8
votes
Accepted
Trying to understand the topology of the Weil group for the local Langlands conjecture
Note that in a topological group $G$, any subgroup $H$ containing an open subgroup $U$ is itself open: we can write $H$ as a disjoint union of $hU$ for a set of coset representatives for $H/U$, and ...
8
votes
Accepted
Equidistribution on $\mathrm{SU}_2$
In the article "On the spectral gap for finitely-generated subgroups of SU(2)" by Jean Bourgain and Alex Gamburd (Invent. Math. 171, No. 1, 83-121 (2008)), they show that free subgroups of $...
7
votes
What is the difference between Hida and Coleman families?
The difference is the generality of the setting: Hida families (first introduced by Hida in the early 80s) apply only to eigencuspforms which are so-called ordinary at $p$ (roughly speaking, the $p$-...
5
votes
Accepted
Generators of the ideal class group
Let $G$ be the ideal class group, and let $H$ be the subgroup generated by the prime ideals of norm at most $3\log^2(d^2)=12\log^2(d)$. Assume that $H$ is a proper subgroup of $G$. Then there is a ...
4
votes
Is there something I am missing about the computation of the $p$-part of the class groups of cyclotomic fields?
[Rather than leaving the comment "Class number formula" for Olivier as a comment, I expand it for other readers of the question, to a partial answer.]
Kummer knew in 1850 that the class ...
4
votes
Accepted
Reference Request: Possible generalizations of the stability of $\gamma$-factors
Let us first take a look at your first question:
I do not think that the generalization you describe is true for m>1. Indeed we can take a characters $\chi_1,\ldots,\chi_n$ of $F^{\times}$ with ...
4
votes
Question about Größencharaktere in imaginary quadratic number fields
Let $P$ be the group of principal fractional ideals $(\alpha)$ where $\alpha \in K^\times$ and let $I$ be the group of all fractional ideals in $K$. I am not going to look at the paper you cite, but ...
4
votes
Class numbers in the unramified biquadratic extensions of number fields
Kuroda's class number formula does not require the extension to be unramified.
It is not true, as you write, that the class number of $K$ can be determined from the class numbers of the $k_i$; you ...
4
votes
Equidistribution on $\mathrm{SU}_2$
It was asked in the comments that I provide some details. I prove slightly more: if $\mu_n$ denotes uniform probability on the sphere of radius $n$ and if $\rho:F_2 \to \mathrm{SU}_2$ is a ...
3
votes
Accepted
Criterion for Ramification of ray class field $K(\mathfrak{p})$ in $\mathfrak{p}$
When the global units surject, then any generator of a principal ideal coprime to ${\mathfrak{m}}$ can always be multiplied by a unit to replace it with a generator of the same principal ideal that is ...
3
votes
Existence of odd mod $p$ Galois representations whose image is $p'$-group
Take an $S_3$-extension $L/K$ in which all primes above $p$ split and which is odd at all real places of $K$. (These are easy to construct, take the Galois closure of $x^3 - a x - b$ where $a$ and $b$...
2
votes
Discrepancy between $\dim H^2(G, \mathrm{ad}(\bar \rho))$ and the number of relations in a minimal presentation of the universal deformation ring $R$
Question 3: No. In fact for any ring $R$ that appears as a deformation ring, $R$ can be a deformation ring of a group with arbitrarily large $h^2$.
It suffices to find, for $G$ a group with ...
2
votes
Accepted
How to compute the asymptotic constant for the count of $S_3$-sextic number fields?
I will do a worked example for the tame case and show that it agrees with the formula from their paper. You should then be able to adapt this to the case $p=2,3$.
Let $p > 3$. I use formula (7) ...
1
vote
Accepted
Defect between modulus and conductor of ray class field
I don't know what "a historical appedage from times before the interpretation of class field theory with topological methods via adelic formalism a la Chevalley" means, but if $\mathfrak{m}$ ...
1
vote
Symmetric power lift of modular forms
This can happen. For example, the dihedral group $D$ of order $10$ has two irreducible (Galois conjugate) representations $U$ and $V$ of dimension $2$ which are not twists of each other. But $\mathrm{...
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