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Results tagged with algebraic-number-theory
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user 2481
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
9
votes
Accepted
If two Hecke characters cut out the same field, are they Galois conjugates?
The answer to both your questions is "no".
Take $K = \mathbf{Q}$. Then there is a unique $\mathbf{Z}_p$-extension of $K$ (contained in $\mathbf{Q}(\zeta_{p^\infty})$) which gives us a surjection $G_K …
- 37k
3
votes
Accepted
PNT for number fields.
This statement is true for all $p$ not dividing $disc(f)$, as you say. Moreover you can write down exactly what the primes dividing $p$ are: they're the ideals $(p, g(\theta))$ for each irreducible fa …
- 37k
11
votes
Accepted
A problem about real quadratic field
This is a consequence of "genus theory". If $K / \mathbf{Q}$ is an abelian extension, then the "genus field" of $K$ is the maximal extension $L / K$ such that $L/K$ is unramified and $L/\mathbf{Q}$ is …
- 37k
2
votes
Can Frobenius traces jump like crazy in non-geometric Galois representations?
"Assuming they're in an algebraic extension of $\mathbb{Q}$ can they
grow exponentially with 𝑝?"?
I'm not sure this statement will have a truth value, because I suspect the assumption never occurs: …
- 37k
7
votes
Accepted
Leopoldt's conjecture for totally real cubic and $S_3$-extensions
Out of curiosity, I thought I'd follow up on znt's suggestion.
Let $K = \mathbf{Q}(\alpha)$ where $\alpha^3 - 13\alpha + 7=0$. Then $K$ is totally real and non-Galois, and the prime 3 is totally ine …
- 37k
9
votes
Accepted
Extending arithmetic functions (and associated Dirichlet series) to arbitrary rings of integers
The answer to your Question 1 is "yes". It's clear that the number of ideals of $\mathcal{O}$ of norm $\le M$ is bounded above by a polynomial in $M$, so one can manipulate Dirichlet series term-by-te …
- 37k
14
votes
Accepted
Non-trivial class number at some finite level in the cyclotomic $\mathbf{Z}_p$-extension of ...
I found the notes of Coates' seminar I alluded to in my comment above. He said the following:
For $n \ge 1$, let $h(n)$ denote the class number of the unique cyclic degree $n$ extension contained in …
- 37k
1
vote
Accepted
Slope decomposition of a product of operators
The answer to your exact question is clearly "no" even if $M$ is finite-dimensional. Any operator on a finite-dimensional space has a slope decomposition, and if $n = 2$ and $U_2 = U_1^{-1}$ then ever …
- 37k
3
votes
Irreducible local Galois representation with arbitrary Hodge-Tate weights
Let me suppose (for simplicity) that $M$ has distinct elements.
Choose a polynomial $P \in \mathbf{Q}_p[X]$ with distinct roots, all of which have the same valuation, with this common valuation being …
- 37k
7
votes
Crystalline Characters
[EDIT: I misunderstood the question -- I thought the poster wanted to know why all crystalline characters are unramified twists of powers of cyclotomic when $L$ or $K$ is $\mathbb{Q}_p$, and wrote out …
- 37k
23
votes
Accepted
Explicit examples of algebraic Hecke characters with infinite image?
The "most obvious" algebraic Hecke characters of a field $K$ are the characters of the ideal class group of $K$, which have trivial infinity-type and trivial conductor. There might be no non-trivial e …
- 37k
6
votes
Accepted
Values of Artin L-functions at negative integers
The question of order of vanishing is quite elementary: the L-function of a Hecke character (i.e. 1-dimensional Artin representation) over any number field has an Euler product, which is convergent an …
- 37k
5
votes
Accepted
Existence of lift of (local) Artin map
No, there does not, except possibly a few small corner cases. The problem is finding somewhere for the torsion in $K^\times$ to go.
If $K$ is a finite extension of $\mathbf{Q}_p$, then there exists a …
- 37k
12
votes
Sign and coefficients of fundamental unit of quadratic field
This might be useful:
Stevenhagen, Peter, The number of real quadratic fields having units of negative norm, Exp. Math. 2, No. 2, 121-136 (1993). ZBL0792.11041.
As Stevenhagen explains, if the discrim …
- 37k
4
votes
Accepted
Order of vanishing of $L$-function and mixed Hodge-structures
There is a good reason why this particular form of the Beilinson conjecture cannot possibly be valid for the particular $i$ and $n$ you mention.
The real $\mathbb{R}$-MHS $H^1(X(\mathbb{C}), \mathbb{R …
- 37k