On the global determinant method

In order to understand the function $\#S(X;B)$ of the variable $B\in\mathbb{R}^{+}$, we will introduce the so-called determinant method to study the number of rational points with bounded height in arithmetic varieties, which was proposed in [24].

In this article, we will construct the global determinant method over an arbitrary number field by Arakelov geometry following the strategy of [12, 13]. As a direct application, we will study the problem of counting rational points in plane curves, and we consider how these upper bounds depend on the degree. Some ideas are inspired by [40, 48, 10].

This paper is organized as following. In §2, we provide some preliminaries to construct the determinant method. In §3, we formulate the global determinant method by the slope method. In §4, we give some useful estimates on the non-geometrically integral reductions, a count of multiplicities over finite fields, the distributions of some particular prime ideals, and the geometric Hilbert-Samuel function. In §5, we provide an explicit upper bound of the determinant and lower bounds of auxiliary hypersurfaces. In §6, we give two uniform upper bounds of rational points of bounded height in plane curves. In §7, under the assumption of GRH, we give some explicit estimates of the distribution of prime ideals of bounded norm in a ring of integers, and explain how to apply these explicit estimates in the global determinant to get more explicit estimates. In Appendix A, we will give an explicit lower bound of a useful function induced by the local Hilbert-Samuel function.

I would like to thank Prof. Per Salberger for introducing his brilliant work [40] to me, and for explaining to me some ingredients of his work. These discussions and suggestions play a significant role in this paper. I would also like to thank Prof. Stanley Yao Xiao for his suggestions on the study of the distribution of prime ideals. In particular, I would like to thank the anonymous referee for all the suggestions on revising the manuscript of the paper. Chunhui Liu was supported by Fundamental Research Funds for the Central Universities FRFCU5710010421.

Let $K$ be a number field, and $\mathcal{O}_{K}$ be its ring of integers. We denote by $M_{K,f}$ the set of finite places of $K$, and by $M_{K,\infty}$ the set of infinite places of $K$. In addition, we denote by $M_{K}=M_{K,f}\sqcup M_{K,\infty}$ the set of places of $K$. For every $v\in M_{K}$ and $x\in K$, we define the absolute value $|x|_{v}=\left|N_{K_{v}/\mathbb{Q}_{v}}(x)\right|_{v}^{\frac{1}{[K_{v}:\mathbb{Q}_{v}]}}$ for each $v\in M_{K}$, extending the usual absolute values on $\mathbb{Q}_{p}$ or $\mathbb{R}$. Here $\mathbb{Q}_{v}$ denotes the $p$-adic field $\mathbb{Q}_{p}$, where $v$ is extended from $p$ under the extension $K/\mathbb{Q}$.

Let $\xi=[\xi_{0}:\cdots:\xi_{n}]\in\mathbb{P}^{n}_{K}(K)$. We define the height of $\xi$ in $\mathbb{P}^{n}_{K}$ as

We also define the logarithmic height of $\xi$ as

which is invariant under the extensions over $K$ (cf. [25, Lemma B.2.1]).

Suppose that $X$ is a closed integral subscheme of $\mathbb{P}^{n}_{K}$, and $\phi:X\hookrightarrow\mathbb{P}^{n}_{K}$ is the closed immersion. For $\xi\in X(K)$, we define $H_{K}(\xi)=H_{K}(\phi(\xi))$, and usually we omit the closed immersion $\phi$ if there is no confusion. Next, we denote

By the Northcott’s property (cf. [25, Theorem B.2.3]), the cardinality $N(X;B)$ is finite for a fixed $B\geqslant 1$.

In this part, we will introduce a function induced by the local Hilbert-Samuel function of schemes at a closed point, and we will use this function in Proposition 3.1. For the motivation and background, see [39, §2] and [13, §3.2].

Let $k$ be a field, and $X$ be a closed subscheme of $\mathbb{P}^{n}_{k}$ of pure dimension $d$, which means all its irreducible components have the same dimension. Let $\xi$ be a closed point of $X$. We denote by

the local Hilbert-Samuel function of $X$ at the point $\xi$ with the variable $s\in\mathbb{N}$, where $\mathfrak{m}_{X,\xi}$ is the maximal ideal of the local ring $\mathcal{O}_{X,\xi}$, and $\kappa(\xi)$ is the residue field of the local ring $\mathcal{O}_{X,\xi}$. For this function, we have the polynomial asymptotic

when $s\rightarrow+\infty$, and we define the positive integer $\mu_{\xi}(X)$ as the multiplicity of point $\xi$ in $X$.

We define the series $\{q_{\xi}(m)\}_{m\geqslant 0}$ as the increasing series of non-negative integers such that every integer $s\in\mathbb{N}$ appears exactly $H_{\xi}(s)$ times in this series. For example, if $H_{\xi}(0)=1$, $H_{\xi}(1)=2,H_{\xi}(2)=4,H_{\xi}(3)=5,\ldots$, then the series $\{q_{\xi}(m)\}_{m\geqslant 0}$ is

Let $\{Q_{\xi}(m)\}_{m\geqslant 0}$ be the partial sum of the series $\{q_{\xi}(m)\}_{m\geqslant 0}$, which is

for all $m\in\mathbb{N}$.

If $X$ is a hypersurface of $\mathbb{P}^{n}_{k}$, then by [26, Example 2.70 (2)], the local Hilbert-Samuel function of $X$ at the point $\xi$ defined in (3) is

In this case, we have the following explicit lower bound of $Q_{\xi}(m)$, which is

This lower bound has the optimal dominant term by the argument in [39, Main Lemma 2.5] and some other subsequent references. In Appendix A, we will provide a detailed proof of this lower bound.

A normed vector bundle over $\operatorname{Spec}\mathcal{O}_{K}$ is all the pairings $\overline{E}=\left(E,\left(\|\raisebox{1.72218pt}{.}\|_{v}\right)_{v\in M_{K,\infty}}\right)$, where:

• —

  $E$ is a projective $\mathcal{O}_{K}$-module of finite rank;

• —

  $\left(\|\raisebox{1.72218pt}{.}\|_{v}\right)_{v\in M_{K,\infty}}$ is a family of norms, where $\|\raisebox{1.72218pt}{.}\|_{v}$ is a norm over $E\otimes_{\mathcal{O}_{K},v}\mathbb{C}$ which is invariant under the action of $\operatorname{Gal}(\mathbb{C}/K_{v})$. We consider a complex place and its conjugation as two different places.

If for all $v\in M_{K,\infty}$, the norms $\left(\|\raisebox{1.72218pt}{.}\|_{v}\right)_{v\in M_{K,\infty}}$ are Hermitian, we say that $\overline{E}$ is a Hermitian vector bundle over $\operatorname{Spec}\mathcal{O}_{K}$. If $\operatorname{rk}_{\mathcal{O}_{K}}(E)=1$, we say that $\overline{E}$ is a Hermitian line bundle.

Suppose that $F$ is a sub-$\mathcal{O}_{K}$-module of $E$. We say that $F$ is a saturated sub-$\mathcal{O}_{K}$-module if $E/F$ is a torsion-free $\mathcal{O}_{K}$-module.

Let $\overline{E}=\left(E,\left(\|\raisebox{1.72218pt}{.}\|_{E,v}\right)_{v\in M_{K,\infty}}\right)$ and $\overline{F}=\left(F,\left(\|\raisebox{1.72218pt}{.}\|_{F,v}\right)_{v\in M_{K,\infty}}\right)$ be two Hermitian vector bundles. If $F$ is a saturated sub-$\mathcal{O}_{K}$-module of $E$ and $\|\raisebox{1.72218pt}{.}\|_{F,v}$ is the restriction of $\|\raisebox{1.72218pt}{.}\|_{E,v}$ over $F\otimes_{\mathcal{O}_{K},v}\mathbb{C}$ for every $v\in M_{K,\infty}$, we say that $\overline{F}$ is a sub-Hermitian vector bundle of $\overline{E}$ over $\operatorname{Spec}\mathcal{O}_{K}$.

We say that $\overline{G}=\left(G,\left(\|\raisebox{1.72218pt}{.}\|_{G,v}\right)_{v\in M_{K,\infty}}\right)$ is a quotient Hermitian vector bundle of $\overline{E}$ over $\operatorname{Spec}\mathcal{O}_{K}$, if for every $v\in M_{K,\infty}$, the module $G$ is a projective quotient $\mathcal{O}_{K}$-module of $E$ and $\|\raisebox{1.72218pt}{.}\|_{G,v}$ is the induced quotient space norm of $\|\raisebox{1.72218pt}{.}\|_{E,v}$.

For simplicity, we will denote by $E_{K}=E\otimes_{\mathcal{O}_{K}}K$ below.

Let $\overline{E}$ be a Hermitian vector bundle over $\operatorname{Spec}\mathcal{O}_{K}$, and $\{s_{1},\ldots,s_{r}\}$ be a $K$-basis of $E_{K}$. We will introduce some invariants in Arakelov geometry below.

Let $\overline{\mathcal{E}}$ be a Hermitian vector bundle of rank $n+1$ over $\operatorname{Spec}\mathcal{O}_{K}$, and $\mathbb{P}(\mathcal{E})$ be the projective space which represents the functor from the category of commutative $\mathcal{O}_{K}$-algebras to the category of sets mapping all $\mathcal{O}_{K}$-algebra $A$ to the set of projective quotient $A$-module of $\mathcal{E}\otimes_{\mathcal{O}_{K}}A$ of rank $1$. Let $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ (or $\mathcal{O}(1)$ if there is no confusion) be the universal bundle, and we denote by $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(D)$ (or $\mathcal{O}(D)$) the line bundle $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)^{\otimes D}$ for simplicity. The Hermitian metrics on $\mathcal{E}$ induce by quotient of Hermitian metrics (i.e. Fubini-Study metrics) on $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ which define a Hermitian line bundle $\overline{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)}$ on $\mathbb{P}(\mathcal{E})$.

For every $D\in\mathbb{N}^{+}$, let

and let $r(n,D)$ be its rank over $\mathcal{O}_{K}$. In fact, we have

For each $v\in M_{K,\infty}$, we denote by $\|\raisebox{1.72218pt}{.}\|_{v,\sup}$ the norm over $E_{D,v}=E_{D}\otimes_{\mathcal{O}_{K},v}\mathbb{C}$ such that

where $\|\raisebox{1.72218pt}{.}\|_{v,\mathrm{FS}}$ is the corresponding Fubini-Study norm.

In this part, we will define a height function of rational points by Arakelov geometry.

Let $\overline{\mathcal{E}}$ be a Hermitian vector bundle of rank $n+1$ over $\operatorname{Spec}\mathcal{O}_{K}$, $P\in\mathbb{P}(\mathcal{E}_{K})(K)$, and $\mathcal{P}\in\mathbb{P}(\mathcal{E})(\mathcal{O}_{K})$ be the Zariski closure of $P$ in $\mathbb{P}(\mathcal{E})$. Let $\overline{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)}$ be the universal bundle equipped with the corresponding Fubini-Study metric at each $v\in M_{K,\infty}$, then $\mathcal{P}^{*}\overline{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)}$ is a Hermitian vector bundle over $\operatorname{Spec}\mathcal{O}_{K}$. We define the height of the rational point $P$ with respect to $\overline{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)}$ as

We keep all the above notations. We choose

where for every $v\in M_{K,\infty}$, $\|\raisebox{1.72218pt}{.}\|_{v}$ is the $\ell^{2}$-norm mapping $(t_{0},\ldots,t_{n})$ to $\sqrt{|v(t_{0})|^{2}+\cdots+|v(t_{n})|^{2}}$. In this case, we use the notations $\mathbb{P}^{n}_{K}=\mathbb{P}(\mathcal{E}_{K})$ and $\mathbb{P}^{n}_{\mathcal{O}_{K}}=\mathbb{P}(\mathcal{E})$ for simplicity. We suppose that $P$ has a $K$-rational projective coordinate $[x_{0}:\cdots:x_{n}]$, then we have (cf. [31, Proposition 9.10])

Let $\psi:X\hookrightarrow\mathbb{P}^{n}_{K}$ be a projective scheme, and $P\in X(K)$. We define the height of $P$ as $h_{\overline{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)}}(\psi(P))$. We will just use the notation $h_{\overline{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)}}(P)$ or $h(P)$ if there is no confusion of the morphism $\psi$ and the Hermitian line bundle $\overline{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)}$.

In this part, we will introduce several height functions of arithmetic varieties, which evaluate their arithmetic complexities.

We refer to some results in [12, 13], which are used in the reformulation of the determinant method by Arakelov geometry. We will also prove a new auxiliary lemma.

In order to benefit the readers, we will provide the details on the construction of certain local homomorphisms, which are introduced in [39, Lemma 2.4], see also [13, §3.2].

Let $X$ be an integral closed subscheme of $\mathbb{P}^{n}_{K}$ and $\mathscr{X}$ be the Zariski closure of $X$ in $\mathbb{P}^{n}_{\mathcal{O}_{K}}$. Let $\mathfrak{p}$ be a maximal ideal of $\mathcal{O}_{K}$ and $\xi\in\mathscr{X}(\mathbb{F}_{\mathfrak{p}})$. In this case, $\mathcal{O}_{\mathscr{X},\xi}$ is a local algebra over $\mathcal{O}_{K,\mathfrak{p}}$. Let $(f_{i})_{1\leqslant i\leqslant m}$ be a family of local homomorphisms of $\mathcal{O}_{K,\mathfrak{p}}$-algebras from $\mathcal{O}_{\mathscr{X},\xi}$ to $\mathcal{O}_{K,\mathfrak{p}}$.

Let $E$ be a free sub-$\mathcal{O}_{K,\mathfrak{p}}$-module of finite type of $\mathcal{O}_{\mathscr{X},\xi}$ and let $f$ be the $\mathcal{O}_{K,\mathfrak{p}}$-linear homomorphism

Since $f_{1}$ is a local homomorphism of $\mathcal{O}_{K,\mathfrak{p}}$-algebras, it must be surjective. Let $\mathfrak{a}$ be the kernel of $f_{1}$, then we have $\mathcal{O}_{\mathscr{X},\xi}/\mathfrak{a}\cong\mathcal{O}_{K,\mathfrak{p}}$. Furthermore, since $\mathcal{O}_{\mathscr{X},\xi}$ is a local ring and we suppose that $\mathfrak{m}_{\xi}$ is its maximal ideal, then we have $\mathfrak{m}_{\xi}\supseteq\mathfrak{a}$. Moreover, since $f_{1}$ is a local homomorphism, we have $\mathfrak{a}+\mathfrak{p}\mathcal{O}_{\mathscr{X},\xi}=\mathfrak{m}_{\xi}$. For each $j\in\mathbb{N}$, $\mathfrak{a}^{j}/\mathfrak{a}^{j+1}$ is an $\mathcal{O}_{\mathscr{X},\xi}/\mathfrak{a}\cong\mathcal{O}_{K,\mathfrak{p}}$-module of finite type.

In order to estimate its rank, we need the following result.