Motivic & Arithmetic probability of a semistable elliptic surface with a Weierstrass torsion section

一个半稳定概形的log-动机上同调及其p进变形理论

A log-motivic cohomology for semistable varieties and its $p$-adic deformation theory

本文提出了一种半稳定概形的log-动机上同调理论，并探讨了其p进变形理论，通过研究Hyodo-Kato类与Hodge滤波的关系，推广了Bloch-Esnault-Kerz的结论。

Differential Forms on Hyperelliptic Curves with Semistable Reduction

Let $C$ be a hyperelliptic curve over a local field $K$ with odd residue characteristic, defined by some affine Weierstrass equation $y^2=f(x)$. We assume that $C$ has semistable reduction and denote by $\mathcal{X} \rightarrow \textrm{Spec}\, \mathcal{O}_K$ its minimal regular model with relative dualizing sheaf $ω_{\mathcal{X}/ \mathcal{O}_K}$. We show how to directly read off a basis for $H^0(\mathcal{X},ω_{\mathcal{X}/\mathcal{O}_K})$ from the cluster picture of the roots of $f$. Furthermore we give a formula for the valuation of $λ$ such that $λ\cdot \frac{dx}{y} \land \dots \land x^{g-1}…

Local and global densities for Weierstrass models of elliptic curves

We prove local results on the $p$-adic density of elliptic curves over $\mathbb{Q}_p$ with different reduction types, together with global results on densities of elliptic curves over $\mathbb{Q}$ with specified reduction types at one or more (including infinitely many) primes. These global results include: the density of integral Weierstrass equations which are minimal models of semistable elliptic curves over $\mathbb{Q}$ (that is, elliptic curves with square-free conductor) is $1/ζ(2)\approx60.79\%$, the same as the density of square-free integers; the density of semistable elliptic curves …

Special Values of Motivic $L$-Functions and Zeta-Polynomials for Symmetric Powers of Elliptic Curves

Let $\mathcal{M}$ be a pure motive over $\mathbb{Q}$ of odd weight $w\geq 3$, even rank $d\geq 2$, and global conductor $N$ whose $L$-function $L(s,\mathcal{M})$ coincides with the $L$-function of a self-dual algebraic tempered cuspidal symplectic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$. We show that a certain polynomial which generates special values of $L(s,\mathcal{M})$ (including all of the critical values) has all of its zeros equidistributed on the unit circle, provided that $N$ or $w$ are sufficiently large with respect to $d$. These special values have arithmetic s…

与半稳定阿贝尔簇相关的强相容系

Strongly compatible systems associated to semistable abelian varieties

本文证明了数域上阿贝尔簇的$\ell$-adic上同调作用在有限扩张后可给出取值于Mumford--Tate群的强相容表示系，并将Weil--Deligne表示的$\ell$独立性从良约化情形推广到半稳定约化情形。

局部域上双曲曲线的算术

Arithmetic of hyperelliptic curves over local fields

本文提出一种描述局部域上双曲曲线性质的聚类图方法，用于编码其伽罗瓦表示、判别式及其他不变量。