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If we consider the action of the $U_p$ operator on overconvergent $p$-adic modular forms, then we can get some information about the field over which the eigenforms are defined by looking at the slopes. For instance, my paper in Math Research Letters (MR2106238) proves that the slopes of $U_2$ acting on 2-adic overconvergent modular forms of level 4 with primitive Dirichlet character are distinct, so the field of definition has to be $\mathbf{Q}_2$. However, there are cases when the slopes fail to be distinct; for instance, in Emerton's thesis it is proved that the lowest slopes of T_2 acting on level 1 forms of weight congruent to 14 modulo 16 are 6 and 6.

For classical modular forms of level 1, we have Maeda's Conjecture which says that the field of definition is essentially as large as it can be; the Hecke polynomial is irreducible with Galois group $S_n$ where $n$ is the dimension. However, there is no reason that this should be true for overconvergent modular forms, and in fact it isn't. Discussions with Robert Coleman led me to the concrete example of 2-adic overconvergent modular forms of tame level 1 and weight 142, where there are two eigenforms of slope 6 which are both defined over the ground field $\mathbf{Q}_2$.

The question is, what should one expect here? Can one tell any more about the field of definition from the slopes than the absolute minimum?

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@Lloyd: as you well know, Maeda's conjecture is about the global field generated by the eigenvalues of a classical newform, but everywhere else in your question you seem to be talking/asking about the local field generated by the eigenvalues. What is happening is that a lot of small rational primes are decomposing into a large number of factors in Maeda's fields. Take for example $\mathbf{Q}(\sqrt{144169})$, the field generated by the level 1 weight 24 cusp forms. All of 2,3,5,7,11,13,17,19 are split in this field! –  Kevin Buzzard Aug 19 '10 at 19:50
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up vote 2 down vote accepted

Professor Buzzard raises the a question of whether every normalized eigenform of level $1$ is defined over a quadratic extension of $\mathbb{Q}_2$.

(This is Question 4.3 of http://www2.imperial.ac.uk/~buzzard/maths/research/papers/conjs.pdf)

In contrast, the multiplicity of the valuation of the set of $2$-adic slopes at level $1$ can be arbitrarily large, as can be observed as follows:

Consider the space $S_k:=S^{new}_k(\Gamma_0(2))$ of newforms of weight $k$. Every newform has slope $(k-2)/2$. Thus, by work of Coleman, the number of slopes of valuation $(k-2)/2$ at level $1$ and weight $k + 2^n$ for sufficiently large $n$ will be at least $\mathrm{dim}(S_k)$, which is unbounded as $k$ increases.

EDIT: The point of the last example is that the answer to the original question is "not much", i.e., there can be many forms of the same slope, but all the forms are defined over a small (or even trivial) extension of $\mathbb{Q}_2$.

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It's true that the dimension of the slope $(k-2)/2$ space in weight $k+2^n$ will be big, but my experience in computing examples is that the actual slopes are all highly congruent to $\pm2^{(k-2)/2}$ and are often defined over $\mathbf{Q}_2$ in this situation. –  Kevin Buzzard Aug 19 '10 at 19:52
    
@BSD: aah-hah! You must hence be one of the bazillion people to whom I stressed that the questions in that paper were not conjectures, but questions :-) On the other hand, I probably computed thousands of examples of coefficient fields over Q_2 and never found a counterexample. Having said that, Gouvea and Mazur made their "precise radius" conjectures based on strong numerical evidence and they weren't quite right :-/ –  Kevin Buzzard Aug 19 '10 at 21:29
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@BSD: I only just saw your comment today about Clay's work. I have a copy of her thesis and Emerton does too, if you need to see it. I guess Calegari and I proved that this constant existed in the case N=1 and p=2. –  Kevin Buzzard Sep 17 '10 at 21:09
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