All Questions
Tagged with galois-representations l-functions
16 questions
8
votes
1
answer
363
views
Evidence for the equivariant BSD conjecture with higher multiplicity
Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(...
- 83
1
vote
0
answers
133
views
Automorphy of the twisted representation
The Artin reciprocity says that if
$$
\chi: \operatorname{Gal}(K/\mathbb Q) \to \mathbb C
$$
is a 1-dimensional representation of a finite Galois extension $K/ \mathbb Q$, then it corresponds to a ...
1
vote
0
answers
131
views
Analytic properties of $L$-functions attached to a compatible system of $\ell$-adic Galois representations
Let $F$ and $E$ be number fields, $G_F$ be the absolute Galois group of $F$, and $S$ be a finite set of primes of $F$. For $\lambda$ a prime of $E$ we denote by $\ell$ its residual characteristic. We ...
- 663
35
votes
1
answer
2k
views
The modularity theorem as a special case of the Bloch-Kato conjecture
In the homepage for the CRM's special semester this year, I found the interesting statement that the modularity theorem (formerly the Taniyama-Shimura-Weil conjecture) is a special case of the Bloch-...
- 37k
7
votes
1
answer
592
views
Which cases of Beilinson-Bloch-Kato for elliptic motives are known?
Let $V$ be a semisimple geometric Galois representation of a number field. Then the Bloch-Kato conjectures state that
$$
\operatorname{ord}_{s=0}{L(V^*(1),s)} = \operatorname{dim}{H^1_f(G_k,V)}-\...
- 37k
6
votes
1
answer
315
views
Proving automorphy of the Galois representations of number fields without considering the residual representation
All the papers proving automorphy of the representations of Galois groups of number fields that I have come across seem to first reduce the representation modulo a prime, prove the automorphy of the ...
- 13k
11
votes
3
answers
944
views
"Extra Euler factors" in one definition of the L-function of a twist of a modular form
Let $(\rho_{f,\lambda})_\lambda$ be the system of Deligne's $\ell$-adic representations attached to a modular newform $f$ (where $\lambda$ runs over the finite places of the number field $K$ generated ...
- 20.8k
6
votes
1
answer
499
views
Galois representation and weight one Hilbert modular form
Let $f$ be a primitive weight one Hilbert modular form for the totally real field $F$ (the weight of $f$ is parallel). Assume that $f$ is $N$-new, $p$-stable and cuspidal. Let $\rho:G_F \rightarrow \...
- 37k
19
votes
2
answers
2k
views
Applications of Artin's holomorphy conjecture
I wonder why the Artin conjecture is so important. The only reason I could figure out is that one could use the holomorphy of Artin L-series and Weil's converse theorem to show modularity of two-...
- 17.6k
28
votes
1
answer
2k
views
Is there a "Langlands philosophy" reason for the fact that the L-function of the Jacobi theta function is (almost) the Riemann zeta function?
The Jacobi theta function $\theta(z) = 1 + 2 \sum_{n = 1}^\infty q^{n^2}$, with $q = e^{\pi i z}$ is a (twisted) modular form with weight $1/2$. It has an associated $L$-function $L(\theta, s) = \sum_{...
- 26k
1
vote
1
answer
226
views
How much extra ramification in a residual representation
Suppose $\rho:G _{\mathbb{Q}} \rightarrow GL_n(\mathbb{Q}_p)$ is a Galois rep. It has a uniquely defined (up to semisimplification residual rep $\bar{\rho}$. $\bar{\rho}$ is unramified where $\rho$ is....
- 829
13
votes
0
answers
284
views
Propagation of modularity and the Artin conjecture
The (still incomplete) solution of the Artin conjecture on dimension $\leq2$ has been a massive research effort that has spanned (knowingly or not) around a century.
A very natural question is, what ...
- 105k
31
votes
1
answer
4k
views
Iwasawa main conjectures vs Bloch-Kato conjectures
Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...
- 10.9k
5
votes
0
answers
288
views
Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?
Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...
- 2,396
2
votes
1
answer
555
views
How do you calculate the Euler factors of the imprimitive symmetric square at primes with bad reduction?
The reference for this question is Coates and Schmidt, Iwasawa theory for the symmetric square.
Let $G = \textrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}))$ and let $D_r \supseteq I_r$ be a ...
- 37k
17
votes
0
answers
1k
views
Special values of Artin L-functions
This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so.
Background: Stark's conjecture interprets ...
- 13k