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What are some general results describing the Krull dimension of the completion of a non-Noetherian ring with a "nice" topology? An example of the sort of "nice" topological ring I'm looking for is a …

I think we can describe $P$ a bit more, using Dirichlet’s unit theorem. Since $\overline{\mathbf{Z}} = \varinjlim_{[K:\mathbf{Q}] < \infty} \mathcal{O}_K$, the same is true for the units. Now Dirichle …

Yes, in general you need to consider all cocharacters. $\mathrm{GL}_n$ has the special property that the dominant coweights are all sums of minuscule cocharacters, i.e. ones of the form $[1,\cdots,1 …

No, these are not the same thing. Formal schemes are to rigid-analytic spaces as $\mathbf{Z}_p$-schemes are to $\mathbf{Q}_p$-schemes. The book Lectures in Formal and Rigid Geometry by Bosch is an e …

I'll assume that you're talking about elliptic curves over $\mathbf{Q}$ - much of what I'm going to say should generalize to totally real fields, replacing modular forms by Hilbert modular forms. I th …

You can see pretty easily that the angle Großencharacter appearing in Hecke's equidistribution theorem cannot arise as the Großencharacter associated to a CM elliptic curve just by thinking about $\in …

The Jacobi theta function $\theta(z) = 1 + 2 \sum_{n = 1}^\infty q^{n^2}$, with $q = e^{\pi i z}$ is a (twisted) modular form with weight $1/2$. It has an associated $L$-function $L(\theta, s) = \sum_ …