Effective joint distribution of eigenvalues of Hecke operators

TL;DR

This paper extends the effective joint distribution results of eigenvalues of Hecke operators to higher level cusp forms using the Eichler-Selberg trace formula, building on prior work with Kuznetsov trace formula.

Contribution

It introduces a new method employing the Eichler-Selberg trace formula to analyze eigenvalue distributions for higher level cusp forms, expanding previous results.

Findings

Effective joint distribution of eigenvalues for higher level cusp forms

Use of Eichler-Selberg trace formula in this context

Extension of Lau and Wang's results to higher levels

Abstract

In 1997, Serre proved that the eigenvalues of normalised $p$-th Hecke operator $T^{'}_p$ acting on the space of cusp forms of weight $k$ and level $N$ are equidistributed in $[-2,2]$ with respect to a measure that converge to the Sato-Tate measure, whenever $N+k \to \infty$. In 2009, Murty and Sinha proved the effective version of Serre's theorem. In 2011, using Kuznetsov trace formula, Lau and Wang derived the effective joint distribution of eigenvalues of normalized Hecke operators acting on the space of primitive cusp forms of weight $k$ and level $1$. In this paper, we extend the result of Lau and Wang to space of cusp forms of higher level. Here we use Eichler-Selberg trace formula instead of Kuznetsov trace formula to deduce our result.

Equations

$$\{{\lambda }_i(n)\},1\le i\le s(N,k)$$

$$T_n^{{}^{\prime }}:=\frac{T_n}{n^{\frac{k-1}{2}}}$$

$$\left\{a_i(n)=\frac{{\lambda }_i(n)}{n^{\frac{k-1}{2}}},1\le i\le s(N,k)\right\},$$

$$d{\mu }_{\infty }=\frac{1}{2\pi }\sqrt{4-x^2}dx.$$

$$\underset{V\to \infty }{\lim }\frac{1}{V}\sum _{p\le V}\varphi \left(\frac{a_i(p)}{p^{\frac{k-1}{2}}}\right)={\int }_{\alpha }^{\beta }\varphi d{\mu }_{\infty }.$$

$$T_p^{{}^{\prime }}(N,k)=\frac{T_p(N,k)}{p^{\frac{k-1}{2}}}$$

$${\mu }_p:=\frac{p+1}{\pi }\frac{\sqrt{1-\frac{x^2}{4}}}{(p^{\frac{1}{2}}+p^{-\frac{1}{2}}{)}^2-x^2}dx$$

$$\frac{1}{s(N,k)}♯\left\{1\le i\le s(N,k):a_i(p)\in [\alpha ,\beta ]\right\}={\int }_{\alpha }^{\beta }{\mu }_p+O\left(\frac{\log p}{\log kN}\right).$$

$$\frac{♯\left\{1\le n\le s(1,k):\left(a_1^{(n)}(p_1),\dots ,a_r^{(n)}(p_r)\right)\in I\right\}}{s^{new}(1,k)}$$

$$={\int }_I\prod _{n=1}^{r}d{\mu }_{p_n}+O\left(\frac{r\log (p_1{p}_2\dots p_r)}{\log k}\right),$$

$$\frac{♯\left\{1\le n\le s(N,k):\left(a_1^{(n)}(p_1),\dots ,a_r^{(n)}(p_r)\right)\in I\right\}}{s(N,k)}$$

$$={\int }_I\prod _{n=1}^{r}d{\mu }_{p_n}+O\left(\frac{r\log (p_1{p}_2\cdots p_r)}{\log (kN)}\right),$$

$$\underset{V\to \infty }{\lim }\frac{1}{V}♯\{n\le V:x_n\in [\alpha ,\beta ]\}=\beta -\alpha .$$

$$\sum _{n\le V}e(m{x}_n)=o(V)\text{as V→∞}.$$

$$\underset{V\to \infty }{\lim }\frac{1}{V}\sum _{n\le V}f(x_n)={\int }_0^{1}f(t)dt.$$

$$\underset{V\to \infty }{\lim }\frac{♯\{n\le V:(x_1^{(n)},x_2^{(n)},\dots ,x_r^{(n)})\in I\}}{V}=\mu (I),$$

$$\sum _{n=1}^{V}e\left(\sum _{i=1}^r{m}_i{x}_n\right)=o(V)\text{mbox}asV\to \infty .$$

$$\underset{V\to \infty }{\lim }\frac{1}{V}\sum _{n\le V}f(x_1^{(n)},x_2^{(n)},\dots ,x_r^{(n)})={\int }_{I}f(x_1,x_2,\dots ,x_r)d{x}_{1}d{x}_2\cdots d{x}_r.$$

$$\underset{n\to \infty }{\lim }\frac{1}{♯A_n}\sum _{t\in A_n}f(t)={\int }_0^{1}f(x)d\mu .$$

$$c_m:=\underset{n\to \infty }{\lim }\frac{1}{♯A_n}\sum _{t\in A_n}e(mt)$$

$$\sum _{m=1}^V|c_m{|}^2=o(V).$$

$$\underset{V\to \infty }{\lim }\frac{1}{V}\sum _{n\le V}f(x_1,x_2,\dots ,x_r)={\int }_{I}f(x_1,x_2,\dots ,x_r)d{x}_{1}d{x}_2\cdots d{x}_r.$$

$$C_{\underline{m}}:=\underset{n\to \infty }{\lim }\frac{1}{{\prod }_{i=1}^r♯A_n}\sum _{(t_1,t_2,\dots ,t_r)\in \Omega }e(m_1{t}_1+m_2{t}_2+\cdots +m_r{t}_r).$$

$$B(△)=\{a{x}^2+bxy+c{y}^2:a,b,c\in \mathbb{Z},a>0,b^2-4ac=△\}.$$

$$b(△)=\{f(x,y)\in B(△):gcd(a,b,c)=1\}.$$

$$f(x,y)\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right):=f(\alpha x+\beta y,\gamma x+\delta y).$$

$$h_w(-3)$$

$$h_w(-4)$$

$$h_w(△)$$

$$\displaystyle A_{1}(n)=\left\{\begin{array}[]{l l}n^{\frac{k}{2}-1}\left(\frac{k-1}{12}\right)\psi(N)&\text{if $n$ is a square}\\ 0&\text{otherwise,}\end{array}\right.$$

$$A_2(n)=-\frac{1}{2}\sum _{\genfrac{}{}{0ex}{}{t\in \mathbb{Z}}{t^2<4n}}\frac{{\varrho }^{k-1}-{\bar{\varrho }}^{k-1}}{\varrho -\bar{\varrho }}\sum _g{h}_w\left(\frac{t^2-4n}{g^2}\right)\mu (t,g,n),$$

$$\mu (t,g,n)=\frac{\psi (N)}{\psi (\frac{N}{N_g})}M(t,n,N{N}_g),$$

Full text

1. Introduction

Let $S(N,k)$ be the space of all holomorphic cusp forms of weight $k$ with respect to $\varGamma_{0}(N).$ For any positive integer $n$, let $T_{n}(N,k)$ be the $n$-th Hecke operator acting on $S(N,k)$. Let $s(N,k)$ denote the dimension of the space $S(N,k)$. For a positive integer $n\geq 1$, let

[TABLE]

denote the eigenvalues of $T_{n}$, counted with multiplicity. For any positive integer $n$, let

[TABLE]

be the normalized Hecke operator acting on $S(N,k)$ with eigenvalues

[TABLE]

counted with multiplicity. By the celebrated theorem of Deligne [4], which proves the famous Ramanujan conjecture, we know that for any prime $p$, such that $p$ coprime to $N$, the eigenvalue of $T_{p}^{{}^{\prime}}$ lies in the interval $[-2,2]$. The recently proved Sato-Tate conjecture by Barnet-Lamb, Geraghty, Harris and Taylor [1], [2] and [5] says that if $a_{i}(p),1\leq i\leq r$ is a $p$-th normalized Hecke eigenvalue, then the family $\{a_{i}(p)\}$ is equidistributed in $[-2,2]$ as $p\to\infty$ with respect to the Sato-Tate measure

[TABLE]

More precisely, the Sato-Tate conjecture states that for any continuous function $\phi:\mathbb{R}\rightarrow\mathbb{R}$, positive integer $V$ and any interval $[\alpha,\beta]\subset[-2,2]$

[TABLE]

In 1997, Serre [11] studied the “vertical” Sato-Tate conjecture by fixing a prime $p$ and varying $N$ and $k.$ In particular, he proved the following theorem:

Theorem 1**.**

Let $p$ be a prime number. Let $\{(N,k)\}$ be a sequence of pairs of positive integers such that $k$ is even and $p$ is coprime to $N$ and $N+k\to\infty.$ Then the family of eigenvalues of the normalized $p$-th Hecke operator

[TABLE]

is equidistributed in the interval $\Omega=[-2,2]$ with respect to the measure

[TABLE]

Remark 2**.**

Also in 1997, Conrey, Duke and Farmer [3] studied a special case of above result by fixing $N=1.$

In 2009, Murty and Sinha [8] investigated the effective/quantitative version of Serre’s results, in which they give explicit estimate on the rate of convergence. They proved the following theorem

Theorem 3**.**

Let p be a fixed prime. Let $\{(N,k)\}$ be a sequence of pairs of positive integers such that $k$ is even, $p$ is coprime to $N$. For an interval $[\alpha,\beta]\subset[-2,2],$

[TABLE]

As a continuation of their paper, in 2010, Murty and Sinha [8] proved a quantitative equidistribution theorem for the eigenvalues of Hecke operators acting on the space $S^{new}(N,k)$. Recently Lau and Wang [7] computed the rate of convergence in Sarnak’s [10] result using the Kuznetsov trace formula. Indeed they proved the joint distribution of eigenvalues of the Hecke operators quantitatively for primitive Maass forms of level 1 and stated that the same hold true for primitive holomorphic cusp forms. More precisely, they proved the following theorem:

Theorem 4**.**

Let $p_{1},p_{2},,...,$ and $p_{r}$ be distinct primes. Let $k$ be a positive even integer such that $r\operatorname{log}\,(p_{1}p_{2},...,p_{r})\leq\delta\operatorname{log}\,k$, for some small absolutely constant $\delta$. Let $a_{i}^{{}^{\prime}}(p_{i})$ be the eigenvalues of normalized Hecke operators $T_{p_{i}}^{{}^{\prime}}$ acting on $S^{new}(1,k)$. For any $I=[\alpha_{n},\beta_{n}]^{r}\subset[-2,2]^{r}$

[TABLE]

where $\displaystyle d\mu_{p_{n}}=\frac{p_{n}+1}{\pi}\frac{\sqrt{1-\frac{x^{2}}{4}}}{\left({p_{n}}^{\frac{1}{2}}+{p_{n}}^{-\frac{1}{2}}\right)^{2}-x^{2}}dx.$

They have remarked that their methods do work for primitive forms in higher level. In this paper we extend their result to the cusp forms of any level $N$ using “Eichler-Selberg trace formula”. Precisely, we prove the following theorem:

Theorem 5**.**

Let $p_{1},p_{2},...,$ and $p_{r}$ be distinct primes. Let $k$ be positive even integer such that $r\operatorname{log}\,(p_{1}p_{2},...,p_{r})\leq\delta\operatorname{log}k$, for some small absolutely constant $\delta$. For $1\leq i\leq r,$ let $a_{i}^{(n)}(p_{i}),1\leq n\leq s(N,k)$ be the eigenvalues of normalized Hecke operators $T_{p_{i}}^{{}^{\prime}}$ acting on $S(N,k)$. For any $I=[\alpha_{n},\beta_{n}]^{r}\subset[-2,2]^{r}$

[TABLE]

where $\displaystyle d\mu_{p_{n}}=\frac{p_{n}+1}{\pi}\frac{\sqrt{1-\frac{x^{2}}{4}}}{({p_{n}}^{\frac{1}{2}}+{p_{n}}^{-\frac{1}{2}})^{2}-x^{2}}dx$ and the implied constant is effectively computable.

2. Equidistribution and its extension

A sequence of real numbers $\{x_{n}\}_{n=1}^{\infty}$ is said to be uniformly distributed or (equidistributed) (mod  1) if for any interval $[\alpha,\beta]\subset[0,1],$ we have

[TABLE]

In 1916, Weyl [12] proved the following criterion for uniform distribution (mod  1). A sequence $\{x_{n}\}_{n=1}^{\infty}$ is uniformly distributed if and only if for any integer $m\neq 0$

[TABLE]

Since the set of trigonometric polynomials is dense in $C^{1}[0,1]$, the above criterion of Weyl is equivalent to the assertion that, for any continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$, we have

[TABLE]

A sequence of tuples $\left\{(x_{1}^{(n)},x_{2}^{(n)},\ldots,x_{r}^{(n)})\right\}$ in ${\mathbb{R}}^{r}$ is said to be uniformly distributed or (equidistributed) (mod  1) if for every $I=\prod_{n=1}^{r}[\alpha_{n},\beta_{n}]\subset[0,1]^{r}$, we have

[TABLE]

where $\mu$ is the usual Lebesgue measure on $\mathbb{R}^{r}$. In the same paper Weyl [12] extended his criterion of equidistribution to the higher dimension as follows:

A sequence of tuples $\left\{(x_{1}^{(n)},x_{2}^{(n)},\ldots,x_{r}^{(n)})\right\}$ in $\mathbb{R}^{r}$ is uniformly distributed if and only if for any integers $m_{1},m_{2},,...,,m_{r}\neq 0$

[TABLE]

Note that as in the one variable case, the set of trigonometric polynomials is also dense in $C([0,1]^{r}),$ the above criterion is equivalent to the following statement:

A sequence of tuples $\left\{(x_{1}^{(n)},x_{2}^{(n)},\ldots,x_{r}^{(n)})\right\}$ in ${\mathbb{R}}^{r}$ is uniformly distributed if and only if for any continuous function $f:{\mathbb{R}}^{r}\rightarrow\mathbb{R},$

[TABLE]

Now we define the set equidistribution as follows:

A sequence of finite multi sets $A_{n}$ with $\sharp A_{n}\to\infty$ is said to be set equidistributed (mod  1) with respect to a probability measure $\mu$ if for every continuous function $f\in C^{1}[0,1]$, we have

[TABLE]

The criterion of Schoenberg and Wiener says that the sequence $\{A_{n}\}$ is set equidistributed with respect to some positive continuous measure if and only if the Weyl limit

[TABLE]

exists and

[TABLE]

For our purpose, let us define the set equidistribution in higher dimension. A tuples of finite multi set say $\Omega=(A_{1}^{(n)},A_{2}^{(n)},\ldots,A_{r}^{(n)})$ with $\sharp A_{i}^{(n)}\to\infty$ as $n\to\infty$ for all $i=1,2,,...,,r$ is said to be set a set equidistributed (mod  1) with respect to a probability measure $\mu$ if for every continuous function $f\in C^{1}([0,1]^{r})$, and $I=[0,1]^{r}$ we have

[TABLE]

With this generalization, we define the Weyl limit as

[TABLE]

3. Eichler Selberg Trace Formula and its Estimations

In this section, we use Eichler-Selberg trace formula, as one of our important tool to prove the main theorem, which is a formula for the trace of $T_{n}$ acting on $S(N,k)$ in terms of class number of binary quadratic forms and few others arithmetic functions. We follow the presentation of [8]. For a non-negative integer $\vartriangle\equiv 0,1$ (mod  4), let $B(\vartriangle)$ be the set of all positive definite binary quadratic forms with discriminant $\vartriangle$. That is

[TABLE]

We denote the set of primitive forms by

[TABLE]

Now we define an action of the full modular group $SL_{2}(\mathbb{Z})$ on $B(\vartriangle)$ as follows:

[TABLE]

Note that the above action takes primitive forms to primitive forms. We know that the above action has finitely many orbits. We define $h(\vartriangle)$ to be the number of orbits of $b(\vartriangle)$. Let $h_{w}$ be defined as follows:

[TABLE]

We define some arithmetical functions which are going to useful to state the Eichler-Selberg trace formula.

Let

[TABLE]

where $\displaystyle\psi(N)=N\prod_{p|N}\left(1+\frac{1}{p}\right)$. Let

[TABLE]

where $\varrho$ and $\bar{\varrho}$ are the zeros of the polynomial $x^{2}-tx+n$, the inner sum runs over positive divisors $g$ of $\frac{(t^{2}-4n)}{g^{2}}\in\mathbb{Z}$ is congruent to [math] or $1\ (mod\ 4)$ and $\mu(t,g,n)$ is given by

[TABLE]

with $N_{g}=gcd(N,g)$ and $M(t,n,k)$ denote the number of solutions of the congruence $x^{2}-tx+n\equiv 0\ \ \pmod{K}$. Now, we let

[TABLE]

where $\phi$ denotes the Euler’s function and in the first summation, if there is a contribution from the term $d=\sqrt{n}$, it should be multiplied by $\frac{1}{2}.$ In the inner sum, we also need the condition that $gcd\ (c,\frac{N}{c})$ divides $gcd(N,\frac{n}{d}-d).$ Finally, let

[TABLE]

Theorem 6**.**

For any positive integer $n$, let $Tr(T_{n})$ be the trace of $T_{n}$ acting on $S(N,k).$ Then, we have,

[TABLE]

We use the above Eichler-Selberg trace formula to prove the following Proposition:

Proposition 7**.**

For any positive integers $m_{1},m_{2},\ldots,m_{r}$ we have,

[TABLE]

Proof.

Consider

[TABLE]

where $B_{i}(m_{1}m_{2}\cdots m_{r})=\displaystyle\frac{A_{i}(p_{1}^{m_{1}}\cdots p_{r}^{m_{r}})}{(p_{1}^{m_{1}}\cdots p_{r}^{m_{r}})^{\frac{k-1}{2}}}$, for all $i=1,2,3,4.$ We use the estimates of [8] and get the estimates for each $B_{i}(m_{1}m_{2}\cdots m_{r}),$ for all $i=1,2,3,4.$ First, we consider $i=1$

[TABLE]

Now, we shall consider $i=2$ as follows.

[TABLE]

For $i=3$, we get,

[TABLE]

Finally, for $i=4$, we get,

[TABLE]

Combining all the estimates above, we get

[TABLE]

where $d(N)$ is the divisor function. ∎

4. The measure $\mu$

Following the definition from Section 2, for non-negative integers $m_{1},m_{2},\ldots,m_{r}$, we get,

[TABLE]

where $c_{m_{i}}$ are the $m_{i}$-th Weyl limit of the family $\left\{\frac{\pm\theta_{i}^{(n)}}{2\pi}\right\}$ and from Theorem 18 of [8], we have

[TABLE]

Note that if $m_{1},m_{2},\ldots,m_{r}$ are all zero then $C_{0,\ldots,0}=1$. Define the measure

[TABLE]

where

[TABLE]

The above $F(x_{1},\ldots,,x_{r})dx_{1}\cdots dx_{r}$ determines a measure on $[0,1]^{r}$ and is the distribution function for the tuples of numbers

[TABLE]

The measure giving the distribution of $(\cos\theta_{1}^{(n)}(p_{1}),\ldots,\cos\theta_{r}^{(n)}(p_{r}))$ is therefore

[TABLE]

Thus, the distribution of the tuples of numbers

[TABLE]

is given by $\prod_{i=1}^{r}\mu_{p_{i}}$ after an easy change of variable.

5. Chebychev polynomials and the trace of Hecke operators

For any integer $n\geq 0$, the $n$-th Chebychev polynomial $X_{n}(x)$ is defined as follows:

[TABLE]

Serre proved the following result in [11].

Lemma 8**.**

We have $T^{{}^{\prime}}_{p^{m}}=X_{m}(T_{p}).$

From [8, page 697], when $m_{i}=1,$ we have,

[TABLE]

Since, for all integers $m\geq 2$,

[TABLE]

we have for $m_{i}\geq 2,$

[TABLE]

Now we prove the following theorem,

Theorem 9**.**

For all non zero positive integers $m_{1},m_{2},\ldots,m_{r}$ consider $\underline{m}=(m_{1},m_{2},\ldots,m_{r}),$ the Weyl limits $C_{\underline{m}}$ are given by

[TABLE]

Moreover,

[TABLE]

Proof.

For any integers $m_{1},m_{2},\ldots,m_{r}\geq 2$, using (5), we have

[TABLE]

where $Z=\{m_{1},m_{2},\ldots,m_{r},m_{1}-2,m_{2}-2,\ldots,m_{r}-2\}$. We know that if a linear operator $T$ is diagonalizable and $\lambda$ is an eigenvalue of $T$, then, for any polynomial $P(x)$, the eigenvalue of $P(T)$ is $P(\lambda)$. Since the Hecke operators $T_{m}$ and $T_{n}$ commutes with each other, there exists an ordered basis such that every Hecke operator can be represented by a diagonal matrix with respect to the basis. Using all the above facts, (5) equals

[TABLE]

Now

[TABLE]

Using (8) and the fact that $s(N,k)C_{\underline{m}}\ \text{behaves like}\ B_{1}(m_{1}\cdots m_{r}),$ we have

[TABLE]

[TABLE]

∎

6. Effective versions

To prove the effective result, we use the following variant of the Erdös-Turán inequality that can be found in [6, Proposition 7.1].

Theorem 10**.**

For any $S=\prod_{n=1}^{r}[\alpha_{n},\beta_{n}]\subset[0,\frac{1}{2}]^{r}$ and a sequence of tuples of number $\{(x_{m_{1}},\ldots,x_{m_{r}})\}\in[0,\frac{1}{2}]^{r},$ we define

[TABLE]

and

[TABLE]

For $I=\prod_{n=1}^{r}[\alpha_{n},\beta_{n}],$ we have

[TABLE]

for any integers $V,M\geq 1$, where

[TABLE]

[TABLE]

for $(m_{1},\ldots,m_{r})\in\mathbb{Z}^{r}$ and

[TABLE]

and

[TABLE]

with $\|F_{t}\|_{\infty}=\max_{x\in[0,1]}|F_{t}(x)|.$

Proof of Theorem 5.

Choose $\theta_{i}^{(n)}(p_{i})\in[0,\frac{1}{2}]$ such that $2\cos 2\pi\theta_{i}^{(n)}(p_{i})=a_{i}^{(n)}(p_{i}).$ Given any sub interval $I\subset[-2,2]$ choose sub interval $I^{{}^{\prime}}\subset[0,\frac{1}{2}]$ so that $\theta_{i}^{(n)}(p_{i})\in I^{{}^{\prime}}$ if and only if $a_{i}^{(n)}\in I.$ By Theorem 10, we have

[TABLE]

Using Proposition 7, we get

[TABLE]

Since the main contribution is coming from $\left(p_{1}^{\frac{3m_{1}}{2}}\cdots p_{r}^{\frac{3m_{r}}{2}}\right)$, by choosing $\displaystyle M=\frac{\operatorname{log}\,kN}{\operatorname{log}\,p_{1}p_{2}\cdots p_{r}}$, we get the required result. $\hfill\Box$

Acknowledgments: The author would like to thank Prof. M. Ram Murty and Dr. Kaneenika Sinha for useful discussions in the earlier version of this manuscript. I am also thankful to Prof. R. Thangadurai and Dr. Jaban Meher for their suggestions to improve the presentation of the paper.