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Results tagged with algebraic-number-theory
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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
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 …
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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 …
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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 …
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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 …
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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 $p$.
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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 …
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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 …
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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 …
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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 …
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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 …
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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 …
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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 …
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 …
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