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Results tagged with algebraic-number-theory
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user 1384
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
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 …
- 41.4k
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 …
- 41.4k
4
votes
Accepted
Are Fredholm hypersurfaces affinoid?
No they're not in general affinoid. The problem is that the zero locus of the power series is computed within a space which is almost never affinoid -- for example in the modular curve case the ambien …
- 41.4k
13
votes
1
answer
729
views
Non-trivial class number at some finite level in the cyclotomic $\mathbf{Z}_p$-extension of ...
An MSc student asked me if I knew an example of a prime $p$ and some finite layer $K_n$ in the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$ (so $[K_n:\mathbf{Q}]=p^n$) which had non-trivial cla …
- 41.4k
12
votes
Accepted
Insolvable number fields ramified only at one (small) prime
Minhyong's comments indicate the issue here. If I want to come up with an extension unramified outside $p$ then why not look at the 2-dimensional mod $p$ representation attached to the $\Delta$ functi …
- 41.4k
13
votes
$A_5$-extension of number fields unramified everywhere
Oh, I know how I would try and build examples. First I would write down a random $A_5$ extension $K$ of $\mathbf{Q}$, ramified at some primes (in fact I would look in a table, e.g. in Buhler's thesis …
- 41.4k
26
votes
2
answers
2k
views
Are there any Hecke operators acting on an elliptic curve with additive reduction that I don...
I could have made this question very brief but instead I've maximally gone the other way and explained a huge amount of background. I don't know whether I put off readers or attract them this way. The …
- 41.4k
1
vote
p-split Hecke characters
I think your $\xi$ had better be algebraic, but perhaps this implicit somehow in your terminology. If $v$ is any finite place of $E$, there is a $p$-adic avatar of $\xi$ with values in $E_v^\times$ (w …
- 41.4k
15
votes
1
answer
843
views
components of E[p], E universal in char p.
I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised.
In this question, in Charles Rezk's answer, I erroneously claim that his construct …
- 41.4k
12
votes
Conjugacy classes in the absolute galois group
Firstly, Frobenius elements aren't even conjugacy classes, as you know. So you had better look at quotients $Gal(K/\mathbf{Q})$ of the Galois group which are unramified outside some set $S$. Now you h …
- 41.4k
8
votes
Does there exist a general theory of "arithmetic complexity"/"arithmetic height"?
In practice what you'll do is look for linear relations with integer coefficients between powers of $\alpha$ (starting with $\alpha^0=1$). If $\alpha$ is algebraic then suddenly you'll find a relation …
- 41.4k
2
votes
Erratum for Cassels-Froehlich
This answer is just to bump this post up to the front page for the final time. I typed up all the errata I heard into one pdf file and put it here. The London Maths Society would like comments, if any …
34
votes
Accepted
Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
EDIT: Hendrik Lenstra emailed me a proof of Conjecture 2. I'll append it below. So Jagy's question is now solved.
OK so I think that Jagy wants to make the following conjecture:
CONJECTURE 1: an i …
- 41.4k
1
vote
Erratum for Cassels-Froehlich
Brian Conrad says:
Page 52, section 8, Definition, displayed expression: put subscripts outside norms.
page 53, 2 lines above section 9: a_N rather than a_n in first norm on right side
Page 56, The …
1
vote
Erratum for Cassels-Froehlich
Hendrik Lenstra says:
Below my 51 errata that I didn't see on your list or in William Stein's mail
yet. Most are of a typographical nature, but some have mathematical
substance. I did at the present …