Cohomological divisibility and point count divisibility

The local-global principle for divisibility in CM elliptic curves

We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime $p$ we give sharp lower bounds on the degree $d$ of a number field over which there exists a CM elliptic curve which gives a counterexample to the local-global principle for divisibility by a power of $p$. As a corollary we deduce that there are at most finitely many elliptic curves (with or without CM) which are counterexamples with $p > 2d+1$. We also deduce that the local-global principle for divisibility by powers of $7$ holds over quadratic f…

On local-global divisibility by $p^2$ in elliptic curves

Let $ p $ be a prime lager than 3. Let $k$ be a number field, which does not contain the subfield of $\mathbb{Q} (ζ_{p^2})$ of degree $p$ over $\mathbb{Q}$. Suppose that $\mathcal{E}$ is an elliptic curve defined over $k$. We prove that the existence of a counterexample to the local-global divisibility by $p^2$ in $\mathcal{E}$, assures the existence of a $k$-rational point of exact order $p$ in $\mathcal{E}$. Using the Merel Theorem, we then shrunk the known set of primes for which there could be a counterexample to the local-global divisibility by $p^2$.

Divisibility on point counting over finite Witt rings

Let $\mathbb{F}_q$ denote the finite field of $q$ elements with characteristic $p$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. In this paper, we investigate the $q$-divisibility for the number of solutions of a polynomial system in $n$ variables over the finite Witt ring $\mathbb{Z}_q/p^m\mathbb{Z}_q$, where the $n$ variables of the polynomials are restricted to run through a combinatorial box lifting $\mathbb{F}_q^n$. The introduction of the combinatorial box makes the problem much more complicated. We prove a …

光滑数的一个推广：多重稠密可除性

An extension of smooth numbers: multiple dense divisibility

本文推广光滑数概念，研究多重稠密可除数，并给出其计数在一致范围内的数量级估计。

关于椭圆曲线中整除性的Hasse原理

On the Hasse principle for divisibility in elliptic curves

研究了数域上椭圆曲线中 $p^n$ 整除性的局部-全局原理，给出了伽罗瓦群生成元条件以判定原理成立并构造反例，推广了 $n=2$ 的已知结果。

代数环面上的局部-整体可除性

Local-global divisibility on algebraic tori

本文完整解决代数环面的局部-整体可除性问题，证明维数 $r<p-1$ 时任意 $p^n$ 可除性成立，并给出高维反例及放宽至 $3(p-1)$ 的条件性结果。

On local-global divisibility by $p^n$ in elliptic curves

Let $p$ be a prime number and let $ k $ be a number field, which does not contain the field $\mathbb{Q} (ζ_p + \bar{ζ_p})$. Let $\mathcal{E}$ be an elliptic curve defined over $k$. We prove that if there are no $k$-rational torsion points of exact order $p$ on $\E$, then the local-global principle holds for divisibility by $p^n$, with $n$ a natural number. As a consequence of the deep theorem of Merel, for $p$ larger than a constant depending only on the degree of $k$, there are no counterexamples to the local-global divisibility principle. Nice and deep works give explicit small constants for…

On the local-global divisibility over abelian varieties

Let $p \geq 2$ be a prime number and let $k$ be a number field. Let $\mathcal{A}$ be an abelian variety defined over $k$. We prove that if ${\rm Gal} ( k ( {\mathcal{A}}[p] ) / k )$ contains an element $g$ of order dividing $p-1$ not fixing any non-trivial element of ${\mathcal{A}}[p]$ and $H^1 ( {\rm Gal} ( k ( {\mathcal{A}}[p] ) / k ), {\mathcal{A}}[p] )$ is trivial, then the local-global divisibility by $p^n$ holds for ${\mathcal{A}} ( k )$ for every $n \in \mathbb{N}$. Moreover, we prove a similar result without the hypothesis on the triviality of $H^1 ( {\rm Gal} ( k ( {\mathcal{A}}[p] ) …

高斯范数的可除性估计与CM椭圆曲线的殆素数阶

Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve

该研究针对椭圆曲线E:y²=x³-x，结合高斯CM筛框架与交叉项，证明了特定素数p下#E(F_p)/8的无平方因子性及素因子个数限制，为相关数论问题提供了可除性估计与分布结果。

Equivariant Cohomology of the Moduli Space of Genus Three Curves with Symplectic Level Two Structure via Point Counts

We make cohomological computations related to the moduli space of genus three curves with symplectic level two structure by means of counting points over finite fields. In particular, we determine the cohomology groups of the quartic locus as representations of the symmetric group on seven elements.

有限域上主极化阿贝尔簇的上同调算术统计

Cohomological Arithmetic Statistics for Principally Polarized Abelian Varieties over Finite Fields

本文研究有限域上主极化阿贝尔簇有理点计数的分布规律，通过计算低维情形的上同调与欧拉示性数并借助上同调稳定性，提出高维极限下点计数渐近行为的猜想。

Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures

We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the étale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the cont…

Variations of Kurepa's left factorial hypothesis

Kurepa's hypothesis asserts that for each integer $n\ge 2$ the greatest common divisor of $!n:=\sum_{k=0}^{n-1}k!$ and $n!$ is $2$. Motivated by an equivalent formulation of this hypothesis involving derangement numbers, here we give a formulation of Kurepa's hypothesis in terms of divisibility of any Kurepa's determinant $K_p$ of order $p-4$ by a prime $p\ge 7$. In the previous version of this article we have proposed the strong Kurepa's hypothesis involving a general Kurepa's determinant $K_n$ with any integer $n\ge 7$. We prove the ``even part'' of this hypothesis which can be considered as…

关于主导有理自映射在射影簇上周期点和前周期点高度有界的性质

On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties

本文推翻了关于周期点高度有界的猜想，并证明了在射影簇上某些主导有理自映射的周期点集高度有界，同时指出前周期点可能不满足这一性质。