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](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=RVCN&pg6=ALLF&pg7=ALLF&pg8=ET&review_format=html&s4=kilford&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=9&mx-pid=2106238)) 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?



Fields of definition for p-adic overconvergent modular eigenforms
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 $\mathbb{Q}\_2$.


