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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
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
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
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
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
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
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
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
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
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
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
2
votes
Erratum for Cassels-Froehlich
Dominique Bernardi points out that the formula for $\phi_a$ is wrong on line 1 of p96. This is a delicate one. The issue is what the definition of the action of $G$ on $A*$ is (NB that starshould be a …
1
vote
Erratum for Cassels-Froehlich
Joseph Oesterlé says:
p 69, l 26 S contains all v with |alpha_v|_v < 1 should be S contains all v with |alpha_v|_v ≠ 1
p 69, l 27 \frac{1}{2}C should be \frac{1}{2C}
p 131 corollary 2. "Let L/K be …
1
vote
Erratum for Cassels-Froehlich
Keith Conrad sent me a nice chunky list here.
1
vote
Erratum for Cassels-Froehlich
Rebecca Bellovin writes:
Here's one I didn't see on the list on mathoverflow: In exercise 2,
part 10, equation (**) (page 353, line 4) should be
$$\prs{\lambda}{b}=\prod_{v\in S}(b,\lambda)_{v} …
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 …