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We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to be smooth, proper, etc).

Given such a family, we can take the $\ell$-adic representation associated to any given fiber, and in this sense we also have a "family" of Galois representations. (Alternatively, by the proper base change theorem in étale cohomology, we can take $R^1f_*(\mathbb{Z}_{\ell})$, which is a sheaf on $Y$, and the stalks of this sheaf are the duals of the above Galois representations).

Now consider different question. Can we have a "family" of cuspidal eigenforms whose associated Galois representations fit into a family in the above sense?

I'll consider the case of weight 2 (though I'm most interested in higher weight). Then such family should lead to a family of RM abelian varieties, i.e. those associated to the weight 2 cusp forms.

Let's go back to the elliptic curve (or abelian variety) side a bit, and think about what this would mean. The level of a modular form corresponds to the conductor of the associated elliptic curve, so the level of the modular forms in such a family should be just as bizarre a function of the base as is the conductor of a family of curves.

To try to engineer such a family, suppose we had a family of elliptic curves that were all known to be modular. I'm most interested in rational families, i.e. with open subsets of projective space as bases. Then if the family is defined over $\mathbb{Q}$, we at least know that the fibers of rational points are modular, and we get a "family" of modular forms over the rational points. What would this "family" look like? I have a feeling it would be pretty strange from the point of view of modular forms.

A different approach is to try to to construct a family of modular curves, then view a family of modular forms as a section of the relative cotangent sheaf or some power thereof. Maybe one could try to make it an "eigensection" of some sort of relative Hecke operators. Of course, the very idea of a family of modular curves seems strange, as there are countably-many modular curves!

In fact, this points to a general problem with this attempt: modular forms are based on discrete data, a discrete set of levels, and a discrete set of eigenforms within each level.

I have a feeling that it's impossible to make this notion work, but please let me know if you have good ideas. In particular, it's possible that experts in modular forms and curves would have more ideas.

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Katz's paper published in the Antwerpen III volume gives a detailed account of modular forms over a ring, more or less along the lines you expect. Of course, the level and the weight are fixed. In the theory of $p$-adic modular forms, one defines analytic families of modular forms. There, the level is basically fixed but the weight varies. Such objects are $q$-expansions $\sum a_n(x)q^n$ parametrized by $p$-adic analytic functions such that at some dense set of $x$, the obtained $q$-expansion is the one of a true modular form. –  ACL Feb 4 '13 at 10:18
and the Galois representations associated to these families will behave like those associated to families of varieties? –  David Corwin Feb 4 '13 at 10:43
At ACL and Davidac897. Katz' modular forms over a ring are in general not families of eigenforms. Moreover, they more or less all come by base changes from families over some "trivial" ring such as $\mathbb Z$, so they are in a sense "constant families". This is not what Davidac897 is looking for. –  Joël Feb 4 '13 at 15:40
Are you aware of Hida's families of modular forms? They might not be exactly what you're interested in since their whole point is to make the weight vary (and hence you can't really force them to be elliptic), but still : you were wondering about the problem of being based on discrete data, and that is an example where it is not annoying, since integers can vary continuously in a $p$-adic setting! –  Julien Puydt Feb 4 '13 at 17:18
@Joël. You're right for Katz's forms. But the analytic $p$-modular forms (of which you're much more aware than I am, of course) are genuinely more general objects. <br /> @Davidac897. Yes: the Galois representations can be defined over the whole family extending the classical representations, at least in weight $\geq 2$ (cf. Mazur & Wiles's paper in Compositio). –  ACL Feb 6 '13 at 17:28

2 Answers 2

$\newcommand\Q{\mathbf{Q}}$ $\newcommand\A{\mathbf{A}}$ $\newcommand\AQ{\A_{\Q}}$

How about instead of considering lisse sheaves, derived pushforwards, étale cohomology, modular forms, blah blah blah, you consider instead the following situation: over the affine line $\Q[t]$, you can consider the equation $x^2 - t$; it's a family of quadratic extensions. If you like, you can turn this into a smooth map of curves (eliminating the bad fibre at $0$) $\pi: X \rightarrow Y$, and you can consider the lisse sheaf $R^0 \pi_* \mathbf{Z}_l$, the stalks of which are Galois representations corresponding to the quadratic character of the associated quadratic extension (along with the trivial character, which I'll suppress). All the Galois representations corresponding to rational points are automorphic/modular for $\mathrm{GL}_1(\AQ)$ - that's a theorem of Gauss called quadratic reciprocity. What does this family look like? Well, it is what it is; the global data is related to the factorization of $t$ in the expected way, and it's not really a "family" is any analytic sense. There is, however, one useful fact to observe about how this family behaves as one varies $t$. Namely, the characters $\chi_t$ are locally constant. That is, if $t$ and $s$ are close in $\Q_v$ for any place $v$, then the local characters $\chi_{t,v}$ and $\chi_{s,v}$ are equal. For example, if $t$ and $s$ are both the same sign, then $\chi_{t,v}(-1) = \chi_{s,v}(-1)$. This is Krasner's lemma (we use smoothness here). It turns out that this "local constancy" of Galois representations is true more generally, I think Kisin proved something along these lines (although you should think of that result as also being Krasner's Lemma).

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Dear Michigan J. Frog: Kisin's result is that the isomorphism class of the fibers over rational points for a lisse $\ell$-adic or lcc abelian sheaf on a $p$-adic analytic space is "locally constant" on the base. It is in his 1999 paper with the title "Local constancy in p-adic families..." on his webpage. The motivation is certainly Krasner's Lemma, but even the lcc case over a disk lies somewhat deeper, requiring either more geometric or function-theortic input (roughly because the "local constancy" is just for an isom. class, without "canonicity"; Kisin captures it via $\pi_1$'s). –  user30379 Feb 13 '13 at 6:34

Perhaps it is easier to explain what's going on here in the context of Galois representations. I know of two almost wholly disjoint ideas that are, confusingly, both described using phrases like "families of Galois representations".

  • One can consider "families of representations" of any group, which are just homomorphisms from G to GL_n(A) for some (usually commutative) ring A, or more generally GL(V) for some locally free sheaf V on a base scheme S, etc. These are "families" in the sense that the image of a group element is a matrix whose entries are functions on Spec(A) (resp. on S, etc). Note that, intuitively, the group is fixed and the coefficients are varying.

  • One can also consider the kind of "geometric" family you mentioned in your question and Michigan J Frog enlarged upon: given a family of geometric objects over a base S, you can do various kinds of relative cohomology to give sheaves on S whose fibres have an action of some kind of Galois group depending on the fibre, and in particular the generic fibre has an action of something like the fundamental group of S. So here the group is, so to speak, varying in the family as well.

The first kind of family, over a p-adic base and with G being a Galois group, comes up a lot in the context of modular forms (Hida, Coleman-Mazur, etc). These can be viewed as sections of a family of sheaves on a subvariety of the rigid-analytic space you get by analytifying the modular curve. Note that we are varying the coefficients and not the group, again.

The second kind of family doesn't come up so much in modular form theory, although it makes a notable appearance in Kato's work on Iwasawa theory for modular forms.

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@David: Speaking about Kato's work, are you refering to his Asterisque paper? Can you expand a bit? Thanks. –  Filippo Alberto Edoardo Feb 6 '13 at 14:44
Yes, I mean the Asterisque paper "P-adic Hodge theory and zeta functions of modular forms" –  David Loeffler Feb 7 '13 at 7:09

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