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Results tagged with nt.number-theory
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user 52824
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
13
votes
Accepted
Is every closed subgroup of $\text{GL}_n(K[[x]])$ finitely generated?
This question has a negative answer is many respects. Firstly, there are simple constructions in the commutative case. Namely, the additive group $K[\![x]\!]$ is an infinite dimensional $\mathbb F_p …
7
votes
Accepted
Compactness of adelic quotients for unipotent groups over global fields
You don't mention if $U$ is assumed to be smooth or connected, but it doesn't matter. In general, if $H$ is any affine group scheme of finite type over a global field $K$ and if $H$ does not contain $ …
- 1,285
6
votes
Accepted
Chinese Remainder Theorem backwards
Since $h_k := \prod_{i=1}^k \phi_{p^i} = (X^{p^k}-1)/(X-1)$, you are trying to compute the exponent of the cokernel
of the inclusion
$$\mathbf{Z}[X]/(h_k) \rightarrow \prod_{i=1}^k \mathbf{Z}[\zeta_{ …
- 1,285
6
votes
Accepted
Theorem 7b of Serre's "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques"
This is ultimately an application of Lang's vanishing theorem for degree-1 Galois cohomology of connected algebraic groups over finite fields (applied to tori).
What follows may look complicated if y …
5
votes
Openness of finite index subgroups of $\mathrm{GL}_n(\prod O_v)$
In view of Laurent's answer, one may ask more generally for a criterion on a connected reductive $K$-group $G$ to ensure that for all compact open subgroups $U$ of $G(\mathbf{A}_K^S)$ (for a finite se …
4
votes
Absolutely irreducible p-adic representation of the absolute Galois group of Q_p
One gets loads of crystalline counterexamples using $p$-divisible groups (and more specifically from any elliptic curve with supersingular reduction).
Let $k$ be a perfect field of characteristic $ …
- 1,285