Suppose that $N$ is prime, and consider the normalized cuspidal Hecke eigenforms of weight 2 and level $\Gamma\_0(N)$.


For such an eigenform $f$, the coefficients generate (an order in) the ring of integers of a totally real field $E$. The corresponding simple abelian variety quotient $A\_f$ of $J\_0(N)$ has dimension $[E:\mathbb{Q}]$. Let $C(N)$ denote the maximal degree $[E:\mathbb{Q}]$ amongst all such eigenforms. It is not too hard to prove that $C(N)$ is unbounded, and not much harder to show that the $\lim \inf$ of $C(N)$ as $N\to \infty$ is infinite. (I don't want to mention the argument here because I don't think it will apply to my question below.)


Is the same result true over $\mathbb{Q}\_2$? Namely, fix an embedding of $\bar{\mathbb{Q}}$ into $\bar{\mathbb{Q}}\_2$. Then, for any $d$, is it true that for all sufficiently large prime $N$ there exists a cuspidal eigenform of weight 2 and level $\Gamma\_0(N)$ whose coefficients generate some field $E/\mathbb{Q}\_2$ with $[E:\mathbb{Q}\_2] > d$?


This would be interesting to know even for $d = 1$.



2-adic Coefficients of Modular Hecke Eigenforms
A natural way to approach this question is to ask for an eigenform $f$ with coefficients in a local field $K$ with residue field degree $\ge 2$. Or, asking for slightly more, with coefficient ring $\mathcal{O}$ admitting a surjective map to a field $\mathbf{F}$ of order divisible by $4$ (this is slighly more since the rings $\mathcal{O}$ can typically be non-trivial orders in $\mathcal{O}\_K$). By Serre's conjecture, this is equivalent to asking for the existence of an irreducible Galois representation


$$\rho: \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \rightarrow \mathrm{GL}\_2(\mathbf{F})$$


with Serre weight and conductor $(2,p)$ (note, no oddness condition!). It's hard to construct such representations, but one natural way is to use induced ($=$ projectively dihedral) representations. 


If $E$ is the corresponding quadratic extension, and $F/E$ the cyclic extension, then the Serre weight and level will be $(2,p)$ providing that $F/E$ is totally unramified and $E = \mathbf{Q}(\sqrt{-1})$, $\mathbf{Q}(\sqrt{p})$, or $\mathbf{Q}(\sqrt{-p})$.
(EDIT: In a previous version of this post, I only thought about extensions that were totally unramified at $2$ for some unknown reason.)
One is in good shape providing that the odd part of the class group of one of the fields $\mathbf{Q}(\sqrt{\pm p})$ is not $(\mathbf{Z}/3 \mathbf{Z})^n$. Since one can't really say much about real quadratic fields, let's think about the imaginary quadratic fields.
It's a theorem, proved by (amongst other people) Lillian Pierce, and Jordan Ellenberg and Akshay Venketesh, that 
the 3-torsion part of the class group is a negligible part of the class group. Thus we are done if $p$ is big *providing* that the $2$-part of the class group is not so big. The $2$-part is trivial if $p$ is $3$ mod $4$, and has order $2$ if $p$ is $5$ mod $8$. However, it is quite possible that the class group of $\mathbf{Q}(\sqrt{-p})$ is $\mathbf{Z}/2^n \mathbf{Z}$, and very rough heuristics suggest that this might happen infinitely often.


[EDIT: LaTeX issues now mostly sorted in the sequel]


If $\mathbf{T}$ denotes the Hecke algebra tensored with $\mathbf{Z}\_2$, then
$\mathbf{T}$ is a free $\mathbf{Z}\_2$-module of rank $n = \mathrm{dim}(S\_2(\Gamma\_0(p)))$. 
It is also a semi-local ring. For each maximal ideal $m$ of $\mathbf{T}$, we are asking that 


$$\mathbf{T}\_{m} \otimes \mathbf{Q}$$ 


is not a number of copies of the 2-adic numbers. If $m$ is the $2$-adic Eisenstein ideal, then in a paper with Matthew Emerton, I prove that 


$$\mathbf{T}\_{m}/2$$


has rank $2^{e-1}-1$, where the $2$-part of the class group of
$\mathbf{Q}(\sqrt{-p})$ (we may assume that $p$ is $1$ modulo $8$) has order $2^{e}$.
If $p$ is $9$ modulo $16$ then $\mathbf{T}$ localized at $m$ is also a domain, and so we are done in this case if $2^{e-1} -1 \ge d$. Even if $p$ is $1$ modulo $16$, this shows that the bulk of $\mathbf{T}/2$ must come from non-Eisenstein primes $m$, since 
$2^e \le h \ll p^{1/2 + \epsilon} \ll p/12 \simeq n$.
So, in the case $d = 1$, one may assume that the odd part of the class group is
$3$-torsion, and then hope that for the corresponding $m$, one can prove that
the corresponding rings $\mathbf{T}\_{m}/2$ can't be too large.


It turns out that the relevant question becomes: Given a $\overline{\rho}$ corresponding to an $S\_3$ extension, and given a finite flat deformation of
$\overline{\rho}$ to $\mathrm{SL}\_2(A)$ for a local ring $A$ killed by $2$, can one bound the rank of $A$ as an $\mathbf{F}\_2$-module? For example, can one prove that $A$ has rank $p^{1 - \epsilon}$? The example of Eisenstein $m$ suggests that one can not neccesarily do better than $p^{1/2 + \epsilon}$. 




---


Here is an answer that shows works for d = 1, and works for general d assuming GRH.


Step I. Let $C$ be the class group of $\mathbf{Q}(\sqrt{-p})$. The group $C$ decomposes as 
$C\_{odd} \oplus C\_{even}$.


Step II. Suppose that $C\_{odd}$ is not annihilated by $2^{2d} - 1$. Then there exists a dihedral representation induced from a character of $C\_{odd}$ which is finite flat at $2$, ordinary at $p$, and has which has image in 


$$\mathrm{SL}\_2(\mathbf{F})$$


for a field $\mathbf{F}$ of degree $>d$. 
Thus we are done unless $C\_{odd} = C\_{odd}[2^{2d} - 1]$.


Step III. If $d = 1$, then then the $2^{2d} - 1 = 3$-torsion of $C\_{odd}$ has small order, namely, of order at most
$$p^{1/2 - \delta}$$
 for some explicit $\delta > 0$. For general $d$, the $2^{2d} -1$-torsion has order at most $p^{\epsilon}$, assuming GRH. 


Step IV: If $p$ is $-1$ modulo $4$ or $5$ modulo $8$, then $C\_{even}$ has order $\le 2$.
This implies by Steps II and III that $C$ has order $\le p^{1/2 - \delta}$, which contradicts the estimate $h \gg p^{1/2 - \epsilon}$.


Step V: We may assume that $p$ is $1$ modulo $8$. Let $|C\_{even}| = 2m$. Using the estimate $h \gg p^{1/2 - \epsilon}$, we deduce that
$m \gg p^{\delta - \epsilon}$ for some explicit $\delta > 0$.


Step VI: Let $\mathbf{T}$ denote the localization of the Hecke algebra at the Eisenstein prime of residual characteristic $2$. It is a consequence of one of the main theorems of FC-ME that one has $\mathbf{T} = \mathbf{Z}\_2[x]/f(x)$, where:


$$f(x) \equiv x^{m - 1} \ \mathrm{mod} \ 2,$$


$$\mathbf{Z}\_2/f(0) \mathbf{Z}\_2 \simeq \mathbf{Z}\_2/n \mathbf{Z}\_2,$$


and $n$ is the numerator of $(p-1)/12$. (Compare the discussion in Mazur's Eisenstein ideal paper, section 19, page 140.)


Step VII: If the coefficients of all forms of level $\Gamma\_0(p)$ define extensions of degree $\le d$, then the normalized $2$-adic valuation of every root of $f(x)$ is at *least* $1/d$. We deduce that


$$\frac{m - 1}{d} \le v\_2(n),$$


Since $v\_2(n) \le \log\_2(n) < \log\_2(p)$, we deduce that


$$ m \le d \log\_2(p).$$


This contradicts the previous estimate $m \gg p^{\delta - \epsilon}$ for sufficiently large $p$.


Remark: For $d = 1$, it should be easy to use the unconditional results to give an explicit lower bound on possible $p$.




---


To summarize, for any $d$, and sufficiently large $p$, there exists either:


(i) A modular form $f$ of level $\Gamma\_0(p)$ with coefficients in an extension of $\mathbf{Q}\_2$ which contains an unramified extension of degree $> d$, and whose residual representation is dihedral,


(ii) A modular form $f$ of level $\Gamma\_0(p)$ which coefficients in an extension of $\mathbf{Q}\_2$ which contains a ramified extension of degree $> d$, and whose residual representation is Eisenstein.




---


Extra: For *any* $p$, the coefficients of an $f$ of level sufficiently divisible by $p$ contains the totally real subfield of the $p^n$th roots of unity.
For any $p$, and sufficiently large $n$, this contains a large extension of the $2$-adic numbers.


Reason why this argument still sucks: Doesn't work at all for primes $> 2$, and uses GRH for $d > 1$.


