[EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely.


As the title says, I would like to understand an isomorphism of Hida from a more geometric perspective than what I normally read. What bothers me is that there are two construction of the universal (ordinary) Hida-Hecke algebra and they turn out to give isomorphic objects: fix a prime $p\geq 5$, and a tame level $N$ prime to $p$.


1. Take the projective limit *over the level* $r$ of the Hecke algebra acting on $S\_k(\Gamma\_1(Np^r),\mathbb{Z}\_p)$ where $k$ is any weight. By applying the usual idempotent, one gets the Hida-Hecke ordinary algebra $h\_k^0(Np^\infty;\mathbb{Z}\_p)$, where I adopt notations as in Hida's paper in Inventiones, 1986, "Galois representations into $\mathrm{GL}(2,\mathbb{Z}\_p[[X]])$...".
2. Consider now the injective limit *over the weight* of the spaces of cusp forms $S\_k(Np;\mathbb{Z}\_p)$. By taking a suitable completion of this injective limit, one sees that the projective limit (over the weight, now) of Hecke algebras acts on the above completion. Applying again the idempotent, we get the Hida-Hecke ordinary algebra $h^0(N,\mathbb{Z}\_p)$.


Theorem 1.1 in the quoted paper by Hida shows that these two algebras are isomorphic (in the most compatible way one can dream of, in particular inducing the same Hecke action on spaces of cusp forms) but his proof is entirely algebraic.


**My question is**: is there a reasonable way to prove this isomorphism geometrically?As Kevin Buzzard suggested, several papers of Katz (and successive work by Coleman-Mazur, Buzzard himself et al.) discuss geometric interpretation of $p$-adic modular forms and $p$-adic families of modular forms. Still, I do not understand how *Hida's isomorphism* comparing the Hecke algebra as acting on the projective limit over the level (so ''at the top of the modular tower'') or on the inductive limit over the weight (so, ''over the first curve $X\_1(Np)$'') can be given a geometric interpretation.



Geometric interpretation of Hida isomorphism
I am not sure what your criteria would be for a proof to be given a geometric interpretation, but the reason why weights "disappear" when we take the inverse limit on the level stems from the contraction property of Hecke operators (at $p$), or informally from the fact that Hecke operators at $p$ diminish the level.


As you know, the proof of the isomorphism between the two different Hecke algebras requires the definition of a map between Hecke algebras acting on forms of weight 2 and forms of weight $k$. Because theses two algebras are sub-algebras of endomorphisms generated by the same abstract elements but acting on different objects, this amounts to constructing a map between the cohomology of (one of the level of) the modular tower with coefficients in the constant sheaf and (one of the level of) the modular tower with coefficients in a sheaf of weight $k$ (or the same thing with the modular tower replaced by the Igusa tower, as in Kevin's answer). This last map is really no big deal: if memory serves, on the sheaves it is just projection on the last component. The remarkable fact is that the map on cohomology then is surjective with a finite kernel (and is an isomorphism in the ordinary case); the proof of this assertion being exactly the contraction property. Note that the proof necessarily requires the choice of a level at some point; how else would you even state the result?


Note for instance that for a tower of more general Shimura varieties, it is not at all obvious how the contraction property will play out: group-theoretic properties of $\operatorname{GL}\_{2}(\mathbb Q\_{p})$ really do play an important role in the proof. See the reference below though, for an answer to these questions.


So in the end, the isomorphism between the two Hecke algebras seems to me to come from the interplay between the cohomology of modular varieties and group-theoretic properties of the Hecke algebra. A very general formulation of this fact can be found in D.Mauger Algèbres de Hecke quasi-ordinaires universelles. Ann. Sci. École Norm. Sup. (4) 37 (2004) (section 2.4 to be precise)


