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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
1
vote
Changing the weight space for an eigenvariety
I think this question is based on a misconception. “Being an eigenvariety” isn’t a rigorously defined property of a space (or map of spaces) which you could prove to hold or to not hold. It’s more lik …
- 37k
5
votes
p-adic L functions from Selmer groups - how canonical are they?
All of your questions are undermined by the same fundamental issue: you cannot talk about "the" p-adic $L$-function in this generality, because there is no sensible definition of what a $p$-adic $L$-f …
- 37k
2
votes
Families of Galois representations over disks
I don't really understand the setup of this question: what is $R\langle x_1/r_1, \dots \rangle$ supposed to mean if $r_i$ is a real number?
That said, if $R = \mathbb{Z}_p$, then $\mathbb{Z}_p[[x_1, \ …
- 37k
4
votes
Accepted
Integration against Eisenstein series can be regarded as a cup product
Yes, that does indeed sound like something I might have said :)
I was referring to some extremely powerful theorems, originally due to Michael Harris, which show that:
The cohomology groups of automo …
- 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
8
votes
Accepted
Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk?
The correct viewpoint is not "$\Lambda$ is like a disc", but "$\Lambda$ is like the functions on a disc".
To see this, ask yourself: given an element $f \in \mathbb{Z}_p[[T]]$, what values can we plug …
- 37k
6
votes
Class number of imaginary quadratic fields
The condition shouldn't be "$n$ is prime" but "$n$ is either 1, 2, or a prime congruent to 3 mod 4". For instance $\mathbb{Q}(-5)$ has class number 2.
The more general statement that the 2-torsion sub …
- 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
10
votes
Accepted
Non-existence of "higher" Artin map
There is no way of reformulating local Langlands for $n > 1$ in terms of such a map.
Local Langlands is a bijection between irreducible smooth representations of $\operatorname{GL}_n(K)$, and $n$-dime …
- 37k
8
votes
Accepted
modularity lifting theorems for non-compact unitary groups
You might like to read the introduction of Harris' 2013 Crelle paper "The Taylor-Wiles method for coherent cohomology" (see link). Here is an excerpt:
In practice, all the higher-dimensional results, …
- 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
2
votes
Accepted
On presentations of universal rings of deformations
Doesn't this kind of prove itself? Pick some elements $\alpha_1, \dots, \alpha_n \in \mathfrak{m}$ which represent $\mathfrak{m} / (p, \mathfrak{m}^2)$. Clearly sending $t_i$ to $\alpha_i$ defines a m …
- 37k
4
votes
Accepted
Sign of the special value at s=0 of Hecke L-functions
If $\chi$ is real-valued, then the question makes sense. Using the functional equation, it reduces to computing the sign of the non-zero real number $L(\chi, 1)$ if $\chi$ is non-trivial, or the resid …
- 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
4
votes
Is there an analogue for Ramanujan–Serre derivative for Hilbert modular forms?
I think there's not a good analogue.
The operator $D_{j, k_j}$, acting on Hilbert modular forms of weight $k = (k_1, \dots, k_n)$ where $n = [F: \mathbf{Q}]$, is the "Maass--Shimura differential opera …
- 37k