Hecke field degrees of cusp forms after Serre's equidistribution theorem: the level aspect without coprimality, the fixed-level weight aspect, and power-saving rates

Serre (1997) proved that the eigenvalues of the normalized Hecke operator $T_p' = T_p/p^{(k-1)/2}$ on the space $S_k(\Gamma_0(N))$ of cusp forms equidistribute in $[-2,2]$ with respect to the $p$-adic Plancherel measure $\mu_p$ as $k+N\to\infty$ over levels prime to $p$, and deduced two arithmetic consequences for the number fields $K_i$ generated by Hecke eigenvalues: for fixed weight $k$ and fixed $r\ge1$, the proportion $s(N,k)_r/s(N,k)$ of eigenforms whose Hecke field has degree exactly $r$ tends to $0$ along levels coprime to any fixed prime (Theorem 5), and the maximal degree $r(N,k)$ tends to $\infty$ with $N$ (Theorem 6, proved using oldforms). He asked what happens outside this regime: whether the coprimality hypothesis can be dropped; what happens at fixed level $N$ as the weight $k\to\infty$; whether the decay is a power saving $s(N,k)_r \ll s(N,k)^\alpha$ with $\alpha<1$; and the newform version of Theorem 6. The later literature resolves the newform version in the main (Royer; Murty–Sinha; Bettin–Perret-Gentil–Radziwiłł: the largest simple Hecke submodule of the new subspace has dimension $\gg_k \log\log N/\log(2p_N)$ for every $N$) and gives exponential-in-$(\log N)^\beta$ multiplicity bounds at prime level, extended by Sarnak–Zubrilina to composite levels that are not ‘super-smooth’ (their §6, Theorem 5) while explicitly excluding primorial-type levels, and stated by them to fall short of Serre's conjectural power saving. The weight aspect at fixed level is bounded below only in Galois-orbit counts (Dieulefait–Pacetti–Tsaknias; Banerjee et al.) and is governed by the still-open Maeda conjecture and its generalizations, and has recent heuristic support at squarefree level for weight 2 (Cowan–Martin). The surviving open core is: (1) the proportion limit over all levels, including levels divisible by every small prime; (2) unboundedness of maximal Hecke-field degree at fixed level, open already at level 1; (3) power-saving bounds for general weight and degree.

Background

For $N\ge1$ and even $k\ge2$, let $S_k(\Gamma_0(N))$ denote the complex vector space of holomorphic cusp forms of weight $k$ for the congruence subgroup $\Gamma_0(N)$ of $\mathrm{SL}_2(\mathbb{Z})$, i.e. matrices $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ with integer entries and $c\equiv0\pmod{N}$; write $s(N,k)=\dim S_k(\Gamma_0(N))$. A cusp form is a holomorphic function on the upper half-plane satisfying the weight-$k$ transformation law under $\Gamma_0(N)$ and vanishing at the cusps; its Fourier expansion $f=\sum_{m\ge1}a(m)q^m$ (with $q=e^{2\pi iz}$) determines it. For a prime $\ell\nmid N$, the Hecke operator $T_\ell$ acts by $T_\ell f=\sum_{m\ge1}(a(\ell m)+\ell^{k-1}a(m/\ell))q^m$, where $a(x)=0$ unless $x$ is a positive integer; the operators $T_n$ with $(n,N)=1$ commute and are simultaneously diagonalizable, so there is a basis $f_1,\dots,f_{s(N,k)}$ with $T_n f_i=x_{i,n}f_i$. The eigenvalues $x_{i,n}$ are totally real algebraic integers, and the field $K_i=\mathbb{Q}(x_{i,n}:(n,N)=1)$ is the Hecke field of $f_i$; Serre writes $s(N,k)_r$ for the number of indices $i$ with $[K_i:\mathbb{Q}]=r$, so $\sum_{r\ge1}s(N,k)_r=s(N,k)$, and $r(N,k)$ for the maximal degree $[K_i:\mathbb{Q}]$.

By the Ramanujan conjecture (proved by Deligne), the eigenvalues of the normalized operator $T_p'=T_p/p^{(k-1)/2}$ lie in $[-2,2]$. Serre's Theorem 1 (1997) states that for any sequence of pairs $(k_\lambda,N_\lambda)$ with $k_\lambda$ even, $k_\lambda+N_\lambda\to\infty$, and $p\nmid N_\lambda$, these eigenvalues equidistribute in $[-2,2]$ for the measure

$$\mu_p=\frac{p+1}{\pi}\cdot\frac{(1-x^2/4)^{1/2}}{(\sqrt p+1/\sqrt p)^2-x^2}\,dx,$$

the $p$-adic Plancherel measure (the pushforward of normalized Haar measure on $\mathrm{SL}_2(\mathbb{Z}_p)$ under the trace map), which is not the classical Sato–Tate measure $\frac{1}{\pi}(1-x^2/4)^{1/2}dx$ and had earlier appeared in the spectral theory of arithmetic graphs. The proof uses the Eichler–Selberg trace formula for the traces of Hecke operators on $S_k(\Gamma_0(N))$.

Atkin–Lehner theory decomposes $S_k(\Gamma_0(N))$ into an old subspace (spanned by forms of lower level and their level-raising copies) and a new subspace $S_k(\Gamma_0(N))^{\mathrm{new}}$ spanned by newforms, i.e. normalized eigenforms not coming from lower level. The Galois conjugates $\sigma(f)$, $\sigma\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, of a newform are again newforms of the same level with conjugate eigenvalues, so the Galois orbit of $f$ has exactly $[K_f:\mathbb{Q}]$ elements; equivalently, the $\mathbb{Q}$-algebra generated by the Hecke operators decomposes into simple factors whose $\mathbb{Q}$-dimensions are these degrees. In weight 2, Shimura (with later refinements by Ribet) identifies these factors with the $\mathbb{Q}$-simple factors of the Jacobian $J_0(N)$ of the modular curve $X_0(N)$: if $J_0(N)\sim\prod_j A_j$ is the decomposition up to $\mathbb{Q}$-isogeny, then $\mathbb{Q}\otimes\mathrm{End}(A_j)$ is a totally real field $E_j$ with $[E_j:\mathbb{Q}]=\dim A_j$, and the multiset of Hecke fields $K_i$ is the multiset of real embeddings of the $E_j$; in particular $s(2,N)_r/r$ counts the $\mathbb{Q}$-simple factors of $J_0(N)$ of dimension $r$.

From his equidistribution theorem Serre deduced, for fixed weight $k$: Theorem 5, that for every fixed prime $p$ and every $r\ge1$, $s(N,k)_r/s(N,k)\to0$ as $N\to\infty$ through levels prime to $p$ (in his words, most Hecke fields have large degree); Theorem 6, that $r(N,k)\to\infty$ as $N\to\infty$, with no coprimality hypothesis, but proved by passing to oldforms; and Theorem 7, that the largest dimension of a $\mathbb{Q}$-simple factor of $J_0(N)$ tends to $\infty$ with $N$. He then asked what happens outside this regime, in four recorded directions: (i) can the hypothesis that $N$ is prime to the fixed prime $p$ be removed in Theorem 5 ("Cela paraît vraisemblable, mais je ne sais pas le démontrer.")? (ii) what happens when the level $N$ is fixed and the weight $k$ varies ("J'ignore ce qui se passe lorsque l'on fixe $N$ et fait varier $k$.")? (iii) is the decay a power saving, $s(N,k)_r/N^\alpha\to0$ for some $\alpha<1$, or even for every $\alpha>0$? (iv) Theorem 5 also holds for the new subspace, but "J'ignore ce qu'il en est du th. 6", since that proof uses oldforms. In the weight aspect at level 1, Maeda's conjecture (recorded by Hida and Maeda in 1997) predicts far more than unboundedness: for every even $k$, the space $S_k(\Gamma_0(1))$ is a single Galois orbit, i.e. one Hecke field of maximal degree $\dim S_k(\Gamma_0(1))$. For the fixed-level weight aspect, an orbit count gives a sufficient condition: $r(N,k)\to\infty$ follows if the number of Galois orbits of eigenforms in $S_k(\Gamma_0(N))$ is $o(\dim S_k(\Gamma_0(N)))$ as $k\to\infty$.

Problem Statement

Fix an even weight $k\ge2$ and a level $N\ge1$, and keep the notation of Serre (1997): $S_k(\Gamma_0(N))$ is the space of cusp forms for $\Gamma_0(N)$, $T_n$ (for $n$ prime to $N$) the Hecke operators, $f_1,\dots,f_{s(N,k)}$ a simultaneous eigenbasis with $T_n f_i=x_{i,n}f_i$, $K_i=\mathbb{Q}(x_{i,n})$ the Hecke fields, $s(N,k)_r=\#\{i:[K_i:\mathbb{Q}]=r\}$, $s(N,k)=\dim S_k(\Gamma_0(N))$, and $r(N,k)$ the maximal degree. Serre's Theorem 5 gives $s(N,k)_r/s(N,k)\to0$ as $N\to\infty$ (for fixed $k$, fixed $r$) along levels prime to a fixed prime $p$, and Theorem 6 gives $r(N,k)\to\infty$ as $N\to\infty$ (for fixed $k$, no coprimality, proved via oldforms). The later literature resolves the newform version of Theorem 6 in the main and quantifies the decay in restricted regimes, leaving the open core of Serre's questions; the three components share one objective — extend the degree-growth theory of Theorems 5–6 over the full $(N,k)$-plane with quantitative control — and components 1 and 3 are linked by the same obstruction (levels divisible by every small prime):

1. Coprimality (Serre's question after Theorem 5). For fixed $k$ and every fixed $r\ge1$, determine whether $s(N,k)_r/s(N,k)\to0$ as $N\to\infty$ over all positive integers $N$, with no coprimality hypothesis. Since the restriction to levels coprime to any fixed prime is Theorem 5 itself, the open content is uniformity over levels divisible by every fixed prime, e.g. primorials; the same regime is where the best unconditional lower bound for the maximal Hecke-field degree of newforms degenerates to an absolute constant.

2. Weight aspect (Serre's remark after Theorem 6). For fixed level $N$ — already for $N=1$ — determine whether $r(N,k)\to\infty$ as $k\to\infty$.

3. Quantitative rate (Serre's second remark in §6.1). For fixed $k$ and $r$, determine whether a power-saving bound holds, i.e. whether $s(N,k)_r=O(s(N,k)^\alpha)$ for some $\alpha<1$ (the source asks for $s(N,k)_r/N^\alpha\to0$ for some $\alpha<1$, or even for every $\alpha>0$), without coprimality or primality restrictions on the levels.

Each of the three components is posed independently and is to be settled in the stated generality, i.e. without additional hypotheses — prime level, fixed Atkin–Lehner sign pattern, restricted sets of levels, or a fixed prime excluded from the level.

References

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