All Questions
77 questions
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Closure of non-closed subset in Ring theory
We say that the ring $R$ is topological, when we are given neighbourhoods $\{O_{\lambda};\lambda \in \Lambda\}$ of $0_R$.
We say ${\cal I}$ is a closed ideal of $R$ if and only if for any sub-index ...
CommunityBot
- 1
5
votes
0
answers
676
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Basic question on p-adic Hodge theory
I am starting to study the rudiments of p-adic Hodge theory and I have the following basic question. Let $\chi$ be the unramified quadratic character of $G_p = \mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{...
- 63.9k
1
vote
1
answer
329
views
Kummer extension of Galois modules
Let $k$ be a field of characteristic $p \geq 0$, $n$ an integer prime to $p$, and $x$ an element of $k \setminus \{0, 1\}$. I have read that the $n^{th}$ root of $1-x$ gives rise to a Galois module $E$...
CommunityBot
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0
votes
1
answer
131
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Radical of modules [closed]
Let $R$ be a local ring with the unique maximal ideal ${\frak m}_R$ and $M$ be a $R$-module. Define
$I(M) \colon= \cap ~({\mathrm{all~ proper~ maximal ~submodules~ of}}~M)$,
where proper means ...
0
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2
answers
2k
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Tensor products of two domains
Let $R$ be an integral noetherian regular local ring. Let $S$ be a noetherian integral domain such that $S/R$ is finite.
That is, $R \subset S$ and the surjection $R^{\oplus n} \twoheadrightarrow S$ ...
- 4,111
1
vote
0
answers
199
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Class number of the cyclotomic tower
Let ${\Bbb Q}(\zeta_{\infty})$ be the field obtained by adjoining all roots of unity. We define
Cl(${\Bbb Q}(\zeta_{\infty})$)$\colon= \underset{m > 1}{\varinjlim}~{\mathrm{Cl}}({\Bbb Z}[\zeta_m])...
- 781
6
votes
1
answer
696
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Selmer Group versus Selmer Variety
For an abelian variety $A$, the $p$-adic Selmer group is defined to be the subset of $H^1(G_k,H_1^{et}(A;\mathbb{Q}_{p}))$ whose restriction to $G_{k_v}$ is in the image of $A(k_v)$ for all places $v$ ...
- 37k
13
votes
4
answers
3k
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When is the Galois representation on the étale cohomology unramified/Hodge-Tate/de Rham/crystalline/semistable?
Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$?
I guess this is true if $...
- 15.4k
1
vote
0
answers
179
views
Spivakovski-Popescu-Neron desingularisation
For $A \colon= {\Bbb F}_p[[X_1,...,X_d]]$, by generalising Popescu-Neron's method, Spivakovski proved that $A$ is written by smooth sub-algebras. That is,
$A \cong \underset{\lambda \in \Lambda}{\...
- 781
1
vote
0
answers
485
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On the coherence of formal power series ring
Let $A = {\Bbb F}_p[[X_1,X_2,...]]$
be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$
$A$ consists of such formal sum elements as $\sum c_{e_1,.....
- 781
1
vote
0
answers
201
views
Number of minimal primes for UFD
Let $R$ be a UFD which is NOT noetherian. It is well-known that $R$
is a Krull ring. Let $I$ be an ideal of $R$ such that the height of $I$
is $d$ which is finite.
Question. Is the number of minimal ...
- 19.6k
13
votes
1
answer
974
views
Which degree does a motivic Galois representation show up in?
Consider a representation $\rho: \operatorname{Gal} (\overline{\mathbb Q} | \mathbb Q ) \to GL_n ( \overline{\mathbb Q}_\ell)$ that is a subrepresentation of $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...
CommunityBot
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45
votes
2
answers
3k
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What are the possible motivic Galois groups over $\mathbb Q$?
Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...
- 3,064
5
votes
0
answers
278
views
Tate's conjecture and symmetry of Hodge-Tate weights
I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf
At page 10 he claims that an indirect ...
- 231
7
votes
0
answers
267
views
Invariant obstructions to gluing Galois representations on elliptic curves
Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point.
...
- 148k
5
votes
0
answers
585
views
Bloch Kato Exponential as formal lie group exponential
Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism
$$DR(V) \...
- 3,555
5
votes
1
answer
755
views
Elliptic curve and Galois representation
For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by
$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = {\...
- 5,019
3
votes
1
answer
437
views
Galois representation attached to $3$-torsion points of an elliptic curve
Let
$ E $ - Elliptic curve defined over $ {\mathbb{Q}} $.
$G_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) $ of $\mathbb{Q}$.
$ E[3] $ - $3$-torsion points ...
- 193
10
votes
0
answers
466
views
Galois action on $E_n$-operads
Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...
CommunityBot
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2
votes
1
answer
338
views
Finite Flat Group Schemes for Modular Forms of Higher Weight
Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from an abelian variety, ...
- 26k
0
votes
0
answers
243
views
Uniqueness of decomposition of completely reducible representations
Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation which is semi-...
CommunityBot
- 1
10
votes
2
answers
961
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What is a(n algebro-geometric) family of modular forms?
We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to be smooth, proper, ...
- 37k
14
votes
1
answer
5k
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Why are Galois Representations so important in Number theory ?
Dear everyone,
Motivation :
From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have ...
6
votes
0
answers
453
views
dimensions of Galois representations appearing in the cohomology
Let's take $D$ to be $n^2$-dimensional central algebra with an involution $*$ over a CM field $E=FE_0$ (where F is totally real, and $E_0$ is imaginary quadratic). Define
$$G(R) = \{ x \in D \otimes _{...
5
votes
1
answer
1k
views
Eichler-Shimura for Shimura curves
Hi,
What is the statement of the Eichler-Shimura relation for Shimura curves? And where
can one find a proof?
Thanks
- 406
36
votes
1
answer
9k
views
Fontaine-Mazur for GL_1
For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...
- 21.3k
8
votes
1
answer
1k
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Learning about Galois representations
My goal was to learn about l-adic representations on some example — I'm a newbie in these topics.
Thus take pt = Spec F_q, ...
- 15.1k