The error term in the Sato-Tate theorem of Birch

TL;DR

This paper derives an explicit error term for the distribution of Frobenius angles in the Sato-Tate theorem for elliptic curves over finite fields, refining the understanding of their statistical behavior.

Contribution

It provides a new explicit error bound in the Sato-Tate distribution for elliptic curves over finite fields, improving previous asymptotic results.

Findings

Error term of O_r(q^{7/4}) established

Quantitative distribution of Frobenius angles confirmed

Enhanced precision in Sato-Tate theorem for elliptic curves

Abstract

We establish an error term in the Sato-Tate theorem of Birch. That is, for $p$ prime, $q=p^r$ we show that $\#\{ (a,b) \in \mathbb{F}_q^2 : \theta_{a,b}\in I\} =\mu_{ST}(I)q^2 + O_r(q^{7/4})$ for any interval $I\subseteq[0,\pi]$ where for an elliptic curve $E: y^2= x^3 +ax +b$, the quantity $\theta_{a,b}$ is defined by $2\sqrt{q}\cos\theta_{a,b} = q+1-E(\mathbb{F}_q)$ and $\mu_{ST}(I)$ denotes the Sato-Tate measure of the interval $I$.

Equations

$$\#\{(a,b)\in {\mathbb{F}}_q^2:{\theta }_{a,b}\in I\}={\mu }_{ST}(I)q^2+O_r(q^{7/4})$$

$$y^2=x^3+ax+b\mathrm{mod}p$$

$$\sum _{a,b=0}^{p-1}{\left|\sum _{x=0}^{p-1}\left(\frac{x^3+ax+b}{p}\right)\right|}^{2R}\sim \frac{1}{R+1}\left(\genfrac{}{}{0ex}{}{2R}{R}\right)p^{R+2}$$

$$a_p(a,b)=-\sum _{x=0}^{p-1}\left(\frac{x^3+ax+b}{p}\right)=2\sqrt{p}\cos {\theta }_{a,b}$$

$$N_I(p):=\#\left\{(a,b)\in (\mathbb{Z}/p\mathbb{Z}{)}^2:p\nmid \Delta (a,b),{\theta }_{a,b}\in I\right\}$$

$$|N_I(p)-{\mu }_{ST}(I)p^2|$$

$$N_I(p)-{\mu }_{ST}(I)p^2=O(p^{7/4})$$

$$N_I(q):=\#\left\{(a,b)\in {\mathbb{F}}_q^2:\Delta (a,b)\ne 0,{\theta }_{a,b}\in I\right\},$$

$$N_I(q)-{\mu }_{ST}(I)q^2=O_r(q^{7/4}).$$

$$E_{a,b}:y^2=x^3+ax+b.$$

$$\begin{array}{rl}{U}_0(x)& =1\\ U_1(x)& =2x\\ U_n(x)& =2x{U}_{n-1}(x)-U_{n-2}(x),n\ge 2.\end{array}$$

$$U_n(x)=\frac{\sin (n+1)\theta }{\sin \theta }.$$

$$\frac{{\rho }^{k-1}-{\bar{\rho }}^{k-1}}{\rho -\bar{\rho }}=q^{\frac{k-2}{2}}{U}_{k-2}\left(\frac{t}{2\sqrt{q}}\right)$$

$$2\cos (n\theta )=\frac{\sin (n+1)\theta }{\sin \theta }-\frac{\sin (n-1)\theta }{\sin \theta }=U_n(\cos \theta )-U_{n-2}(\cos \theta ),$$

$${\hat{S}}_{M,J}^{\pm }(0)={\int }_0^1{S}_{M,J}^{\pm }(x)dx=\beta -\alpha \pm \frac{1}{M+1},$$

$$\left|{\hat{S}}_{M,J}^{\pm }(m)-{\hat{\chi }}_J(m)\right|\le \frac{1}{M+1}.$$

$${\hat{S}}_M^{\pm }(0)=(\beta -\alpha )\pm \frac{1}{M+1}$$

$${\hat{S}}_M^{\pm }(m)+{\hat{S}}_M^{\pm }(-m)=\frac{\sin 2\pi m\beta -\sin 2\pi m\alpha }{m\pi }+O\left(\frac{1}{M+1}\right).$$

$$\begin{array}{rl}{N}_I(q)=\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}{\chi }_{I^{\prime }}\left(\frac{{\theta }_{a,b}}{2\pi }\right)& \le \sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}\left(\sum _{|m|\le M}{\hat{S}}_M^{+}(m)e\left(m\frac{{\theta }_{a,b}}{2\pi }\right)+\sum _{|m|\le M}{\hat{S}}_M^{+}(m)e\left(-m\frac{{\theta }_{a,b}}{2\pi }\right)\right)\\ & =\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}\sum _{|m|\le M}{\hat{S}}_M^{+}(m)2\cos \left(m{\theta }_{a,b}\right)\\ & =2{\hat{S}}_M^{+}(0)\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}1+\sum _{0<|m|\le M}{\hat{S}}_M^{+}(m)\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}\left[U_m(\cos {\theta }_{a,b})-U_{m-2}(\cos {\theta }_{a,b})\right],\end{array}$$

$$U_r(\cos {\theta }_{a,b})=\frac{\sin (r+1){\theta }_{a,b}}{\sin {\theta }_{a,b}}$$

$$\begin{array}{rl}{N}_I(q)& \le q^2\left(2{\hat{S}}_M^{+}(0)-({\hat{S}}_M^{\pm }(2)+{\hat{S}}_M^{\pm }(-2))\right)+\sum _{m=1}^2({\hat{S}}_M^{\pm }(m)+{\hat{S}}_M^{\pm }(-m))\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}{U}_m(\cos {\theta }_{a,b})\\ & +\sum _{3\le m\le M}({\hat{S}}_M^{\pm }(m)+{\hat{S}}_M^{\pm }(-m))\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}\left[U_m(\cos {\theta }_{a,b})-U_{m-2}(\cos {\theta }_{a,b})\right].\end{array}$$

$$2{\hat{S}}_M^{\pm }(0)-({\hat{S}}_M^{\pm }(2)+{\hat{S}}_M^{\pm }(-2))={\mu }_{ST}(I)+O\left(\frac{1}{M+1}\right)$$

$$\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}{U}_m(\cos {\theta }_{a,b})=\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}\frac{\sin (m+1){\theta }_{a,b}}{\sin {\theta }_{a,b}}$$

$$\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}\frac{\sin (m+1){\theta }_{a,b}}{\sin {\theta }_{a,b}}=(q-1)\sum _{|t|\le 2\sqrt{q}}H(t^2-4q)\frac{\sin (m+1){\theta }_t}{\sin {\theta }_t},$$

$$\frac{\sin (m+1){\theta }_t}{\sin {\theta }_t}=U_m\left(\frac{t}{2\sqrt{q}}\right),$$

$$\begin{array}{rl}\mathrm{Tr}{T}_k(q)& =\frac{k-1}{12}{q}^{\frac{k-2}{2}}\delta (q,2)-\frac{1}{2}\sum _{|t|\le 2\sqrt{q}}\frac{{\rho }^{k-1}-{\bar{\rho }}^{k-1}}{\rho -\bar{\rho }}H(t^2-4q)-\frac{1}{2}\sum _{d{d}^{\prime }=q}\min (d,d^{\prime }{)}^{k-1}\\ & =\frac{k-1}{12}{q}^{\frac{k-2}{2}}\delta (q,2)-\frac{1}{2}\sum _{|t|\le 2\sqrt{q}}{q}^{\frac{k-2}{2}}\frac{\sin (k-1){\theta }_t}{\sin {\theta }_t}H(t^2-4q)-\frac{1}{2}\sum _{d{d}^{\prime }=q}\min (d,d^{\prime }{)}^{k-1},\end{array}$$

$$\mathrm{Tr}{T}_k(q)\ll k{q}^{\frac{k-1}{2}},$$

$$\sum _{|t|\le 2\sqrt{q}}H(t^2-4q)\frac{\sin (k-1){\theta }_t}{\sin {\theta }_t}\ll rk{q}^{1/2},$$

$$\sum _{\genfrac{}{}{0ex}{}{a,b\in {\mathbb{F}}_q}{\Delta (a,b)\ne 0}}\frac{\sin (m+1){\theta }_{a,b}}{\sin {\theta }_{a,b}}\ll rm{q}^{\frac{3}{2}}.$$

$$N_I(q)-q^2{\mu }_{ST}(I)\ll \frac{q^2}{M}+rM{q}^{3/2}.$$

Full text

[TABLE]

for any interval $I\subseteq[0,\pi]$ where for an elliptic curve $E:y^{2}=x^{3}+ax+b$, the quantity $\theta_{a,b}$ is defined by $2\sqrt{q}\cos\theta_{a,b}=q+1-E(\mathbb{F}_{q})$ and $\mu_{ST}(I)$ denotes the Sato-Tate measure of the interval $I$.

1. Introduction

In 1968, Birch [2] proved that the Sato-Tate conjecture holds for the family of elliptic curves

[TABLE]

as $(a,b)$ varies over elements in $(\mathbb{Z}/p\mathbb{Z})^{2}$ such that $4a^{3}+27b^{2}\not\equiv 0\,\rm$mod  $p$, where $p$ is a fixed prime. Let $(\frac{\cdot}{p})$ denote the Legendre symbol. More precisely, he proved that

[TABLE]

as $p\to\infty$. These are the moments predicted by the Sato-Tate conjecture and by standard probability theory one can deduce the relevant distribution. This moment calculation then implies the Sato-Tate distribution for the angles $\theta_{a,b}$ where we write

[TABLE]

where, if $E(\mathbb{F}_{p})$ denotes the number of $\mathbb{F}_{p}-$ points (including the point at infinity) on an elliptic curve $y^{2}=x^{3}+ax+b$, then $a_{p}(a,b)$ is $p+1-E(\mathbb{F}_{p})$. There were two key ingredients in Birch’s proof. The first was Deuring’s theorem [8] that there are a total of $H(t^{2}-4p)$ isomorphism classes of elliptic curves over $\mathbb{F}_{p}$ with $p+1-t$ points where $H(N)$ denotes the Hurwitz-Kronecker class number. The second ingredient is the Eichler-Selberg trace formula (see [10, Appendix]) which gives the trace of the $n$-th Hecke operator acting on the space of holomorphic cusp forms of even weight $k\geq 4$ with respect to the full modular group.

A natural question that arises from Birch’s paper is the order of the error term in the Sato-Tate distribution. More precisely, let us fix an interval $I=[\alpha,\beta]\subseteq[0,\pi]$. We want to count the number

[TABLE]

and give a quantitative estimate for

[TABLE]

where the Sato-Tate measure of the interval $I$ is given by $\mu_{ST}(I)=\frac{2}{\pi}\int_{I}\sin^{2}(\theta)d\theta.$

Using the discrepancy estimate of Niederreiter [19], Banks and Shparlinski [1] noted that

[TABLE]

would follow from the work of Katz [11] extended to their setting. This extension is not routine and appears in the work of Michel [14] where he deals with the case of one-parameter families of elliptic curves. There is also a related paper of Fisher [9]. The work of Michel [14] relies heavily on Weil II of Deligne [6]. For many of us working in classical analytic number theory, Weil II and its cohomological mysteries present formidable prerequisites that often represent a “black box” whose pronouncements must be accepted on faith. On the other hand, using the moment estimates in Birch’s proof, (1) was estimated by Miller and Murty in [15] as well, where only a log saving over the trivial estimate of $p^{2}$ was obtained. The goal of this paper is to show that the estimate (2) can be deduced using classical techniques, from just the Ramanujan-Petersson conjecture, now a theorem due to Deligne (which is implied by Weil I [5] and [7]) as well as the two key ingredients of Birch [2] mentioned earlier. The result is in fact, true in the more general case of elliptic curves over a finite field $\mathbb{F}_{q}=\mathbb{F}_{p^{r}}$. Let

[TABLE]

where we now have $a_{q}(a,b)=q+1-E(\mathbb{F}_{q})=2\sqrt{q}\cos\theta_{a,b}.$ We prove the following.

Theorem 1**.**

Assume the notation above. We then have

[TABLE]

We note that the result of Banks and Shparlinski, (and that in Theorem 1) gives a true error term only when the size of the interval $I$ is greater that $p^{-1/4+\varepsilon}$. This was improved (on average) by Baier and Zhao in [3] and recently by David, Koukoulopoulos and Smith [4] where the effective version of Birch’s theorem is shown to hold for intervals $I$ of length $\geq p^{-1/2+\varepsilon}$ although in these cases, the saving is only a power of a logarithm over the main term.

Acknowledgements

The authors would like to thank Igor Shparlinski for his helpful comments and the anonymous referee for suggestions that improved the exposition of this article.

2. Preliminaries

2.1. Isomorphism classes of elliptic curves

We briefly discuss the ingredients from the theory of counting elliptic curves, which will be needed for the proof. For more details, see [13] or [20].

For $p\neq 2,3$, consider the elliptic curve over $\mathbb{F}_{q}$ in Weierstrass form

[TABLE]

Analogous to the case of $\mathbb{F}_{p}$, the number of $\mathbb{F}_{q}$ points on $E$, given by $E(\mathbb{F}_{q})$ is $q+1-a_{q}(a,b)$. We have Hasse’s bound $|a_{q}(a,b)|\leq 2\sqrt{q}$. Two curves $E=E_{a,b}$ and $E^{\prime}=E_{a^{\prime},b^{\prime}}$ over $\mathbb{F}_{q}$ are isomorphic if there is an element $u\in\mathbb{F}_{q}^{*}$ such that $a^{\prime}=u^{4}a$ and $b^{\prime}=u^{6}b$. An automorphism of $E$ is an isomorphism from $E$ to $E$. Clearly, isomorphism of elliptic curves is an equivalence relation and the size of the equivalence class of $E$ is given by $\frac{\#\mathbb{F}_{q}^{*}}{\#\text{Aut }E}$.

For $t^{2}<4q$, Deuring [8] essentially showed that the number isomorphism classes of elliptic curves $E$ with $q+1-t$ points, weighted by $\#(\text{Aut }E)^{-1}$ is $H(t^{2}-4q)$, where $H(N)$ is the Hurwitz-Kronecker class number (see [13] for a detailed description of these numbers). Thus, for $t^{2}<4q$, the total number of curves $E$ over $\mathbb{F}_{q}$ with $q+1-t$ points is $(q-1)H(t^{2}-4q)$.

2.2. Chebyshev polynomials

The Chebyshev polynomials of the second kind, $U_{n}(x)$ for integers $n\geq 0$ are defined recursively in the following way:

[TABLE]

If $x=\cos\theta$, then the polynomials can be written explicitly as

[TABLE]

In our application of these polynomials, $x=\cos\theta_{a,b}$. It is not hard to see that

[TABLE]

where $\rho$ and $\bar{\rho}$ are the roots of the equation $y^{2}-ty+q=0$. Observe that

[TABLE]

a fact that will be needed later.

2.3. Beurling-Selberg Polynomials

The Beurling-Selberg polynomials have been frequently used to obtain effective results on equidistribution and in this note, we use this method to study the quantity $N_{I}(p)$. We give a brief introduction to these polynomials, see [16, Chapter 1] for a detailed exposition. Let $\chi_{J}(x)$ denote the characteristic function of the interval $J=[\alpha,\beta]\subseteq[0,1]$ and $M\geq 1$ be an integer. One can construct trigonometric polynomials $S_{M,J}^{-}(x)$ and $S_{M,J}^{+}(x)$ of degree less than or equal to $M,$ respectively called the minorant and majorant Beurling-Selberg polynomials for the interval $J$ such that

• (a)

For all $x\in\mathbb{R},\,S_{M,J}^{-}(x)\leq\chi_{I}(x)\leq S_{M,J}^{+}(x)$

• (b)

[TABLE]

• (c)

For $0<|m|\leq M,$

[TABLE]

Henceforth, we will supress the $J$ in the subscript of the Fourier coefficients with the understanding that the definition of these approximating polynomials depends on the interval $J$.

For $J=[\alpha,\beta]$ we will also use the following estimates, which follow from properties (b) and (c) listed above.

[TABLE]

and for $m>0,$

[TABLE]

These polynomials were used in [18] to study the ‘vertical’ Sato-Tate distribution in the case of modular forms.

3. Proof of the main theorem

Let $I\subseteq[0,\pi]$. We consider the angles $\left\{\frac{\theta_{a,b}}{2\pi},-\frac{\theta_{a,b}}{2\pi}\right\}$ and count when they occur in $I^{\prime}=I/2\pi$. Approximating using Beurling-Selberg polynomials, we have

[TABLE]

where, as noted in Section 2.2,

[TABLE]

denotes the $r$-th Chebyshev polynomial of the second kind evaluated at $\cos(\theta_{a,b})$. Note that $U_{0}(x)=1$. Therefore,

[TABLE]

Using (4) and (5), we see that

[TABLE]

where $\mu_{ST}(I)=\frac{2}{\pi}\int_{I}\sin^{2}\theta d\theta$ denotes the Sato-Tate measure of $I\subseteq[0,\pi]$. It remains to estimate the sums

[TABLE]

for $m=1,\ldots,M$. If we write $a_{q}(a,b)=t=2\sqrt{q}\cos\theta_{t}$ where $|t|\leq 2\sqrt{q}$, then we have

[TABLE]

where we group the curves into isomorphism classes, as discussed in Section 2.1. If $m$ is odd, writing

[TABLE]

we see that the sum in (8) is zero. This follows from the fact that in the polynomial $U_{m}\left(\frac{t}{2\sqrt{q}}\right),$ the parity of the powers of $t$ that appear is the same as that of $m$, so the terms corresponding to $t$ and $-t$ cancel each other when $m$ is odd.

On the other hand, using the Eichler-Selberg Trace formula (see [10] or [12]) we have for even $k\geq 4$,

[TABLE]

where $\delta(q,2)$ is $1$ when $q$ is a square and zero otherwise. Using the Ramanujan-Petersson bound for Hecke eigenvalues, we have

[TABLE]

using the fact that the dimension of the space of cusp forms of weight $k$ and full level grows like $k$. Therefore,

[TABLE]

where $q=p^{r}$. Going back to (8), letting $k=m+2$, we deduce that for even $m=2,\ldots,M$,

[TABLE]

Using (7), (11) and the estimate $(\widehat{S}^{\pm}_{M}(m)+\widehat{S}^{\pm}_{M}(-m))\ll 1/m$ (evident from (5)) in (6), we get

[TABLE]

Letting $M=\left\lfloor p^{1/4}r^{-1/2}\right\rfloor$, we get

[TABLE]

Using $\widehat{S}^{-}_{M}(m)$ the lower bound estimation is similar.

4. Concluding remarks

It is interesting to consider to what extent our error term is best possible. For example, for $q=p$, the sum in (9) is essentially $p^{-\frac{k-1}{2}}\operatorname{\text{ Tr }}T_{k}(p)$. If we accept the prediction that the Sato-Tate distribution corresponding to distinct Hecke eigenforms as discussed in [17], then we can arrange all the Fourier coefficients appearing in Tr $T_{k}(p)$ to be arbitrarily close to $2p^{\frac{k-1}{2}}$ simultaneously, for infinitely many primes $p$. Thus the estimate in (10) cannot be improved for all primes $p$. This does not however say anything about the combined error term in (12). Thus, the question of the optimal error term becomes an intruiging problem for future research.