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Let $p$ be a prime and consider the ($p$-deprived) Hecke algebra $\mathbb{T}$ which the projective limit of the Hecke $\mathbb{Z}_p$-algebras $\mathbb{T}_k$ which act on modular forms of level $1$ with coefficients in $\mathbb{Z}_p$ and of weight at most $k$. (See $\S$2.1 of https://www.math.uchicago.edu/~emerton/pdffiles/padic.pdf)

Is the Krull dimension of $\mathbb{T}$ known when $p = 2$ or $3$? Is anything conjectured? The only references I can find deal with cases where the corresponding residual representation is absolutely irreducible, which excludes these cases.

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When I posted this, I was under the impression that the Gouvea-Mazur infinite fern argument only holds when the residual representation is absolutely irreducible. Now I have learned that with this was because pseudo-deformations were not so well understood when GM wrote their article, but now the argument can be made to work for N = 1 and p = 2,3. So in fact in each of these cases, the Hecke algebra is a power series ring over $\mathbb{Z}_p$ in three variables.

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