Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with algebraic-number-theory
Search options not deleted
user 5015
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
10
votes
On a minimal algebraic number field which satisfies the principal ideal theorem
The answer to the second question is also "no". Take $k= \mathbb{Q}(\sqrt{-5})$. Then the only non-trivial class capitulates in $H=k(i)$ and it also does in $K=k(\sqrt{-3})$, yet $H$ and $K$ are not i …
- 8,945
5
votes
Accepted
What is the exact meaning of the real period in the $p$-adic formulation of BSD?
Mazur-Tate-Teitelbaum have written their p-adic BSD paper for a modular form of even weight $k\geq 2$ so unless we are dealing with a elliptic curve, we might want to avoid choosing a period. They vie …
- 8,945
10
votes
Accepted
discriminant of subfield of $\mathbb{Q}(\zeta_p)$
The Führerdiskriminantenproduktformel tells you that is it the product of conductors of characters, but all but the trivial character must have conductor $p$.
- 8,945
10
votes
Accepted
class number of prime degree field with prime conductor
Maybe I am making a mistake here, but let me try:
Let $H$ be the Hilbert class field of $K$. Then $H\cap \mathbb{Q}(\zeta_p)=K$ as otherwise one prime in there should be totally ramified and unramifi …
- 8,945
2
votes
class group size of cyclotomic field subextension
The $p$-primary part of the class group of $\mathbb{Q}_1$, and in fact all $\mathbb{Q}_n$ in the cyclotomic tower of $\mathbb{Q}$, is trivial for all $p$. This is contained in Proposition 13.22 of Was …
- 8,945
5
votes
Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is know...
Much more than Iwasawa's original theorem is known by now. First of all, the $p$-primary part of the class group stays trivial in the field $K_n=\mathbb{Q}(\mu_{p^{n+1}})$ unless it is already non-tri …
- 8,945
8
votes
Accepted
Capitulation in cyclotomic extensions
Assume $p$ is an irregular prime for which
Vandiver's conjecture holds, e.g. $p<12'000'000$. This conjecture asserts that $p$ does not divide the $+$-part of the class group.
Then there is no capitu …
- 8,945
7
votes
Accepted
When is this localization map injective, if at all?
Often it is, but not always. For instance if $K=\mathbb{Q}$ then the map is injective if and only if the rank of $E(\mathbb{Q})$ is at most $1$. This is because the $p$-adic elliptic logarithm of a no …
- 8,945
5
votes
Accepted
Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension
For any $\mathbb{Z}_p$-extension the ramification is concentrated among the primes above $p$. Those that are ramified are totally ramified.
For the cyclotomic $\mathbb{Z}_p$-extension all places abov …
- 8,945
2
votes
Vanishing of the degree 2 cohomology of a p-adic field with coefficients Q/Z and action of t...
I don't think the action of Frobenius $\phi$ is trivial. The inf-res five term exact sequence reduces to a short exact sequence
$$ 0\to \operatorname{Hom}\bigl(G/I, \mathbb{Q}/\mathbb{Z}\bigr)\to \ope …
- 8,945
7
votes
Accepted
Is there something I am missing about the computation of the $p$-part of the class groups of...
[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 group of $\mat …
- 8,945
11
votes
Accepted
Oesterlé's unpublished bound on Uniform Boundedness
Yes, this is published as appendix A to chapter 3 in Derickx' PhD thesis available here: https://openaccess.leidenuniv.nl/handle/1887/43186 . The thesis contains, of course, many more interesting resu …
- 8,945
3
votes
applications of Tate-Poitou duality
Three examples of the use of the Poitou-Tate duality:
All parity results (Dokchitsers, Mazur-Rubin, Nekovar, ...) for elliptic curve use this duality somewhere.
The duality is also crucial for Eule …