint "On Serre's modularity conjecture for 2-dimensional mod p representations of G_Q unramified outside p", for the case of trivial character. More precisely, the modularity lifting result needed at a crucial step is the one for the case of a deformation corresponding to a $p$-adic semistable (in the sense of Fontaine) Galois representation attached to a semistable abelian variety, but in Khare's preprint it is applied a le…

Non-abelian Abel's theorems and quaternionic rotation

In order to compute with $l$--adic sheaves or crystals on a line over $\mathbb{F} _q$ a low-technology alternative to the traditional computation with the Hecke operators on the automorphic side could be helpful. A program which has evolved over the years in our discussions with M. Kontsevich centers around the concept that, in the geometric case, there must exist certain multiplication laws on the Galois--representation side that could be thought of as precursors of the automorphic lifts: non-abelian Abel's theorems, and their restrictions to diagonal, Clausen identities. To a varying extent,…

Hurwitz四元数证明拉格朗日与雅可比四平方定理的推广

Generalizing Hurwitz's quaternionic proof of Lagrange's and Jacobi's four-square theorems

本文推广Hurwitz利用四元数序证明四平方定理的技术至数域，通过确定序的因式分解理论与轨道相交条件，结合四元数序类数最新结果，得到对应二次型的普遍性及表示数公式，并证明Götzky四平方定理。

Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications

We prove two formulas in the style of the Gross-Zagier theorem, relating derivatives of L-functions to arithmetic intersection pairings on a unitary Shimura variety. We also prove a special case of Colmez's conjecture on the Faltings heights of abelian varieties with complex multiplication. These results are derived from the authors' earlier results on the modularity of generating series of divisors on unitary Shimura varieties.

计算某些零维 Shimura 曲线上的点数

Counting points on some genus zero Shimura curves

本文利用零维 Shimura 曲线上有界高度点的计数，研究数域 $K$ 上具潜在四元数乘法的阿贝尔曲面数量。

On the modularity level of modular abelian varieties over number fields

Let f be a weight two newform for Gamma_1(N) without complex multiplication. In this article we study the conductor of the absolutely simple factors B of the variety A_f over certain number fields L. The strategy we follow is to compute the restriction of scalars Res_{L/\Q}(B), and then to apply Milne's formula for the conductor of the restriction of scalars. In this way we obtain an expression for the local exponents of the conductor N_L(B). Under some hypothesis it is possible to give global formulas relating this conductor with N. For instance, if N is squarefree we find that N_L(B) belongs…

构造算术西格尔模形式：实乘法阿贝尔曲面的西塔提升与显式方法

Constructing Arithmetic Siegel Modular Forms: Theta Lifting and Explicit Methods for Real Multiplication Abelian Surfaces

该研究提出显式可计算蓝图，利用酉对偶对$(\text{U}(2,2), \text{Sp}_4)$的西塔对应构造实乘法阿贝尔曲面的向量值西格尔模形式，通过局部施瓦茨函数、分歧素数测试向量构造及计算流程，为朗兰兹函子性实例提供路线图。

实数域上具有四元数乘法的阿贝尔曲面的超特殊素数

Superspecial primes for QM abelian surfaces over real number fields

本文将Baba和Granath关于特定四元数乘法阿贝尔曲面超特殊素数的结果，扩展到模域为任意实嵌入数域且弱化局部条件的情形，证明依赖志村曲线Heegner除子相交理论。

The arithmetic of QM-abelian surfaces through their Galois representations

This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with several explicit examples.

具有虚乘法的阿贝尔三维簇的剩余表示的计算

Computing residual representations of abelian threefolds with imaginary multiplication

本文研究虚二次域上具有虚乘法的阿贝尔三维簇的剩余伽罗瓦表示，提出大像判据与高效算法，应用于大量曲线并实现酉半相似群为伽罗瓦群，同时发现新的 Picard 曲线族。

Modular abelian varieties over number fields

The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the zeta function $L(B/K;s)$ is equivalent to the product of zeta functions of non-CM newforms for congruence subgroups $Γ_1(N)$. The characterization involves the structure of End(B), isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illu…

On the local-global divisibility of torsion points on elliptic curves and ${\rm GL}_2$-type varieties

Let $p$ be a prime number and let $k$ be a number field. Let $E$ be an elliptic curve defined over $k$. We prove that if $p$ is odd, then the local-global divisibility by any power of $p$ holds for the torsion points of $E$. We also show with an example that the hypothesis over $p$ is necessary. We get a weak generalization of the result on elliptic curves to the larger family of ${\rm GL}_2$-type varieties over $k$. In the special case of the abelian surfaces $A/k$ with quaternionic multiplication over $k$ we obtain that for all prime $p$, except a finite number depending on $A$, the local-gl…

论G函数的$v$进赋值 III

$G$-functions and quaternionic multiplication in $\mathcal{A}_2$

本研究接续系列前作，探究单参数主极化阿贝尔曲面相关G函数的取值关系，建立其在四元数乘法阿贝尔曲面对应点上的阿基米德与$p$进取值联系，并给出对$\u27e8$2$\u27e9$中Zilber-Pink猜想的应用。

非分裂Cartan曲线与伪椭圆曲线的一个刻画

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。