p-adic 与实三次曲面上的直线

三次超曲面四维簇、Kuznetsov分支与Chow动机

Cubic fourfolds, Kuznetsov components and Chow motives

本文证明了具有Fourier-Mukai导出等价Kuznetsov分支的两个光滑三次超曲面四维簇，其Chow动机作为Frobenius代数对象同构，并推得l-adic上同调Frobenius代数间存在Galois等变同构，同时讨论了与K3曲面导出等价的情形。

Counting rational points on smooth cubic curves

We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are uniform in the sense that they only depend on the rank of the corresponding Jacobian.

Rigid analytic p-adic Simpson correspondence for line bundles

The p-adic Simpson correspondence due to Faltings is a p-adic analogue of non-abelian Hodge theory. The following is the main result of this article: The correspondence for line bundles can be enhanced to a rigid analytic morphism of moduli spaces under certain smallness conditions. In the complex setting, Simpson shows that there is a complex analytic morphism from the moduli space for the vector bundles with integrable connection to the moduli space of representations of a finitely generated group as algebraic varieties. We give a p-adic analogue of Simpson's result.

Cyclic coverings of the $p$-adic projective line by Mumford curves

Exact bounds for the positions of the branch points for cyclic coverings of the $p$-adic projective line by Mumford curves are calculated in two ways. Firstly, by using Fumiharu Kato's *-trees, and secondly by giving explicit matrix representations of the Schottky groups corresponding to the Mumford curves above the projective line through combinatorial group theory.

Memory efficient hyperelliptic curve point counting

In recent algorithms that use deformation in order to compute the number of points on varieties over a finite field, certain differential equations of matrices over p-adic fields emerge. We present a novel strategy to solve this kind of equations in a memory efficient way. The main application is an algorithm requiring quasi-cubic time and only quadratic memory in the parameter n, that solves the following problem: for E a hyperelliptic curve of genus g over a finite field of extension degree n and small characteristic, compute its zeta function. This improves substantially upon Kedlaya's resu…

三次曲面上点类的非结合Moufang环的一个例子

An example of a non-associative Moufang loop of point classes on a cubic surface

本文通过构造一个特定的三次曲面，证明了其点类模$(1-θ)^3$形成的交换Moufang环是非结合的，从而解决了Manin在50多年前提出的关于非结合Moufang环存在性的问题。

Line bundles and p-adic characters

For a certain class of vector bundles E on abelian varieties A over local fields containing all line bundles algebraically equivalent to zero we define a canonical representation of the Tate module of A on the fibre of E in the zero section. This extends an old construction of Tate for line bundles to vector bundles of higher rank. We also compare this construction to the theory of parallel transport for vector bundles on p-adic curves developed in mathAG/0403516. Relations with the Hodge-Tate decomposition are also explained.

$R$-等价于立方面I：已有非平凡普遍等价的情况

$R$-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence

本文研究了具有非平凡普遍等价的立方面，发现2-进数立方面在特定条件下$R$-等价为平凡或指数2，并验证了特定例子的平凡性，同时探讨了生成AI在研究中的应用。

Rapoport-Zink uniformization for the moduli space of polarized K3 surfaces

We compute the image of the $p$-adic period map for polarized K3 surfaces with supersingular reduction. This gives rise to a Rapoport-Zink type uniformization of their moduli space by an explicit open rigid analytic subvariety of a local Shimura variety of orthogonal type. In contrast to the case of Rapoport-Zink uniformization of Shimura varieties and in analogy to the complex case, the uniformizing domain does not carry an action of a $p$-adic Lie group, but only of a discrete subgroup. We briefly sketch how the same arguments can be applied to obtain a uniformization for the moduli space of…

$p$-adic Hurwitz numbers

We introduce stable tropical curves and use these to count covers of the $p$-adic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the covers.

Commutative algebraic groups and $p$-adic linear forms

Let $G$ be a commutative algebraic group defined over a number field $K$ that is disjoint over $K$ to $\mathbb G_a$ and satisfies the condition of semistability. Consider a linear form $l$ on the Lie algebra of $G$ with algebraic coefficients and an algebraic point $u$ in a $p$-adic neighbourhood of the origin with the condition that $l$ does not vanish at $u$. We give a lower bound for the $p$-adic absolute value of $l(u)$ which depends up to an effectively computable constant only on the height of the linear form, the height of the point $u$ and $p$.

On the sheaf cohomology of some $p$-adic period domains with coefficients in certain line bundles

Let $p$ be a prime. This papers aims at investigating sheaf cohomology of a broader class of $p$-adic period domains, other then the Drinfeld's upper half space (cf. \cite{O2}). Concretely, we let $\mathbf{G}$ be a split connected reductive group over $\mathbb{Q}_p$ and restrict our attention to the $p$-adic period domain $\mathscr{F}^{\mathrm{wa}}$ which parametrizes the weakly admissible filtrations on the trivial $\mathbf{G}$-isocrystal inside a complete flag variety $\mathscr{F}$. Then, we consider sheaf cohomology of $\mathscr{F}^{\mathrm{wa}}$ with coefficients in vector bundles which ar…

A geometric $p$-adic Simpson correspondence in rank one

For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb Q_p$ we give a geometrisation of the $p$-adic Simpson correspondence of rank one in terms of analytic moduli spaces: The $p$-adic character variety is canonically an étale twist of the moduli space of topological torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$-line bundles: We develop a theory of topological torsion subsheaves of $v$-sheaves and apply this to the diamantine $v$-Pica…

Continuous rational functions on real and $p$-adic varieties

This note studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a subvariety? Which linear equations can be solved using such functions?

1-动机的p进周期猜想：积分与线性关系

P-adic Period Conjectures for 1-motives: Integration and Linear Relations

本文为1-动机建立p进周期理论，构造p进积分配对与分层周期空间，提出并证明深度1和2的周期猜想，并发展p进解析子群定理，为研究p进周期线性关系及超越性开辟新途径。

Continuous rational functions on real and $p$-adic varieties II

This note replaces two earlier preprints (1101.3737 by Kollár) and (1211.6681 by Nowak). It studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a subvariety? Which linear equations can be solved using such functions? v.2: Section 3 simplified and typos corrected.

Rational points of elliptic surfaces and Zariski $N$-ples for cubic-line, cubic-conic-line arrangements

In this paper, we continue the study of the relation between rational points of rational elliptic surfaces and plane curves. As an application, we give first examples of Zariski pairs of cubic-line arrangements that do not involve inflectional tangent lines.

Lines on the secant cubic hypersurfaces of Severi varieties

The secant varieties of Severi varieties provide special examples of (singular) cubic hypersurfaces. An interesting question asks when a given cubic hypersurface is projectively equivalent to a secant cubic hypersurface. Inspired by the ''geometric'' Torelli theorem for smooth cubic hypersurfaces due to F. Charles, we study the geometry of lines on secant cubics and describe the Fano variety of lines. Then we verify the ''geometric'' Torelli theorem for the case of secant cubics. Namely, a cubic hypersurface is isomorphic to a secant cubic if and only if their Fano varieties of lines are isomo…

通过p进霍奇理论与朗兰兹对应构造4个穿孔的复射影直线上的$\text{GL}_2$型阿贝尔簇族及其在Painleve VI方程代数解中的应用

Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation

本研究借助p进霍奇理论与朗兰兹对应，构造了4穿孔射影直线上无穷多族带特定坏约化的$\text{GL}_2$型阿贝尔簇，结合相关猜想证明其可由椭圆曲线泛族挠截面参数化，且可导出Painleve VI方程的代数解。

Independence of $\ell$-adic Galois representations over function fields

Let $K$ be a finitely generated extension of $\mathbb{Q}$. We consider the family of $\ell$-adic representations ($\ell$ varies through the set of all prime numbers) of the absolute Galois group of $K$, attached to $\ell$-adic cohomology of a smooth separated scheme of finite type over $K$. We prove that the fields cut out from the algebraic closure of $K$ by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question asked by Serre in 1991.