# Congruences between modular forms and the eigencurve construction

This question might be too conceptual.
Congruences between modular forms (due to Shimura, Hida, etc) are really amazing. I know that the eigencurve construction are closely related to these relations. The basic reference is "The Eigencurve" by Coleman and Mazur. Besides, I think "A brief introduction to the work of Haruzo Hida" by Mazur is a good introduction.

It seems that we at first prove the congruence and then interpolate them into a family, like the Eisenstein family, and the property of the eigencurve are deduced from properties of modular forms.

So I wonder conversely, can the eigencurve construction explain (prove) more congruences between modular forms (not just these used in building the eigencurve)? Or other interesting facts about modular forms? For example, see section 5 of "A brief introduction to the work of Haruzo Hida".

I type slowly, sorry...

• Please improve (clarify) your question by adding more detail, context, and references. Oct 15 '15 at 1:46

Here's a theorem about congruences between modular forms:

Theorem. Let $f$ be a (normalised) eigenform of weight $k$ and level $\Gamma = \Gamma_1(N) \cap \Gamma_0(p)$. Then for any $r$, and any $k'$ sufficiently close (*) to $k$, there exists an eigenform of weight $k'$ and level $\Gamma$ that is congruent to $f$ modulo $p^r$.

This is an incredibly powerful theorem; it's virtually self-evident once you know the eigencurve exists; and I don't think I know of any way of proving it without constructing the eigencurve in the process.

(*) Here "close" means that $k'$ has to be congruent to $k$ modulo $(p-1) p^j$, for some $j$ depending on $f$, $p$ and $r$.
• And is there no way prove this theorem without constructing the eigencurve? At least if $j=0$ it's just multiplying with the Hasse invariant (Eisenstein series I mean). Maybe I shall post another MO question...... Oct 16 '15 at 3:51
• No, it is much more powerful than just multiplying by Eisenstein series, because it's showing that there are eigenforms of weight $k'$ highly congruent to $f$, not just any old forms of that weight. Oct 16 '15 at 6:18