Wooley: The Circle Method

Introduction

The circle method is the Hardy-Littlewood method (1920s), ``any method involving Harmonic analysis that counts solutions of Diophantine questions'', including Kloosterman methods.

Example. Consider a homogeneous polynomial $F(x_1,\dots,x_s) \in \mathbb{Z}[x_1,\dots,x_s]$ of degree $d$. We count

$$N_F(B) = \char93 \{(x_1,\dots,x_s) \in [-B,B]^s: F(x)=0\}$$

$$= \int_0^1 G(\alpha)\,d\alpha,$$

where

$$G(\alpha)=\sum_{\vert x\vert \leq B} e(\alpha F(x_1,\dots,x_s)),$$

and $e(z)=e^{2\pi iz}$.

Let $\psi(B) \to \infty$ as $B \to \infty$ as slowly as you like, $\psi(B) < B^{d/2}$. We look at

$$\mathfrak{M}(q,a)=\{\alpha \in [0,1]:\vert q\alpha-a\vert \leq \psi(B)B^{-d}\},$$

and let

$$\mathfrak{M}=\bigcup_{\substack{0 \leq a \leq q \leq \psi(B) \\ \gcd(a,q)=1}} \mathfrak{M}(q,a),$$

the major arcs. Then

$$\int_\mathfrak{M}G(\alpha) d\alpha \sim v_\infty \prod_p v_p B^{s-d},$$

the product of local densities, where $v_\infty$ is the volume of the real manifold defined by $F(x)=0$ in $[-1,1]^s$, and

$$v_p=\lim_{h \to \infty} p^{h(1-s)}\char93 \{F(x)=0~(\textup{\text{mod}}~{p^n}):x \in (\mathbb{Z}/p^h\mathbb{Z})^s\}.$$

This particular statement is true in a very broad sence, provided that $s$ is not small and that the geometry of $F=0$ is not too wild, e.g. nonsingular.

For $\mathfrak{m}=[0,1) \setminus \mathfrak{M}$, the minor arcs, then $G(\alpha)$ should be ``randomly'' behaved, so one tries to show: $G(\alpha)=o(B^{s-d})$ when $\alpha \in \mathfrak{m}$. If true, then

$$N_F(B)=\int_\mathfrak{M}G(\alpha)\,d\alpha + \int_\mathfrak{m}G(\alpha)\,d\alpha \sim v_\infty \prod_p v_p B^{s-d} + o(B^{s-d}).$$

For this one needs non-singular $\mathbb{R}$ and $\mathbb{Q}_p$-points. When this method works, one gets weak approximation and the Hasse principle.

In particular, this will not work for varieties which fail the Hasse principle. The basic techniques work for all number fields $K/\mathbb{Q}$, or even for $\mathbb{F}_q[t]$ or other function fields.

Scope of the Circle Method

The circle method works with ``sufficiently many'' variables.

Proposition. [Birch 1957] Given forms $F_1,\dots,F_r \in \mathbb{Q}[x_1,\dots,x_s]$, of respectively odd degrees $d_1,\dots,d_r$, and provided that $s>s_0(d_1,\dots,d_r)$ is large enough, then there exists a rational point on $F_1=\dots=F_r=0$.

This method diagonalizes each of the forms, but at a great cost:

$$s_0(d_1,\dots,d_r) \leq \psi_{(d-5)/2}(d_1+\dots+d_r)$$

where $d=\max d_i$, and $\psi_0(x)=\exp(x)$, $\psi_1(x)=(\underbrace{\exp \circ \dots \circ \exp}_{42\log x})(x)$, $\psi_2(x)=(\underbrace{\psi_1 \circ \dots \circ \psi_1}_{42\log x})(x)$ and so on.

Example. For $d=3$, one has $s_0(3)=15$ (Davenport 1963); $s_0(\underbrace{3,\dots,3}_r)=(10r)^5$ (Schmidt 1984). $s_0(3,3)=831$ (Dietmann-W).

Proposition. [Birch 1962] Let $F(x) \in \mathbb{Z}[x_1,\dots,x_s]$ be homogeneous of degree $d$. Let $V=\{F(x)=0\}$. Then whenever $s-\dim(V_\textup{sing}) > (d-1)2^d$, one has $N_F(B)$ asymptotic to a product of local densities as before.

The difficulty of this result depends on the singular locus being reasonably small in dimension. This holds for any number field, and it is probable that this holds for a function field assuming the characteristic is sufficiently large.

Proposition. [Heath-Brown 1983, Hooley 1988] For $s_0(3)=8$, we have the Hasse principle for nonsingular cubic forms.

We now turn to some simpler situations.

Proposition. [Brudem-W] If $F=\Phi_1(x_1,x_2) + \dots + \Phi_{s/2}(x_{s-1},x_s)$, $\Phi_i \in \mathbb{Z}[x,y]$ binary, homogeneous of degree $d$, then $N_F(B)$ is asymptotic to the product of local densities whenever

$$s> \begin{cases} 2^d, & d = 3,4 \\ (17/16)2^d, & 5 \leq d \leq 10 \\ 2d^2\log d+\dots, & d\text{ large}. \end{cases}$$

For a diagonal form $a_1x_1^d+\dots+a_sx_s^d=0$, work by Hua, Vaughan, Heath-Brown, the same conclusion holds for

$$s> \begin{cases} 2^d, & d = 3 \leq d \leq 5 \\ (7/8)2^d & 6 \leq d \leq 8 \\ d^2\log d+\dots, & d\text{ large}; \end{cases}$$

One also has the weaker statement that $N_F(B)$ is greater than a constant times the product of local densities in the cases that

$$s \geq \begin{cases} 7, & d=3 \\ 12, & d=4 \\ \vdots \\ d(\log d+\log\log d+2+o(1)), & d \text{ large}. \end{cases}$$

Presumably: $s>2d$ should suffice for the method to work.

Simultaneous equations: we expect need $s_0(d)$ variables for $1$ form of degree $d$ makes it look we need $rs_0(d)$ variables for $r$ forms of degree $d$. For $\sum_{j=1}^{s} a_{ij}x_j^d=0$, ( $1 \leq i \leq r$), the number of variables required is given by: if the forms are in general position, and $s>(3r+1)2^{d-2}$, then we have an asymptotic formula. For $r$ diagonal cubics, $s \geq 6r+3$. For $2$ diagonal cubics, one has the Hasse principle whenever $s \geq 13$ (Brudeur, W).

Keys to Success

We have the major arcs

$$\mathfrak{M}(q,a)=\{\alpha \in [0,1): \vert q\alpha-a\vert \leq \psi(B)B^{-d}\}$$

with

$$\mathfrak{M}=\bigcup_{\substack{0 \leq a \leq q \leq \psi(B) \\ \gcd(a,q)=1}} \mathfrak{M}(q,a).$$

For $\alpha=a/q$ a rational number, we have

$$G(\alpha) =\sum_{\vert x\vert \leq B}e(\alpha F(x))= \sum_{r_1=1}^q \dots \... ...iv r_i ~(\textup{\text{mod}}~{q}) \\ 1 \leq i \leq s}} e((a/q)F(r_1,\dots,r_s))$$

$$= (B/q)^s \sum_{r_1=1}^q \dots \sum_{r_s=1}^{q} e((a/q)F(r)) + O((B/q)^{s-1} q^s)$$

$$\sim q^{-s} S(q,a) B^s.$$

One can handle the case $\alpha=(a/q)+\beta$ for $\beta$ small by using the mean value theorem,

$$G(\alpha)=q^{-s}S(q,a)v(\beta) + O((q(1+B^d\vert\beta\vert))^s)$$

and

$$v(\beta)=\int_{-B}^B \dots \int_{-B}^{B} e(\beta F(\gamma_1,\dots,\gamma_s))\,d\gamma.$$

One can apply Poisson summation and Kloosterman methods to get the error to be of type $B^{ds/4}$.

For the minor arcs, we want to show $\int_\mathfrak{m}G(\alpha)\,d\alpha=o(B^{s-d})$. One has Weyl differencing: letting $f(\alpha)=\sum_{\vert x\vert \leq B}e(\alpha x^d)$, we have

$$\vert f(\alpha)\vert^2 = \sum_{\vert x\vert \leq B}\sum_{\vert y\... ... e(\alpha(x^d-y^d)) = \sum_{h \in I} \sum_{y \in I(h)} e(\alpha((y+h)^d-y^d))$$

where now $(y+h)^d-y^d=hp_{d-1}(y,h)$. Repeating in this way, one can get down to sums of linear polynomials. Provided $\alpha \in \mathbb{R}$, $a \in \mathbb{Z}$, $q \in \mathbb{N}$, $\gcd(a,q)=1$, with $\vert\alpha-a/q\vert \leq q^{-2}$, then

$$\vert f(\alpha)\vert \ll B^{1+\epsilon}(a^{-1}B^{-1}+qB^{-d})^{2^{1-d}}.$$

Finally, there is recent work of Heath-Brown and Skorobogatov: For $at^{\ell}(1-t)^m=N(x)$, $N$ a norm form of degree $k$, then the Brauer-Manin obstruction is the only one to weak approximation and the Hasse principle. One uses descent to $cN(y)+dN(z)=\lambda w^k$, where the circle method gives weak approximation and the Hasse principle. One can generalize this to the case

$$aL_1(x)^{\ell_1} L_{2r}(x)^{\ell_{2r}}=N(v),$$

where $L_i(x) \in \mathbb{Q}[x_1,\dots,x_r]$ linear forms, $\gcd(\ell_1,\dots,\ell_{2r})=1$. Again we have that the Brauer-Manin obstruction is the only one, and one has descent to

$$\sum_{j=1}^{2r}c_{ij}N(y_i)=\lambda_i w^k$$

for $(1 \leq i \leq r)$.

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