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Geometric interpretation of Hida familiesisomorphism

[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 families from a more geometric perspective than what I normally read. What bothers me is that there are two construction of the universal Hecke (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 $p$.

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). Hida interpretes the space of cusp formsof weight $k$ and level $Np^r$ as global section of a sheaf of differentials ''twisted $k-2$ times'' over the closed modular curve $X_1(Np^r)$, as usual, ) but the limit process his proof is purely entirely algebraic- at least as far as I understand.

My question is: is there a reasonable way to interpret a Hida family as a section prove this isomorphism geometrically?As Kevin Buzzard suggested, several papers of some sheaf Katz (and successive work by Coleman-Mazur, Buzzard himself et al.) discuss geometric interpretation of differentials $\Omega_?^?$ over the pro-scheme p$-adic modular forms and $\varprojlim X_1(Np^r)$? Does this make easier to understand why, up there, there is no notion p$-adic families of a weight anymore? Would Theorem 1.1 above follow from this interpretationmodular forms. Still, perhaps by producing a huge sheaf 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 $X_1(Np)$, and whose sections are automatically isomorphic to those of $\Omega_?^?$? And, ultimately: does this make any sense at all?X_1(Np)$'') can be given a geometric interpretation.
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As the title says, I would like to understand Hida families from a more geometric perspective than what I normally read. What bothers me is that there are two construction of the universal Hecke algebra: 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 algebra 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). Hida interpretes the space of cusp forms of weight $k$ and level $Np^r$ as global section of a sheaf of differentials ''twisted $k-2$ times'' over the closed modular curve $X_1(Np^r)$, as usual, but the limit process is purely algebraic - at least as far as I understand.

My question is: is there a reasonable way to interpret a Hida family as a section of some sheaf of differentials $\Omega_?^?$ over the pro-scheme $\varprojlim X_1(Np^r)$? Does this make easier to understand why, up there, there is no notion of a weight anymore? Would Theorem 1.1 above follow from this interpretation, perhaps by producing a huge sheaf over the modular curve $X_1(Np)$, and whose sections are automatically isomorphic to those of $\Omega_?^?$? And, ultimately: does this make any sense at all?
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Geometric interpretation of Hida families

As the title says, I would like to understand Hida families from a more geometric perspective than what I normally read. What bothers me is that there are two construction of the universal Hecke algebra: 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 algebra ordinary $h^0(N,\mathbb{Z}_p)$.

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). Hida interpretes the space of cusp forms of weight $k$ and level $Np^r$ as global section of a sheaf of differentials ''twisted $k-2$ times'' over the closed modular curve $X_1(Np^r)$, as usual, but the limit process is purely algebraic - at least as far as I understand.

My question is: is there a reasonable way to interpret a Hida family as a section of some sheaf of differentials $\Omega_?^?$ over the pro-scheme $\varprojlim X_1(Np^r)$? Does this make easier to understand why, up there, there is no notion of a weight anymore? Would Theorem 1.1 above follow from this interpretation, perhaps by producing a huge sheaf over the modular curve $X_1(Np)$, and whose sections are automatically isomorphic to those of $\Omega_?^?$? And, ultimately: does this make any sense at all?