How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces

The proportion of plane cubic curves over ${\mathbb Q}$ that everywhere locally have a point

We show that the proportion of plane cubic curves over ${\mathbb Q}_p$ that have a ${\mathbb Q}_p$-rational point is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. As a consequence, we obtain the density of plane cubic curves over ${\mathbb Q}$ that have points everywhere locally; numerically, this density is shown to be $\approx 97.3\%$.

A positive proportion of locally soluble hyperelliptic curves over $\mathbb Q$ have no point over any odd degree extension

A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has a point over every completion of $\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$ are locally soluble but have no points over any odd degree extension of $\mathbb Q$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $k > 0$, the proportion of locally soluble hyperelliptic curves over $\mathbb Q$ of genus $g$ having no points over any odd degree extension of $\mathbb Q$ of degree at most $k$ tends …

分裂二次曲面上一族二次曲面的局部可解性

Local solubility for a family of quadrics over a split quadric surface

本文考察由分裂二次曲面上有理点参数化的对角二次曲面，分析其处处局部可解情形的密度。

The proportion of genus one curves over $\mathbb{Q}$ defined by a binary quartic that everywhere locally have a point

We consider the proportion of genus one curves over $\mathbb{Q}$ of the form $z^2=f(x,y)$ where $f(x,y)\in\mathbb{Z}[x,y]$ is a binary quartic form (or more generally of the form $z^2+h(x,y)z=f(x,y)$ where also $h(x,y)\in\mathbb{Z}[x,y]$ is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime $p$ is given by a fixed degree-$9$ rational function of $p$ for all odd $p$ (and for the generalised equation, the sa…

The proportion of failures of the Hasse norm principle

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.

分裂二次曲面上的对角二次型的局部可解性的渐近性质

Asymptotics For Local Solubility Of Diagonal Quadrics Over A Split Quadric Surface

本文证明了由分裂二次曲面上特定有理点参数化的处处局部可解对角二次曲面密度的渐近性质。

三元三次型的局部可解性

Local solubility of ternary cubic forms

研究三元三次型 $ϕ_{a,b}(x,y,z)$ 的局部可解性，给出了满足特定边界与温和条件的整数对 $(a,b)$ 数量，使得该形式在所有 $p$ 进数域 $\mathbb{Q}_p$ 中均有非零解的渐近公式。

亏格为1的曲线的二次扭曲

Quadratic twists of genus one curves

本文研究整系数四次首一不可约多项式所定义亏格1曲线的二次扭曲，刻画处处局部可解扭曲参数的计数渐近式，并用数域的 Dedekind zeta 函数表示其 Dirichlet 级数。

Tamagawa products of elliptic curves over $\mathbb{Q}$

We explicitly construct the Dirichlet series $$L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product $m.$ Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)=0.5053\dots.$ As a corollary, we find that $L_{\mathrm{Tam}}(-1)=1.8193\dots$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}.$ We give an application of these results to c…

整体函数域中的 Hasse 范数原理

The Hasse Norm Principle in Global Function Fields

本文研究整体函数域扩张 $L/\mathbb{F}_q(t)$ 中 Hasse 范数原理的失效情形，并量化处处为局部范数却非全局范数的 $d$ 次多项式比例。

三次广义 Kummer 曲面的超越 Brauer 群

Transcendental Brauer groups of cubic generalised Kummer surfaces

本文研究由三次曲线构造的广义 Kummer K3 曲面，通过联系其与积曲面的超越 Brauer 群，显式计算有理对角三次曲线情形，并据此探讨处处局部可解曲线的 Galois 三次点存在性。