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\title[Geometry of Numbers]{Geometry of Numbers}
\subtitle{TCC module: Term 3 2024}
\author{Simon L Rydin Myerson}
\institute{University of Warwick}
\date{For link to course webpage see \url{https://maths.fan}\\Manin's conjecture, $x\cdot y=0$}

\begin{document}

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\titlepage
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\begin{frame}
\frametitle{Idea of application}

\begin{corollary}
\(L^\perp\) has rank \(n-d\) and
\(\det(L^\perp) = \det(L)\).
\end{corollary}


\begin{itemize}
	\item $ Z(N)=
	\#\{x,y\in\mathbb Z^n\setminus \{0\} : x\cdot y =0 , \|x\|\cdot\|y\|\leq N\},$ let \(L = L(x/\operatorname{gcd}(x))\).
	
	\item 
 \(\#\{y\in L^\perp: \|y\|\leq \frac N{\|x\|}\} = (\frac N{\|x\|})^{n-d}\frac{\operatorname{Vol}(B(0;1))}{\det L}(1+O_n(\frac{\lambda_{n-d}(L^\perp)}{N/\|x\|}))\) if \(\frac N{\|x\|}>\lambda_d(L^\perp)\).
 \item WLOG \(\|x\|\leq \|y\|\). We will (eventually) prove that most lattices are `balanced', in the sense that \(\lambda_i(M) \asymp \det(M)^{1/\operatorname{rank}(M)}\). One has to be careful: for given constants in \(\asymp\), a positive proportion of lattices violate this.
 \item So for \(n\geq 3\), \(Z(N)\) we will prove
 \[
 \sum_{x\in\mathbb Z^n\setminus \{0\}, \|x\|\leq N} (N/\|x\|)^{n-1} \frac{\operatorname{gcd}(x)}{\|x||}
 \sim c_n N^{n-1}\log N
 \]
for some explicit \(c_n>0\).
\item Different proofs were given by Franke-Manin-Tschinkel (89), Thunder (93), Robbiani (01), and Spencer (08). Morally speaking we follow Thunder.
\end{itemize}
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\begin{frame}{Rational points}
We can understand \(x,y\in\mathbb Z^n\setminus \{0\} : x\cdot y =0\) as rational points on a projective variety.

$\mathbb{P}^{n-1}$ has rational points \([x]\): \(x\in \mathbb Z^n\), $(x_1,\dotsc,x_n)=1$, with height \(h(P)=\|x\|\).

Moreover $\mathbb{P}^{n-1}\times\mathbb{P}^{n-1} $ has rational points 
\[
\{([x],[y])
:x,y\in\mathbb Z^n, \gcd(x)=\gcd(y)=1\},
\]
with height \( \|x\|\cdot\|y\|\).

The equation \(x\cdot y =0\) defines a hypersurface \(H\) in  $\mathbb{P}^{n-1}\times\mathbb{P}^{n-1} $, with the  number of points of height \(\leq N\) given by
\[
\frac{1}{4}\#\{x,y\in\mathbb Z^n : \gcd(x)=\gcd(y)=1, \|x\|\cdot\|y\|\leq N\}
\]
\[
=
\frac{1}{4}
\sum_{d_1,d_2\in \mathbb N}
\mu(d_1)\mu(d_2)
Z(N/d_1d_2).
\]
Here \(\mu(d)=\pm1\) is the M\"obius function, and the proof uses the identity \(\sum_{d \mid m} \mu(d) = \mathbf{1}_{m=1}\) valid for \(\|m\|\in \mathbb Z\setminus \{0\}\)
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\begin{frame}{The Manin-Peyre conjecture}

\begin{block}{Conjecture}
	Let \(V\) be a Fano variety and let \(H\) be an anticanonical height on \(V(\mathbb Q)\). There is a thin subset \(T\subset V(\mathbb Q)\) such that
	\[
	\#
	\{
	P\in V(\mathbb Q)\setminus T: H(P)\leq X
	\}
	\sim c_{MP} X (\log X)^{\operatorname{rank}\operatorname{Pic}_{\mathbb{Q}}(V)-1},
	\]
	for a certain explicit constant \(c_{MP}\geq 0\).
\end{block}

E.g. for a hypersurface in \(\mathbb P^{n-1}\), \(H([x]) = \|x\|^{n-\deg V}\).

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\begin{frame}
\frametitle{Rational points: a heuristic}
	
	Let $\vec{f}(\vec{x}) \in \mathbb Z[x_1,\dotsc,x_n]^R$ be a system of homogeneous forms. Let $V(\vec{f})\subset \mathbb{P}^{n-1}$ be the zero locus. Rational points: $(x_1,\dotsc,x_n)=1$, $\vec{f}=\vec{0}$ with height \(h(P)=\|x\|\).
	\bigskip
	
	We assume $V(\vec{f})$ is irreducible$/\overline{\mathbb Q}$, and that the $f_i$ generate $I(V(\vec{f}))$.
	\bigskip
	
	
	Let $U\subset V(\vec{f})$ be a sufficiently small Zariski open subset. Define $$M(B) = \operatorname{measure}\{ \vec{t}\in\mathbb R^n : \|\vec{f}(\vec{t})\|\leq 1 \}.$$

\begin{enumerate}
	\item If $M(B)\ll 1$ as $B\to\infty$, we hope:
	\bigskip
	
	\begin{itemize}
		\item The degree of $V(\vec{f})$ is large and/or its dimension is small,
		\bigskip
		
		\item The number of rational points on $U$ is finite.
	\end{itemize}

\end{enumerate}
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\begin{frame}
\frametitle{Rational points heuristically, II}

\begin{enumerate}
\setcounter{enumi}{1}	\item If $M(B) \to \infty$  but $M(B)\ll_\epsilon B^\epsilon$ as $B\to\infty$, we hope:
\bigskip

	\begin{itemize}
		\item $V(\vec{f})$ has some kind of group structure,
		\bigskip
		
		\item $\{P\in U(\mathbb Q):h(P)\leq B\}\sim C(\log B)^{r/2}$ for suitable $C\geq 0,$ $r\in \mathbb N$, $h$.
		\bigskip
		
	\end{itemize}
	\item If for some $\delta>0$ we have $M(B) \gg B^\delta$ as $B\to\infty$, we hope:
	\bigskip
	
	\begin{itemize}
		\item The degree of $V(\vec{f})$ is small and/or its dimension is large,
		\bigskip
		
		\item $\{P\in U(\mathbb Q):h(P)\leq B\}\sim CB(\log B)^{r-1}$ for suitable $C,$ $r$, $h$.
	\end{itemize}
\end{enumerate}
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%\end{block}
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\begin{frame}
\frametitle{The Hardy-Littlewood heuristic}
	\begin{itemize}
		\item \(\vec{f}(\vec{x})\in \mathbb Z[x_1,\dotsc,x_s]^R\) will be a system of \(R\) homogenous  forms of degrees \(d_i\) in \(s>2\sum d_i\) variables with integer coefficients.
	%	\item We count solutions of \(\vec{f}=\vec{0}\) in integers of size \(B\), where \(B\) is big.
		
		\item We assume:  \(\vec{\alpha}\cdot \vec{f}=\sum^R \alpha_i F_i\) is nonzero and indefinite for  all \(\vec{\alpha} \in \mathbb R^R\setminus\{\vec{0}\}\).
		\item We study \(Z_{\vec{f}}(B) = \#\{\vec{x}\in\mathbb Z^s: \vec{f}(\vec{x})=\vec{0},\,\|\vec{x}\|\leq B\}\).
		%    		\vspace{-.7em}
	\end{itemize}

\begin{block}{Heuristic}
	\begin{itemize}
		\item Model  \(\vec{x}\) by a random real vector  \(\vec{X}\), and  model \(f_i(\vec{x})\) by \(\lfloor f_i(\vec{X})\rfloor\).
		%	
		\item That is, let \(\vec{X}\) be a uniform random variable on \([-B,B]^s\). Maybe \(Z_{\vec{f}}(B) \sim (2B)^s\cdot \mathbb P[\vec{f}(\vec{X})\in [0,1)^R],\) which is typically \(\sim c_{\vec{f}} B^{s-2\sum d_i}\).
		
		\item But: if \(R=1\), \(f(\vec{x})= x_1^2+x_2^2-3x_3^2\), then \(Z_{{f}}(B) =1\).
		\item Fix: let \(\vec{X}_p\) be uniformly distributed on \(\mathbb Z_p^s\). Predict
		\[
		\qquad\mathllap{Z_{\vec{f}}(B)}-1 = (1+o(1))(2B)^s\cdot \mathbb P[\vec{f}(\vec{X})\in [0,1)^R] \prod_p  \lim_{N\to \infty } p^{nR}\mathbb P[p^N\mid \vec{f}(\vec{X}_{p})].
		\]
	\end{itemize}
\end{block}
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Predict
	\[
	\qquad\mathllap{Z_{\vec{f}}(B)}-1 = (1+o(1))(2B)^s\cdot \mathbb P[\vec{f}(\vec{X})\in [0,1)^R] \prod_p  \lim_{N\to \infty } p^{nR}\mathbb P[p^N\mid \vec{f}(\vec{X}_{p})].
	\]
	\bigskip
	
	Let $ V $ be a smooth projective variety in $ \mathbb P^n_{\mathbb Q} $ defined by $ f_1 = \dotsb = f_r = 0 $.
	\bigskip
	
	We can assume that $ f_i $ has integral coefficients with gcd 1.	Then $ V $ has a reduction mod $ p $, a variety $ V_p $ in $ \mathbb P^n_{\mathbb F_p} $.
	
	\bigskip
	Let $ N_p $ be the number of $ \mathbb F_p $-points on $ V_p $.
	The natural guess for $ N_p $ is that it is roughly $ p^{\dim V} $. Indeed one could project onto the first $ D $ co-ordinates and in some sense this gives a way to parametrise most of the points on $ V_p $.
	
	\bigskip
	For all but finitely many \(p\), we have $\lim_{N\to \infty} p^{nR}\mathbb P[p^N\mid \vec{f}(\vec{X}_{p})] = N_p/p^{\dim V}.$
	
	\bigskip
	If the product $\prod N_p/p^{\dim V}$ diverges, we actually have a natural way to correct this prediction too.
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\begin{frame}{An extra power of log}
	
	The following is an explanation of what is written in section 2.1 of \url{https://archive.mpim-bonn.mpg.de/id/eprint/2194/1/preprint_1993_79.pdf}, which assumes some slightly more technical maths (and is also in French). I will look around to see if there is a good introduction to some of these ideas.
	
	We look at
	\[
	F(s) = \prod_p \bigg( 1 - p^{1-s}\frac{N_p-p^{\dim V}}{p^{\dim V}} \bigg)^{-1}.
	\]
	This is well defined for $ \Re s > 1 $, where $ \Re s $ is the real part. What is more, it is (up to a bounded multiplicative factor) the same as the ``Artin L-function $ L_S(s,\operatorname{Pic} V_{\bar{Q}}) $ of the Picard group of $ V_{\bar{Q}} $", which has a meromorphic continuation to the whole complex plane. And, crucially, $ L_S(s,\operatorname{Pic} V_{\bar{Q}}) $ has a pole at $ s=1 $ with multiplicity equal to the rank of $ \operatorname{Pic} V_{\bar{Q}} $.
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\begin{frame}$ Z(N)=
\#\{x,y\in\mathbb Z^n\setminus \{0\} : x\cdot y =0 , \|x\|\cdot\|y\|\leq N\},$ let \(L = L(x/\operatorname{gcd}(x))\).

\begin{align*}Z(N) = \sum_{x\in\mathbb Z^n\setminus \{0\}, \|x\|\leq \sqrt{N} }&
\#\{y:y\cdot x=0,\|x\|\leq \| y\|\leq  \sqrt{N}/\|x\| \}
\\{}+{}&\#\{y:y\cdot x=0,\|x\|< \| y\|\leq  \sqrt{N}/\|x\| \}
\end{align*}


	Let \(B(R)=\{y\in\mathbb R^n:\|y\| \leq R\}\) or \(\{y\in\mathbb R^n:\|y\| < R\}\), let \(\gcd(z)=1\).
	
	\bigskip
	
	\(\#L(z)^\perp\cap B(R) = \frac{\operatorname{Vol}(B(R))}{\|z\|} +O(1+\frac{R}{\lambda_1(L(z)^\perp)})^{n-1}\)
	
	\bigskip
	
\(\sum_{\|z\|\leq \sqrt{N}/g,\gcd(z)=1}
	\#L(z)^\perp\cap D(N/d\|z\|)
	\) 
	
	\(-\sum_{\|z\|\leq \sqrt{N}/g,\gcd(z)=1}
	\#L(z)^\perp\cap D(d\|z\|)
	\) 
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