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Results tagged with algebraic-number-theory
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user 2284
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
20
votes
Accepted
Infinitely many number fields of class number 1
We don't know that there are infinitely many number fields with Class Number one, so a fortiori we don't know any explicit infinite family of such number fields.
- 10.9k
8
votes
Accepted
What is the difference between Hida and Coleman families?
The difference is the generality of the setting: Hida families (first introduced by Hida in the early 80s) apply only to eigencuspforms which are so-called ordinary at $p$ (roughly speaking, the $p$-a …
- 10.9k
3
votes
Does Beilinson's conjecture on values L-functions work for smooth projective varieties over ...
In addition to François's answer, I'll address the second question.
Are there any differences between the case over $\mathbb Q$ and a number fields $L$?
The main difference - which can be dealt …
- 10.9k
1
vote
Accepted
Steinberg components of local deformation rings
If I understand correctly, I believe that you want $r:\Gamma\longrightarrow\operatorname{GL}_2(R)$ to factor through $R^{\operatorname{St}}$ if $r$ is a non-trivial extension of $\beta$ by $\alpha$ wi …
- 10.9k
4
votes
Conductor of Galois representation attached to newform
In fact much more than the equality of conductor is true: the local Galois representation $\rho_{F,\lambda}|G_{\mathbb Q_{p}}$ obtained by restricting $\rho_{F,\lambda}$ to the decomposition group at …
- 10.9k
11
votes
Accepted
Class field towers
Let $\ell$ be an odd prime and $m$ an integer such that
$$|\{p|m,\ p\equiv1\operatorname{mod} \ell\}|\geq8.$$
Then Y.Furuta proved that $\mathbb Q(\zeta_m)$ admits an infinite unramified $\ell$-class …
- 10.9k
4
votes
Properties of Mod $\ell^m$ Galois representation associated to modular form
Write $L$ for the finite Galois extension of $\mathbb Q$ with Galois group $G_{\mathbb Q}/\operatorname{Ker}\rho_{F,v}^m$. Then $\rho_{F,v}^m(\operatorname{Frob}_p)$ is the identity in $\operatorname{ …
- 10.9k
27
votes
Accepted
Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?
I am not sure I really agree with the following quote (which is the opening paragraph of Modular forms and Galois cohomology by H.Hida) because I suspect that a mathematician valuing creativity and ve …
- 10.9k
1
vote
Relation between the Selmer group and the ideal class group
Franz Lemmermeyer's answer points out to a connection. Let me point out in the converse direction that believable heuristics suggest that if $E(\mathbb Q)=0$, then $\operatorname{Sel}_p(E)$ should be …
- 10.9k
13
votes
Accepted
Some questions on the $p$-adic properties of special $L$-values
1) What generalizations of the Kummer congruences are known?
This is somewhat imprecise as a question and in particular, I would dispute a little your assertion that
This is probably the same …
- 10.9k
12
votes
2
answers
386
views
Is there something I am missing about the computation of the $p$-part of the class groups of...
Well, the answer of the question in the title in certainly Yes, many things in fact, but let me be more precise.
In 1958, Serre gave a Bourbaki talk on the recent works of Iwasawa on class groups in t …
- 10.9k
5
votes
Is there something I am missing about the computation of the $p$-part of the class groups of...
Recently, I stumbled coincidentally on the paper
Computation of invariants in the theory of cyclotomic fields K. Iwasawa and C. Sims J. Math. Soc. Japan Vol.18 (1966)
This explains in full details how …
- 10.9k
8
votes
Accepted
Proving automorphy of the Galois representations of number fields without considering the re...
The canonical answer to that question is certainly the world of so called converse theorems, whose basic ideas go back to Hecke's remark that an holomorphic $L$-function satisfying a suitable function …
- 10.9k
34
votes
4
answers
3k
views
$A_5$-extension of number fields unramified everywhere
So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and solvabl …
- 10.9k
5
votes
Conceptual understanding of the Gross-Zagier theorem.
In my current (no very deep) understanding, there are two possible ways to make the proof of the Gross-Zagier more conceptual.
The first is to recognize in each terms of the equation products of loc …
- 10.9k