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14 votes
Accepted

Totally ramified subextension in a finite extension of $\mathbf{Q}_p$

This is not a complete answer, but perhaps it's a roadmap to a counterexample. My strategy is to consider some non-Galois $K/\mathbf{Q}_p$ for which the result is true, and let's make some deductions …
Kevin Buzzard's user avatar
14 votes
3 answers
3k views

Computing (on a computer) higher ramification groups and/or conductors of representations.

I am supervising an undergraduate for a project in which he's going to talk about the relationship between Galois representations and modular forms. We decided we'd figure out a few examples of weight …
Kevin Buzzard's user avatar
6 votes
Accepted

Hilbert Symbols, Norms, and p-adic roots of unity

I think I can construct an explicit counterexample with $a\in\mathbb{Q}_p$. Choose a compatible sequence $\zeta_{p^m}$ of $p^m$th roots of unity in $\overline{\mathbb{Q}}_p$. Write $q=p^n$ with $n\g …
Kevin Buzzard's user avatar
10 votes
Accepted

Type of place versus type of unitary group

Things are perhaps a bit messier than you hope. In particular it is not true that the unitary group is non-quasi-split if and only if $v$ ramifies. Disclaimer: I did not know the answer to this questi …
Kevin Buzzard's user avatar