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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
50
votes
13
answers
6k
views
Erratum for Cassels-Froehlich
Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).
IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has …
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
27
votes
"Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
I am pretty sure that when different number theorists say "one of the main goals of number theory is to understand Gal(Q-bar/Q)" they may well mean different things.
One example of what someone might …
- 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
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
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
13
votes
Why do congruence conditions not suffice to determine which primes split in non-abelian exte...
OK how's about this to finish (I don't think either argument posted so far deals with this case). Say $K/\mathbf{Q}$ is finite and (away from a finite set of exceptions) $p$ splits completely in $K$ i …
- 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
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
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
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
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
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
5
votes
Erratum for Cassels-Froehlich
Ok so it looks like I misjudged this and the community seem happy to have the question here, at least at present. So I figured I'd pass on the comments which Serre sent the LMS.
p.135, part b) of L …
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