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edited conditions on rings in $\mathcal{X}$
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This isn't an answer, just some idle thoughts about the following closely related problem:

Which complete local Noetherian rings $A$ can occur as the quotient of the universal deformation ring $R$ of a representation $\rho: G_{\mathbb{Q},S} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$?

Such a universal ring $R$ should always have dimension $4$ and be a complete intersection, at least conjecturally.

So straight away the following question occurs:
is the set $\mathcal{X}$ of isomorphism classes of complete, local, Noetherian, complete intersectionintegral, $4$-dimensional $W(\mathbb{F}_q)$-algebras actually uncountable? (Conditions slightly edited)

I don't know the answer but if indeed $\mathcal{X}$ is uncountable, that mightseems to be a problem: It seems unlikely that they Only finitely many of the elements of $\mathcal{X}$ could all occur as quotients of the countable seteach of universal deformation rings.the countably many universal $R$s (Although now I think about it I'm not so sure that couldn't happen.namely, one for each minimal prime ideal of $R$).

On the other hand, I think one might be able to verify that the set of universal deformation rings is dense in $\mathcal{X}$ (for some reasonable topology). For example, I believe that any Artin local $A$ is a quotient of an $R$ as above. This should follow from the method of Taylor and Wiles: they show that you can get a formal power series ring, in as many variables as you want, as a limit of such deformation rings $R$.

This isn't an answer, just some idle thoughts about the following closely related problem:

Which complete local Noetherian rings $A$ can occur as the quotient of the universal deformation ring $R$ of a representation $\rho: G_{\mathbb{Q},S} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$?

Such a universal ring $R$ should always have dimension $4$ and be a complete intersection, at least conjecturally.

So straight away the following question occurs:
is the set $\mathcal{X}$ of isomorphism classes of complete, local, Noetherian, complete intersection, $4$-dimensional $W(\mathbb{F}_q)$-algebras actually uncountable?

I don't know the answer but if indeed $\mathcal{X}$ is uncountable, that might be a problem: It seems unlikely that they could all occur as quotients of the countable set of universal deformation rings. (Although now I think about it I'm not so sure that couldn't happen.)

On the other hand, I think one might be able to verify that the set of universal deformation rings is dense in $\mathcal{X}$ (for some reasonable topology). For example, I believe that any Artin local $A$ is a quotient of an $R$ as above. This should follow from the method of Taylor and Wiles: they show that you can get a formal power series ring, in as many variables as you want, as a limit of such deformation rings $R$.

This isn't an answer, just some idle thoughts about the following closely related problem:

Which complete local Noetherian rings $A$ can occur as the quotient of the universal deformation ring $R$ of a representation $\rho: G_{\mathbb{Q},S} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$?

Such a universal ring $R$ should always have dimension $4$ and be a complete intersection, at least conjecturally.

So straight away the following question occurs:
is the set $\mathcal{X}$ of isomorphism classes of complete, local, Noetherian, integral, $4$-dimensional $W(\mathbb{F}_q)$-algebras actually uncountable? (Conditions slightly edited)

I don't know the answer but if indeed $\mathcal{X}$ is uncountable, that seems to be a problem: Only finitely many of the elements of $\mathcal{X}$ could occur as quotients of each of the countably many universal $R$s (namely, one for each minimal prime ideal of $R$).

On the other hand, I think one might be able to verify that the set of universal deformation rings is dense in $\mathcal{X}$ (for some reasonable topology). For example, I believe that any Artin local $A$ is a quotient of an $R$ as above. This should follow from the method of Taylor and Wiles: they show that you can get a formal power series ring, in as many variables as you want, as a limit of such deformation rings $R$.

added 60 characters in body
Source Link
Passerby
Passerby
  • 21
  • 1
  • 2

This isn't an answer, just some idle thoughts about the following closely related problem:

Which complete local Noetherian rings $A$ can occur as the quotient of the universal deformation ring $R$ of a representation $\rho: G_{\mathbb{Q},S} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$?

Such a universal ring $R$ should always have dimension $4$ and be a complete intersection, at least conjecturally.

So straight away there is a problem the following question occurs: 
is the set $\mathcal{X}$ of isomorphism classes of complete, local, Noetherian, complete intersection, $4$-dimensional $W(\mathbb{F}_q)$-algebras actually uncountable?

I don't know the answer but if indeed $\mathcal{X}$ is uncountable, that is might be a severe problem -- I can hardly imagine: It seems unlikely that onethey could describe, in any sensible way,all occur as quotients of the countable set which arise asof universal deformation rings. (Although now I think about it I'm not so sure that couldn't happen.)

On the other hand, I think one might be able to verify that the set of universal deformation rings is dense in $\mathcal{X}$ (for some reasonable topology). For example, I believe that any Artin local $A$ is a quotient of an $R$ as above. This should follow from the method of Taylor and Wiles: they show that you can get a formal power series ring, in as many variables as you want, as a limit of such deformation rings $R$.

This isn't an answer, just some idle thoughts about the following closely related problem:

Which complete local Noetherian rings $A$ can occur as the quotient of the universal deformation ring $R$ of a representation $\rho: G_{\mathbb{Q},S} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$?

Such a universal ring $R$ should always have dimension $4$ and be a complete intersection, at least conjecturally.

So straight away there is a problem: is the set $\mathcal{X}$ of isomorphism classes of complete, local, Noetherian, complete intersection, $4$-dimensional $W(\mathbb{F}_q)$-algebras actually uncountable?

I don't know the answer but if indeed $\mathcal{X}$ is uncountable, that is a severe problem -- I can hardly imagine that one could describe, in any sensible way, the countable set which arise as universal deformation rings.

On the other hand, I think one might be able to verify that the set of universal deformation rings is dense in $\mathcal{X}$ (for some reasonable topology). For example, I believe that any Artin local $A$ is a quotient of an $R$ as above. This should follow from the method of Taylor and Wiles: they show that you can get a formal power series ring, in as many variables as you want, as a limit of such deformation rings $R$.

This isn't an answer, just some idle thoughts about the following closely related problem:

Which complete local Noetherian rings $A$ can occur as the quotient of the universal deformation ring $R$ of a representation $\rho: G_{\mathbb{Q},S} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$?

Such a universal ring $R$ should always have dimension $4$ and be a complete intersection, at least conjecturally.

So straight away the following question occurs: 
is the set $\mathcal{X}$ of isomorphism classes of complete, local, Noetherian, complete intersection, $4$-dimensional $W(\mathbb{F}_q)$-algebras actually uncountable?

I don't know the answer but if indeed $\mathcal{X}$ is uncountable, that might be a problem: It seems unlikely that they could all occur as quotients of the countable set of universal deformation rings. (Although now I think about it I'm not so sure that couldn't happen.)

On the other hand, I think one might be able to verify that the set of universal deformation rings is dense in $\mathcal{X}$ (for some reasonable topology). For example, I believe that any Artin local $A$ is a quotient of an $R$ as above. This should follow from the method of Taylor and Wiles: they show that you can get a formal power series ring, in as many variables as you want, as a limit of such deformation rings $R$.

Source Link
Passerby
Passerby
  • 21
  • 1
  • 2

This isn't an answer, just some idle thoughts about the following closely related problem:

Which complete local Noetherian rings $A$ can occur as the quotient of the universal deformation ring $R$ of a representation $\rho: G_{\mathbb{Q},S} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$?

Such a universal ring $R$ should always have dimension $4$ and be a complete intersection, at least conjecturally.

So straight away there is a problem: is the set $\mathcal{X}$ of isomorphism classes of complete, local, Noetherian, complete intersection, $4$-dimensional $W(\mathbb{F}_q)$-algebras actually uncountable?

I don't know the answer but if indeed $\mathcal{X}$ is uncountable, that is a severe problem -- I can hardly imagine that one could describe, in any sensible way, the countable set which arise as universal deformation rings.

On the other hand, I think one might be able to verify that the set of universal deformation rings is dense in $\mathcal{X}$ (for some reasonable topology). For example, I believe that any Artin local $A$ is a quotient of an $R$ as above. This should follow from the method of Taylor and Wiles: they show that you can get a formal power series ring, in as many variables as you want, as a limit of such deformation rings $R$.