A note on the dimension of the largest simple Hecke submodule

Sandro Bettin Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy bettin@dima.unige.it , Corentin Perret-Gentil Centre de recherches mathématiques, Université de Montréal, Montréal, Canada corentin.perretgentil@gmail.com and Maksym Radziwiłł Caltech, Department of Mathematics, 1200 E California Blvd, Pasadena, CA, 91125, USA maksym.radziwill@gmail.com

Abstract.

For $k\geq 2$ even, let $d_{k,N}$ denote the dimension of the largest simple Hecke submodule of $S_{k}(\Gamma_{0}(N);\mathbb{Q})^{\text{new}}$ . We show, using a simple analytic method, that $d_{k,N}\gg_{k}\log\log N/\log(2p)$ with $p$ the smallest prime co-prime to $N$ . Previously, bounds of this quality were only known for $N$ in certain subsets of the primes. We also establish similar (and sometimes stronger) results concerning $S_{k}(\Gamma_{0}(N),\chi)$ , with $k\geq 2$ an integer and $\chi$ an arbitrary nebentypus.

1. Introduction

For an integral weight $k\geq 2$ and a level $N\geq 1$ , the anemic Hecke $\mathbb{Q}$ -algebra

\mathbb{T}:=\mathbb{Q}[T_{n}:(n,N)=1],

generated by the Hecke operators $T_{n}$ , acts on the space of cusp forms $S_{k}(\Gamma_{0}(N))$ .

Simple Hecke submodules of $S_{k}(\Gamma_{0}(N))$ of dimension $d$ correspond to $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ -orbits of size $d$ of (arithmetically) normalized eigenforms $f\in S_{k}(\Gamma_{0}(N))$ . When $k=2$ , the work of Shimura also gives a correspondence with simple factors of dimension $d$ of the Jacobian $J_{0}(N)$ of the modular curve $X_{0}(N)$ . Thus it is interesting to ask about the dimension $d_{k,N}$ of the largest simple Hecke submodule of $S_{k}(\Gamma_{0}(N))$ , or equivalently the maximal degree of Hecke fields of normalized eigenforms.

Maeda [HM97] postulated that $S_{k}(\Gamma_{0}(1))$ is a simple Hecke module for all even $k\geq 12$ . This deep conjecture implies among other things that $L(\tfrac{1}{2},f)\neq 0$ for all $f\in S_{k}(\Gamma_{0}(1))$ , see [CF99]. When $N>1$ , there is an obstruction to simplicity due to the Atkin–Lehner involutions, but numerical evidence suggests that this is the only asymptotic barrier when $N$ is square-free. This led Tsaknias [Tsa14] to suggest the following generalization of Maeda’s conjecture (see also [DT16] for non-square-free levels):

Conjecture 1.

For $k\geq 2$ even and large enough and $N$ square-free, the number of Galois orbits of newforms in $S_{k}(\Gamma_{0}(N))$ is $2^{\omega(N)}$ . In particular, for any fixed $\varepsilon>0$ we have

d_{k,N}\gg_{k,\varepsilon}N^{1-\varepsilon}.

That is, there exists a constant $c(k,\varepsilon)>0$ depending at most on $k$ and $\varepsilon$ such that $d_{k,N}>c(k,\varepsilon)N^{1-\varepsilon}$ for all square-free $N\geq 1$ .

There is a massive gap between Conjecture 1 and the unconditional results. Through an equidistribution theorem for Hecke eigenvalues, Serre [Ser97] was the first to establish that $d_{k,N}\rightarrow\infty$ as $k+N\rightarrow\infty$ . Subsequently, by making Serre’s equidistribution theorem effective, Royer [Roy00] and Murty–Sinha [MS09] showed that $d_{k,N}\gg_{k,p}\sqrt{\log\log N}$ for any $p\nmid N$ . In the particular case where $N$ lies in a restricted set of primes, this bound has been improved by several authors. Extending a method of Mazur to all even weights, Billerey and Menares [BM16, Theorem 2] obtained that $d_{k,N}\gg_{k}\log{N}$ when $N\geq(k+1)^{4}$ is in a explicit set primes of lower natural density $\geq 3/4$ . When the lower bound is fixed in advance and one looks for a level with a given number of prime divisors attaining it, see also [DJUR15]. When $N\equiv 7\pmod{8}$ is prime, Lipnowski–Schaeffer [LS18, Corollary 1.7] also showed that $d_{2,N}\gg\log\log N$ , which can be significantly improved for $N$ in certain subsets of the primes under certain well-known conjectures and heuristics.

In this paper we show that bounds of Lipnowski–Schaeffer quality can be obtained for all levels and integer weights. Our method is however, analytic and we believe simpler than the one in [LS18].

Theorem 1.

Let $k\geq 2$ even and $N\geq 1$ be integers. Then the dimension of the largest simple Hecke submodule of $S_{k}(\Gamma_{0}(N))^{\text{new}}$ is

d_{k,N}\gg_{k}\frac{\log\log{N}}{\log(2p_{N})},

as $N\rightarrow\infty$ , where $p_{N}$ denotes the smallest prime co-prime to $N$ .

Since the vast majority of integers $N$ have a small co-prime factor, this bound is essentially asserting that $d_{k,N}\gg_{k}\log\log N$ . Theorem 1 appears to be the first bound of “ $\log\log N$ strength” for any even weight $k\geq 4$ , and in the case $k=2$ , without restriction on the level.

We state below a more general and precise form of Theorem 1 that holds in the presence of a nebentypus.

Theorem 2.

Let $k\geq 2$ and $N\geq 1$ be integers. Let $p\nmid N$ and let $\chi:(\mathbb{Z}/N)^{\times}\to\mathbb{C}^{\times}$ be a homomorphism such that $\chi(-1)=(-1)^{k}$ . Then the maximum size of the $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ -orbits of newforms $f\in S_{k}(\Gamma_{0}(N),\chi)$ is

\geq\frac{2}{(k-1)\log(4p)}\cdot\log\left(\frac{\log N}{2\pi\log p}\right)

for all sufficiently large $N$ (in terms of $k$ ).

By definition, the same lower bound holds for the maximum degree of the Hecke fields $K_{f}$ of newforms $f$ (see Section 2). Note that $K_{f}$ always contains the cyclotomic field $\mathbb{Q}(\zeta_{\operatorname{ord}(\chi)})$ generated by the values of $\chi$ (a consequence of the Hecke relations at $p^{2}$ , see Lemma 2 below), so the trivial lower bound in both cases is $\varphi(\operatorname{ord}(\chi))$ .

Remark 1.

The result of Billerey–Menares mentioned above actually shows that when $\ell\geq(k+1)^{4}$ belongs to an explicit set $\mathcal{L}$ of primes with lower density $\geq 3/4$ , there exists a normalized eigenform $f\in S_{k}(\Gamma_{0}(\ell))^{\text{new}}$ with $\deg(K_{f})\gg_{k}\log{\ell}$ . Hence, for $\varepsilon>0$ , if an integer $N$ has a prime factor $\ell>N^{\varepsilon}$ that lies in $\mathcal{L}$ , then $\deg(K_{f})\gg_{k,\varepsilon}\log{N}$ for some $f\in S_{k}(\Gamma_{0}(\ell))\hookrightarrow S_{k}(\Gamma_{0}(N))$ . Hence, Theorem 2 with “newform” replaced by the weaker conclusion “normalized eigenform” would follow from [BM16, Theorem 2] for almost all integers $N$ .

In certain special situations it can be shown that the degree of the number field $K_{f}$ is large for all newforms $f\in S_{k}(\Gamma_{0}(N),\chi)$ . For instance, when $p^{r}\mid N$ Brumer [Bru95, p.3, Theorem 5.5, Remark 5.7] showed that $K_{f}$ contains the maximal real subfield of the $p^{s}$ -th roots of unity, where $s=\lceil{\frac{r}{2}-1-\frac{1}{p-1}}\rceil$ (see also [Mat10, CE04]).

We exhibit a similar phenomenon which sometimes allows to significantly improve on Theorem 2 and the trivial bound $\deg K_{f}\geq\varphi(\operatorname{ord}\chi)$ , when $k$ is odd, depending on the nebentypus $\chi$ and the factorization of $N$ .

Theorem 3.

Let $k\geq 3$ be an odd integer, $N\geq 1$ be square-free, $\chi:(\mathbb{Z}/N)^{\times}\to\mathbb{C}^{\times}$ be a homomorphism such that $\chi(-1)=(-1)^{k}$ , and decompose

N_{2}=\prod_{\begin{subarray}{c}p\mid N\\ \chi_{p}=1\end{subarray}}p,\hskip 28.45274pt\chi=\prod_{p\mid N}\chi_{p},\hskip 14.22636pt\text{with}\hskip 14.22636pt\chi_{p}:(\mathbb{Z}/p)^{\times}\to\mathbb{C}^{\times}.

Then, for any newform $f\in S_{k}(\Gamma_{0}(N),\chi)$ ,

\deg{K_{f}}\geq\varphi(\operatorname{ord}(\chi))\cdot 2^{\omega(N_{2})-\omega((N_{2},2\operatorname{ord}(\chi)))-1},

In particular, if $(N_{2},2\operatorname{ord}(\chi))=1$ , then

\deg{K_{f}}\geq\varphi(\operatorname{ord}(\chi))\cdot 2^{\omega(N_{2})-1}.

For example, given $\varepsilon>0$ and $k\geq 3$ odd, for a “typical” square-free integer $N$ and $\chi$ a random quadratic character mod $N$ (resp. the trivial character), we get

\deg K_{f}\gg_{\varepsilon}(\log{N})^{\frac{\log{2}}{2}-\varepsilon}\qquad(\text{resp. }\gg_{\varepsilon}(\log{N})^{\log{2}-\varepsilon})

for all newforms $f\in S_{k}(\Gamma_{0}(N),\chi)$ . In fact it is possible to extend Theorem 3 to the case of non-square-free $N$ , but we maintain this restriction to keep the exposition simple.

A short outline of the proofs

We will now say a few words about the proof of these theorems and the limitations of our method of proof.

The proof of Theorem 1 and Theorem 2 proceeds by observing that if we can find a newform $f$ for which the eigenvalue $a_{f}(p_{N})$ is abnormally small in absolute value but non-zero, then the degree of the corresponding Hecke field $K_{f}$ needs to be large (see Proposition 1). We then use the equidistribution of Hecke eigenvalues (in the form of Murty–Sinha) to prove the existence of such an $f$ . This contrasts with the previous analytic approaches in which one probed (using the equidistribution of Hecke eigenvalues) the neighborhood of every algebraic integer up to a certain height.

The proof of Theorem 3 proceeds by first noticing that by strong multiplicity one, the number field $\mathbb{Q}(a_{f}(n):n\geq 1)$ coincides with $K_{f}=\mathbb{Q}(a_{f}(n):(n,N)=1)$ . Subsequently we focus exclusively on the ramified primes $p\mid N$ . For $k$ odd, the coefficient of $f$ at $p\mid N_{2}$ is equal to $\sqrt{p}$ multiplied by a factor lying in a small extension of $K_{f}$ (the eigenvalue of an Atkin–Lehner operator). Considering all these divisors yields the factor $2^{\omega(N_{2})}$ .

Limitations of the method

The best result that the method of proof of Theorem 1 and Theorem 2 can theoretically deliver is for each $k$ even and $N\geq 1$ the existence of an $f\in S_{k}(\Gamma_{0}(N))$ such that $\deg K_{f}\gg_{k}\log N$ . To see this consider for simplicity $k$ fixed and $N$ odd. Then we expect that the coefficients $a_{f}(2)$ with $f$ varying in $S_{k}(\Gamma_{0}(N))$ behave as a collection of roughly $\asymp_{k}N^{1+o(1)}$ random numbers distributed according to the Sato-Tate law. Therefore by linearity of expectation for any given $\varepsilon>0$ we expect that there exists a form $f\in S_{k}(\Gamma_{0}(N))$ with $0<|a_{f}(2)|\ll_{k}N^{-1+\varepsilon}$ and moreover that this is best possible up to the factor $N^{\varepsilon}$ . Plugging this into Proposition 1 would result in a lower bound $\deg K_{f}\gg_{k}\log N$ for some $f\in S_{k}(\Gamma_{0}(N))$ . Note that the existence of a $\delta>0$ such that for all $k$ fixed and $N$ odd there exists an $f\in S_{k}(\Gamma_{0}(N))$ with $0<|a_{f}(2)|\ll_{k}N^{-\delta}$ would be also enough to obtain the lower bound $\deg K_{f}\gg_{k}\log N$ for some $f\in S_{k}(\Gamma_{0}(N))$ .

Acknowledgments

The work of the first author is partially by PRIN 2015 “Number Theory and Arithmetic Geometry”. The third author would like to acknowledge the support of a Sloan fellowship. We would like to thank Nicolas Billerey, Armand Brumer, and Ricardo Menares for comments on the manuscript. We would like to thank the referees for a careful reading of the paper and useful suggestions.

2. Proof of Theorem 1 and Theorem 2

Throughout let $k\geq 2$ and $N\geq 1$ be integers, and $\chi:(\mathbb{Z}/N)^{\times}\to\mathbb{C}^{\times}$ a homomorphism such that $\chi(-1)=(-1)^{k}$ . Let $f\in S_{k}(\Gamma_{0}(N),\chi)$ be a normalized eigenform with Fourier expansion

f(z):=\sum_{n\geq 1}a_{f}(n)e(nz),\qquad a_{f}(1)=1,\qquad e(z):=e^{2\pi iz}.

Given a prime $p\nmid N$ , we also define (for reasons that will become clear when proving Lemma 1)

a_{f}^{\prime}(p)=\frac{a_{f}(p)}{2p^{\frac{k-1}{2}}\sqrt{\chi(p)}}\in\mathbb{R},

for a fixed choice of square root.

Since simple Hecke submodules of $S_{k}(\Gamma_{0}(N))$ of dimension $d$ correspond to $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ -orbits of size $d$ of (arithmetically) normalized eigenforms $f\in S_{k}(\Gamma_{0}(N))$ (see [DI95]), it suffices to obtain lower bounds for

\max_{f\in S_{k}(\Gamma_{0}(N),\chi)}\deg K_{f},\qquad K_{f}=\mathbb{Q}\left(a_{f}(n):(n,N)=1\right),

where $f$ runs over newforms, to prove Theorems 1 and 2.

The first input to our argument is a simple lemma from diophantine approximation, that allows to pass from small values of $|a_{f}(p)|$ to lower bounds for the degree of the Hecke field.

Proposition 1.

If $p\nmid N$ and $a_{f}(p)\neq 0$ , then

\deg\mathbb{Q}(a_{f}(p))\geq\frac{2}{k-1}\cdot\frac{\log{\frac{1}{|a_{f}^{\prime}(p)|}}}{\log(4p)}.

Proof.

Since $a_{f}(p)$ is an algebraic integer [DI95, Corollary 12.4.5], its norm is a nonzero integer. Thus if we denote by $g$ the degree of $a_{f}(p)$ and by $a_{f,1}(p),\ldots,a_{f,g}(p)$ all of the conjugates of $a_{f}(p)$ (including $a_{f}(p)$ itself), then,

\prod_{i=1}^{g}|a_{f,i}(p)|\geq 1.

By Deligne’s proof of the Ramanujan–Petersson conjecture for $f$ [Del71], $a_{f}(p)$ is the sum of two $p$ -Weil numbers of weight $k-1$ , so $|a_{f,i}(p)|\leq 2p^{\frac{k-1}{2}}$ for all $i$ . Therefore,

\prod_{i=1}^{g}|a_{f,i}(p)|\leq|a_{f}(p)|\left(2p^{\frac{k-1}{2}}\right)^{g-1}

and the claim follows. ∎

Remark 2.

If $\Gamma_{f}\leq\operatorname{Gal}(K_{f}/\mathbb{Q})$ is the group of inner twists of $f$ (see [Rib80, Section 3], [Rib85, Section 3]), then the proof of Proposition 1 shows that the lower bound can actually be improved by a factor of $|\Gamma_{f}|$ (or even $|\Gamma_{f}|^{2}$ if $\chi(p)\in\mathbb{Q}^{\times}$ ). In the case $k=2$ , $\chi=1$ , $N$ square-free, there are no nontrivial inner twists, but otherwise it is believed that $|\Gamma_{f}|$ could become large; if $\chi^{2}\neq 1$ , there is always a nontrivial inner twist given by conjugation.

We will now use the equidistribution of Hecke eigenvalues to exhibit a newform $f\in S_{k}(\Gamma_{0}(N),\chi)^{\text{new}}$ for which $a_{f}(p)$ is abnormally small, yet non-zero. This will therefore give a lower bound for the degree of $\mathbb{Q}(a_{f}(p))$ and thus a lower bound for the degree of $K_{f}$ .

Lemma 1.

Let $p\nmid N$ . There exists a newform $f\in S_{k}(\Gamma_{0}(N),\chi)^{\text{new}}$ such that,

(1)

0<|a_{f}^{\prime}(p)|\leq\frac{\pi}{2}\cdot\frac{p+1}{p}\cdot\frac{\log p}{\log N}

for all sufficiently large $N$ (in terms of $k$ ).

Proof.

Let $B_{k}(\Gamma_{0}(N),\chi)$ be the $\overline{\mathbb{Q}}$ -basis of $S_{k}(\Gamma_{0}(N),\chi)^{\text{new}}$ composed of the $d_{k,N,\chi}$ newforms at level $N$ . For $(n,N)=1$ , let us also normalize Hecke operators acting on $S_{k}(\Gamma_{0}(N),\chi)^{\text{new}}$ as $T_{n}^{\prime}:=T_{n}/(2n^{\frac{k-1}{2}}\sqrt{\chi(n)})$ . By [Ser97, Sections 5.1, 5.3], the normalized eigenvalues $(a_{f}^{\prime}(p))_{f\in B_{k}(\Gamma_{0}(N),\chi)}$ are distributed in $[-1,1]$ as $N\to\infty$ according to a measure converging to the Sato–Tate measure as $p\to\infty$ .

For $A\in(0,1)$ , let us give a lower bound on

\displaystyle C_{k,N,\chi}(A)

\displaystyle:=

\displaystyle\frac{|\{f\in B_{k}(\Gamma_{0}(N),\chi):0<|a^{\prime}_{f}(p)|\leq A\}|}{d_{k,N,\chi}}.

If the nebentypus is trivial and we do not necessarily want to find a form that is new, we can directly apply [MS09, Theorem 19] to get (3) below. In general, [MS09, Theorem 8, Lemma 17, Section 10] show that for any $M\geq 1$ ,

(2)		$\displaystyle\left\|C_{k,N,\chi}(A)-2\int_{0}^{A}F(-x)dx\right\|$
	$\displaystyle\leq\frac{1}{M+1}+\sum_{1\leq\|m\|\leq M}\left(\frac{1}{M+1}+\min\left(2A,\frac{1}{\pi\|m\|}\right)\right)\left\|\frac{\operatorname{tr}\left(T^{\prime}_{p^{\|m\|}}-T^{\prime}_{p^{\|m\|-2}}\right)}{d_{k,N,\chi}}-c_{m}\right\|,$

where $c_{m}=\lim_{k+N\to\infty}\operatorname{tr}(T^{\prime}_{p^{|m|}}-T^{\prime}_{p^{|m|-2}})/d_{k,N,\chi}$ and $F(x)=\sum_{m\in\mathbb{Z}}c_{m}e(mx)$ , with the convention that $T^{\prime}_{n}=0$ if $n<1$ . The Eichler–Selberg trace formula for $S_{k}(\Gamma_{0}(N),\chi)$ [Ser97, (34)] and [Ser97, Section 5.3] gives that,

	$\displaystyle\operatorname{tr}T^{\prime}_{p^{m}}$	$\displaystyle=$	$\displaystyle\sum_{N_{1}\mid N}d^{*}(N/N_{1})\Big{(}A_{\text{main}}(k,N_{1},T^{\prime}_{p^{m}})+A_{\text{ell}}(k,N_{1},\chi,T^{\prime}_{p^{m}})$
			$\displaystyle\hskip 85.35826pt+A_{\text{hyp}}(k,N_{1},\chi,T^{\prime}_{p^{m}})+\delta_{\begin{subarray}{c}k=2\\ \chi=1\end{subarray}}A_{\text{par}}(k,N_{1},T^{\prime}_{p^{m}})\Big{)},$

for any $m\geq 1$ , with the main, elliptic, hyperbolic and parabolic terms given in [Ser97, (35, 39, 45, 47)], and where $d^{*}$ is the multiplicative function defined by $d^{*}(\ell)=-2$ , $d^{*}(\ell^{2})=1$ , and $d^{*}(\ell^{\alpha})=0$ for $\ell$ a prime and $\alpha\geq 3$ an integer. By [Ser97, (35)],

\sum_{N_{1}\mid N}d^{*}(N/N_{1})A_{\text{main}}(k,N_{1},T^{\prime}_{p^{m}})=\frac{\psi(N)^{\text{new}}(k-1)}{12}\cdot p^{-m/2}\cdot\delta_{m\text{ even}},

where $\psi(N)^{\text{new}}=\sum_{N_{1}\mid N}d^{*}(N/N_{1})N_{1}\prod_{\ell\mid N_{1}}(1+1/\ell)$ , and by [MS09, Section 9],

F(x)=\frac{\psi(N)^{\text{new}}(k-1)}{12d_{k,N,\chi}}\cdot\frac{2(p+1)}{\pi}\cdot\frac{\sqrt{1-x^{2}}}{p+2+1/p-4x^{2}}.

By [Ser97, (44, 46, 48)], we find as in [MS09, (8)] that for any $N_{1}\mid N$ ,

$\displaystyle\|A_{\text{ell}}(k,N_{1},\chi,T^{\prime}_{p^{m}})\|$	$\displaystyle\leq$	$\displaystyle\frac{4e}{\log{2}}\cdot 2^{\omega(N_{1})}p^{3m/2}\log(4p^{m/2}),$
$\displaystyle\|A_{\text{hyp}}(k,N_{1},\chi,T^{\prime}_{p^{m}})-A_{\text{hyp}}(k,N_{1},\chi,T^{\prime}_{p^{m-2}})\|$	$\displaystyle\leq$	$\displaystyle\sqrt{N_{1}}\tau(N_{1}),$
$\displaystyle\|A_{\text{par}}(k,N_{1},T^{\prime}_{p^{m}})-A_{\text{par}}(k,N_{1},T^{\prime}_{p^{m-2}})\|$	$\displaystyle\leq$	$\displaystyle p^{m/2}.$

Moreover, we note that $|d^{*}(n)|\leq 2^{\omega(n)}\leq\tau(n)\ll_{\varepsilon}n^{\varepsilon}$ for all integers $n$ (see [Ser97, (52)]). Hence, this yields with (2)

(3)		$\displaystyle C_{k,N,\chi}(A)\geq$	$\displaystyle\frac{\psi(N)^{\text{new}}(k-1)}{12d_{k,N,\chi}}\cdot\frac{4(p+1)}{\pi}\int_{0}^{A}\frac{\sqrt{1-x^{2}}}{p+2+1/p-4x^{2}}dx-\frac{1}{M+1}$
(4)			$\displaystyle-c(\varepsilon)N^{\varepsilon}\left(\frac{4e}{\log{2}}\cdot\frac{p^{3M/2}}{d_{k,N,\chi}}\cdot\log(4p^{M/2})-\frac{\sqrt{N}}{d_{k,N,\chi}}-\delta_{\begin{subarray}{c}k=2\\ \chi=1\end{subarray}}\cdot\frac{p^{M/2}}{d_{k,N,\chi}}\right),$

for any $\varepsilon>0$ , with $c(\varepsilon)>0$ a constant depending only on $\varepsilon$ . As in [Ser97, (61, 62)],

d_{k,N,\chi}=\frac{k-1}{12}\cdot\psi(N)^{\text{new}}+O\left(N^{1/2+\varepsilon}\right),

therefore given $\varepsilon<1/100$ positive, as long as $M\leq(2/3-3\varepsilon)\log(N)/\log p$ , all the three terms in (4) are less than $c^{\prime}(\varepsilon)N^{-\varepsilon/100}$ for all $N$ and some constants $c^{\prime}(\varepsilon)$ depending only on $\varepsilon$ ¹¹1The choice of $M$ is motivated by the fact that the growth of (4) is dominated by the first term in (4) which is roughly of size $N^{\varepsilon}p^{3M/2}N^{-1}$ . Thus it is sufficient to choose $M$ so that this term is negligible, that is $p^{3M/2}N^{\varepsilon-1}\ll N^{-\varepsilon}$ ..

By a Taylor expansion at $x=0$ ,

\int_{0}^{A}\frac{\sqrt{1-x^{2}}}{p+2+1/p-4x^{2}}dx=\frac{p}{(p+1)^{2}}\cdot A\cdot(1+O(A)),

therefore

\displaystyle C_{k,N,\chi}(A)

\displaystyle\geq

\displaystyle\frac{4}{\pi}\cdot\frac{p}{p+1}\cdot A(1+O(A))-\frac{1}{M+1}-\frac{c^{\prime}(\varepsilon)}{N^{\varepsilon/100}}.

Hence, given $\varepsilon>0$ , choosing $A$ so that,

\frac{4}{\pi}\cdot\frac{p}{p+1}\cdot A>\Big{(}\frac{3}{2}+\varepsilon\Big{)}\cdot\frac{\log p}{\log N}>\frac{1+\varepsilon}{M+1}

ensures that $C_{k,N,\chi}(A)>0$ for all sufficiently large $N$ . In particular fixing a sufficiently small $\varepsilon>0$ we see that for all $N$ large enough any

A>\frac{\pi}{2}\cdot\frac{p+1}{p}\cdot\frac{\log p}{\log N}

is acceptable.

∎

Theorem 1 and Theorem 2 now follows from combining Proposition 1 and Lemma 1 and specializing accordingly.

3. Proof of Theorem 3

For $k\geq 2$ and $N\geq 1$ square-free, let $f\in S_{k}(\Gamma_{0}(N),\chi)$ be a newform. We factor the character $\chi$ as $\prod_{p\mid N}\chi_{p}$ with $\chi_{p}:(\mathbb{Z}/p)^{\times}\rightarrow\mathbb{C}^{\times}$ a character modulo $p$ . The idea behind Theorem 3 is inspired by [CK06], where Choie and Kohnen show that the non-diagonalizability of a “bad” Hecke operator $T_{p}$ (i.e. with $p\mid N$ ) implies that $\sqrt{p}\in\mathbb{Q}(a_{n}(f):n\geq 1)$ , and hence that this field has degree at least $2^{s}$ if $s$ such operators are non-diagonalizable.

Let

N_{2}=\prod_{\begin{subarray}{c}p\mid N\\ \chi_{p}=1\end{subarray}}p

and write $N=N_{1}N_{2}$ , with $(N_{1},N_{2})=1$ since $N$ is square-free. It follows that $\chi=\chi_{N_{1}}\chi_{N_{2}}$ with $\chi_{N_{1}}$ a primitive character of modulus $N_{1}$ and $\chi_{N_{2}}=1$ the principal character modulo $N_{2}$ . Our argument is based on the Atkin–Lehner operators

\displaystyle W_{p}

\displaystyle:S_{k}(\Gamma_{0}(N),\chi)\to S_{k}(\Gamma_{0}(N),\overline{\chi}_{p}\chi_{N/p})\ ,\ p\mid N

where $\chi_{N/p}=\prod_{\ell\mid N/p}\chi_{\ell}$ and on the properties of the pseudo-eigenvalues $\lambda_{p}(f)$ studied by Atkin and Li [Li74, AL78]. Examining these elements gives bounds on the degrees of Fourier coefficients $a_{f}(p)$ at “bad” primes $p\mid N_{2}$ . In turn, this yields lower bounds on $\deg{K_{f}}$ since:

Lemma 2.

We have $K_{f}=\mathbb{Q}(a_{f}(n):n\geq 1)$ .

Proof.

Let $K:=\mathbb{Q}(a_{f}(n):n\geq 1)$ and let $L$ be its Galois closure. By the Hecke relations $a_{f}(p)^{2}=a_{f}(p^{2})-p^{k-1}\chi(p)$ for all $p\nmid N$ , we have the tower of extensions $\mathbb{Q}(\zeta_{\operatorname{ord}\chi})\subset K_{f}\subset K\subset L$ . By Galois theory, it suffices to show that $\operatorname{Gal}(L/K_{f})\subset\operatorname{Gal}(L/K)$ . To that effect, let $\sigma\in\operatorname{Gal}(L/K_{f})$ . By the fact that $\chi^{\sigma}=\chi$ and [DI95, Corollary 12.4.5], $f^{\sigma}$ is a newform in $S_{k}(\Gamma_{0}(N),\chi)$ whose Fourier coefficients coincide with those of $f$ at all integers co-prime to $N$ . By strong multiplicity one [DI95, Theorem 6.2.3], $f=f^{\sigma}$ , so that $\sigma$ fixes all coefficients of $f$ , i.e. $\sigma$ fixes $K$ . ∎

Recall that for $p\mid N$ , the pseudo-eigenvalue $\lambda_{p}(f)\in\mathbb{C}$ is defined by the equation

W_{p}f=\lambda_{p}(f)g,

where $g\in S_{k}(\Gamma_{0}(N),\overline{\chi}_{p}\chi_{N/p})$ is a newform (see [AL78, p.224]) given by

(5)

a_{g}(\ell)=\begin{cases}\overline{\chi}_{p}(\ell)a_{f}(\ell)&:\ell\neq p\\ \chi_{N/p}(p)\overline{a_{f}(p)}&:\ell=p\end{cases}

for primes $\ell$ ([AL78, (1.1)]).

In general, we only know that the pseudo-eigenvalue $\lambda_{p}(f)$ is algebraic with modulus 1 ([AL78, Theorem 1.1]). However, under additional assumptions on $\chi$ , we have the following information on its field of definition:

Lemma 3.

Let $p\mid N_{2}$ . Then, $\lambda_{p}(f)\in\mathbb{Q}(\zeta_{2\operatorname{ord}(\chi)})$ .

Proof.

From the identity $W_{p}^{2}=\chi_{p}(-1)\overline{\chi}_{N/p}(p)\mathrm{id}$ ([AL78, Proposition 1.1]), we get that

(6)

\lambda_{p}(f)\lambda_{p}(g)=\chi_{p}(-1)\overline{\chi}_{N/p}(p)=\pm\overline{\chi}_{N/p}(p).

Since $p\mid N_{2}$ we have $\chi_{p}=1$ , so that $g\in S_{k}(\Gamma_{0}(N),\chi)$ , and $a_{g}(\ell)=a_{f}(\ell)$ for all prime $\ell\neq p$ , by (5). By strong multiplicity one, we get $g=f$ . By (6), we obtain $\lambda_{p}(f)^{2}=\overline{\chi}_{N/p}(p)$ and thus the claim. ∎

The next ingredient is the explicit determination of $\lambda_{f}(p)$ in terms of $a_{f}(p)$ by Atkin and Li.

Lemma 4.

Let $p\mid N_{2}$ . Then $a_{f}(p)\neq 0$ and

\lambda_{p}(f)=-\frac{p^{k/2-1}}{a_{f}(p)}.

Proof.

The fact that $a_{f}(p)\neq 0$ is [Li74, Theorem 3(ii)], and the formula for the eigenvalue is [AL78, Theorem 2.1]. ∎

Proof of Theorem 3.

By Lemmas 2, 3 and 4, we get

\displaystyle\{p^{k/2}:p\mid N_{2}\}

\displaystyle\subset

\displaystyle K_{f}(\zeta_{2\operatorname{ord}(\chi)}).

Since $L:=\mathbb{Q}(\zeta_{\operatorname{ord}(\chi)})\subset K_{f}$ , we have

	$\displaystyle[K_{f}:\mathbb{Q}]$	$\displaystyle\geq$	$\displaystyle\frac{1}{2}\cdot[K_{f}(\zeta_{2\operatorname{ord}(\chi)}):\mathbb{Q}]$
		$\displaystyle=$	$\displaystyle\frac{1}{2}\cdot[K_{f}(\zeta_{2\operatorname{ord}(\chi)}):L]\cdot\varphi(\operatorname{ord}(\chi)),$

where the last factor is the trivial bound.

The square roots of odd primes $p\mid\operatorname{ord}(\chi)$ belong to $L$ . On the other hand, for $S:=\{\sqrt{p}:p\mid N_{2},\,p\nmid 2\operatorname{ord}(\chi)\}\subset K_{f}(\zeta_{2\operatorname{ord}(\chi)})$ , we have

[K_{f}(\zeta_{2\operatorname{ord}(\chi)}):L]\geq[L(S):L]=2^{|S|}

by [Hil98, Theorem 87], and the claim follows. ∎

Remark 3.

Since the character $\chi_{p}$ is primitive for $p\mid N_{1}$ , [Li74, Theorem 3(ii)] and [AL78, Theorem 2.1, Proposition 1.4] show that $\lambda_{p}(f)=p^{k/2-1}g(\chi_{p})/a_{f}(p)$ , with $g(\chi_{p})$ the Gauss sum attached to $\chi_{p}$ . The degree of $p^{k/2-1}g(\chi_{p})$ over $\mathbb{Q}$ can be determined precisely, however we have no information about the field of definition of $\lambda_{p}(f)$ , except the fact that it is a root of unity. If we could show that it belongs to a small extension of $K_{f}$ , in the same way as we did for $\lambda_{p}(f)$ with $p\mid N_{2}$ , then we could add a factor as large as $\operatorname{ord}(\chi)$ to the lower bound of Theorem 3, including when $k$ is even.

References

[AL78] A.O.L. Atkin and W. W. Li. Twists of newforms and pseudo-eigenvalues of $W$ -operators. Inventiones Mathematicae, 1978.
[BM16] N. Billerey and R. Menares. On the modularity of reducible ${\rm mod}\,l$ Galois representations. Math. Res. Lett., 23(1):15–41, 2016.
[Bru95] A. Brumer. The rank of $J_{0}(N)$ . Astérisque, (228):3, 41–68, 1995. Columbia University Number Theory Seminar (New York, 1992).
[CE04] F. Calegari and M. Emerton. The Hecke algebra $T_{k}$ has large index. Math. Res. Lett., 11(1):125–137, 2004.
[CF99] J. B. Conrey and D. W. Farmer. Hecke operators and the nonvanishing of $L$ -functions. In Topics in number theory (University Park, PA, 1997), volume 467 of Math. Appl., pages 143–150. Kluwer Acad. Publ., Dordrecht, 1999.
[CK06] Y. Choie and W. Kohnen. Diagonalizing “bad” Hecke operators on spaces of cusp forms. In W. Zhang and Y. Tanigawa, editors, Number theory: tradition and modernization, pages 23–26. Springer US, Boston, MA, 2006.
[Del71] P. Deligne. Formes modulaires et représentations $l$ -adiques. In Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363, volume 175 of Lecture Notes in Math., pages Exp. No. 355, 139–172. Springer, Berlin, 1971.
[DI95] F. Diamond and J. Im. Modular forms and modular curves. In Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994), volume 17 of CMS Conf. Proc., pages 39–133. Amer. Math. Soc., Providence, RI, 1995.
[DJUR15] L. V. Dieulefait, J. Jiménez Urroz, and K. A. Ribet. Modular forms with large coefficient fields via congruences. Res. Number Theory, 1:Art. 2, 14, 2015.
[DT16] L. Dieulefait and P. Tsaknias. Possible connection between a generalized Maeda’s conjecture and local types. preprint arXiv:1608.05285, 2016.
[Hil98] D. Hilbert. The theory of algebraic number fields. Springer Berlin Heidelberg, 1998.
[HM97] H. Hida and Y. Maeda. Non-abelian base change for totally real fields. Pacific J. Math., (Special Issue):189–217, 1997. Olga Taussky-Todd: in memoriam.
[Li74] W. W. Li. Newforms and Functional Equations. Mathematische Annalen, 212, 1974.
[LS18] M. Lipnowski and G. J. Schaeffer. Detecting large simple rational Hecke modules for $\Gamma_{0}(N)$ via congruences. International Mathematics Research Notices, page rny190, 2018.
[Mat10] MathOverflow. Galois orbits of newforms with prime power level, May 2010. https://mathoverflow.net/questions/24923.
[MS09] M. R. Murty and K. Sinha. Effective equidistribution of eigenvalues of Hecke operators. Journal of Number Theory, 129(3):681–714, 2009.
[Rib80] K. A. Ribet. Twists of modular forms and endomorphisms of abelian varieties. Mathematische Annalen, 253(1):43–62, 1980.
[Rib85] K. A. Ribet. On $\ell$ -adic representations attached to modular forms. II. Glasgow Mathematical Journal, 27:185–194, 1985.
[Roy00] E. Royer. Facteurs $\mathbb{Q}$ -simples de $J_{0}(N)$ de grande dimension et de grand rang. Bull. Soc. Math. France, 128(2):219–248, 2000.
[Ser97] J-P. Serre. Répartition asymptotique des valeurs propres de l’opérateur de Hecke $T_{p}$ . Journal of the American Mathematical Society, 10(1):75–102, 1997.
[Tsa14] P. Tsaknias. A possible generalization of Maeda’s conjecture. In Gebhard Böckle and Gabor Wiese, editors, Computations with modular forms, volume 6, pages 317–329. Springer International Publishing, Cham, 2014.