All Questions
23 questions
3
votes
1
answer
407
views
Twists of elliptic curves
I have a few questions regarding twists of elliptic curves.
In the context of the Shafarevich group, I see people refer to the group of twists of an elliptic curve $E/\mathbb{Q}$ by $H^1(\mathbb{Q}, ...
- 2,907
5
votes
0
answers
234
views
Triviality of $\unicode{1064}(T_pE ⊗ T_pE)$ for elliptic curves and Bogomolov's lemma
Consider the case of an elliptic curve $E$ over Q, and let $S$ be a finite set of primes including all places of bad reduction and a place $p$ of good reduction.
Bogomolov's Lemma says that when $p$ ...
- 2,907
2
votes
0
answers
149
views
Absolute Bloch-Kato Cohomology
The étale cohomology $R\Gamma_{\mathrm{ét}}(X;\mathbb{Z}_p(n))$ of a scheme $X/K$ can be computed by a Hochschild-Serre spectral sequence with terms of the form $H^i(K;H^j(X_{\overline{K}};\mathbb{Z}...
- 15.4k
1
vote
0
answers
128
views
Representability of twists of projective schemes
Let $K$ be a perfect field, and let $S$ be a projective $K$-scheme. Denote by $\text{Twist}(S/K)$ the set of twists of $S$ up to $K$-isomorphism. These are (apriori) sheaves $\mathcal{F}$ on the ...
- 2,907
1
vote
0
answers
140
views
Kernel of restriction map in Galois cohomology
Let $S$ be the algebraic group $SL_2/\mathbb{Q}_p$ with a $G=G_{\mathbb{Q}}$ action, (acts by conjugation with a representation $\rho: G\longrightarrow GL_2$.)
Let $G_p$ be the decomposition group at ...
- 2,907
1
vote
1
answer
198
views
Crystalline fibre of a morphism of Galois cohomology stacks
Let $K = \mathbb{Q}_p$, $G = G_K$ its absolute Galois group. Let
$$1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$$
be a split exact sequence of (not necessarily abelian) group ...
- 2,907
2
votes
1
answer
333
views
Equivalence between twists of a curve and torsors of its automorphism group
Let $X$ be a curve defined over a number field $K$, and let $G_K$ be the absolute Galois group of $K$. Let $\text{Aut}(X)$ be the group of $\overline{K}$-defined automorphisms of $X$, and consider the ...
- 2,907
2
votes
0
answers
107
views
Extensions of groups with a $G$-action
Let $1\longrightarrow A\longrightarrow \mathcal{G}\longrightarrow R\longrightarrow 1$ be an exact sequence of algebraic group schemes, with $\mathcal{G}$ being an extension of $R$, an affine reductive ...
- 2,907
2
votes
1
answer
410
views
Galois cohomology of Tate modules
Let $E,E'$ be a pair of elliptic curves defined over $\mathbb{Z}$. Let $T_p[E], T_p[E']$ be their associated ($p$-adic) Tate modules. These are Galois representations for the absolute Galois group of $...
- 2,907
1
vote
0
answers
182
views
Crystalline exact sequence in Galois cohomology
Let $G$ be the absolute Galois group of $\mathbb{Q}_p$, and let $1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$ be a short exact sequence of (non-abelian) algebraic group ...
- 2,907
3
votes
1
answer
225
views
Deformations of Galois cohomology
Let $M = (\mathbb{Z}_p)^2$ be a Galois representation, with Galois action given by $\rho: G\longrightarrow SL_2(\mathbb{Z}_p)$. I am trying to understand how sensitive the Galois cohomology group $H^1(...
- 2,907
4
votes
0
answers
259
views
Galois cohomology of the Serre group in the proof of the fundamental theorem of CM
I am working through J.S. Milne's note on the fundamental theorem of complex multiplication over $\mathbb{Q}$. Let $E$ be a CM-field Galois over $\mathbb{Q}$, and $S^E$ the Serre group corresponding ...
- 570
2
votes
0
answers
281
views
Galois cohomology of cyclotomic extension
Let $K$ be a complete discrete valuation ring with algebraically closed residue field $F$ of characteristic $p > 0$. Suppose ${\Bbb Q}_p \subset K$ and the absolute ramification index v$_{\pi_K}(p) ...
- 781
2
votes
1
answer
466
views
Do $PGL_n$-torsors induce elements of the Brauer group
Let $K$ be a field and let $n\geq 2$. If $n=2$, then the set of $K$-isomorphism classes of $PGL_n$-torsors is in bijection with the $n$-torsion of the Brauer group of $K$.
Is this only for $n=2$?
Is ...
- 23
6
votes
1
answer
695
views
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 $v$ ...
- 15.4k
0
votes
0
answers
283
views
Normalizer of non-split tori
Let $\mathbb{G}$ be a connected reductive group over $\mathbb{C}$. Let $G:=\mathbb{G}(\mathbb{C}(\!(t)\!))$. Let $T$ be a maximal torus in $G$.
Question: What do we know about the normalizer $N_G(T)$...
- 2,751
13
votes
2
answers
1k
views
Galois cohomologies of an elliptic curve
I asked this question on Mathematics Stack Exchange but did not get any answer and I was suggested to post the question here.
I am studying basic theory of elliptic curves. I encountered Galois ...
user46846
5
votes
1
answer
310
views
Twists of projective automorphisms
Let $X$ be a projective variety over a perfect field $k$. Recall that a twist of $X$ is a variety $Y$ over $k$ such that $$X_{\bar k} \cong Y_{\bar k}.$$
The twists of $X$ are classified by the Galois ...
- 21.1k
9
votes
1
answer
781
views
Variant of Hilbert 90 for Galois extensions
Let $K/\mathbb F_q(x)$ be a finite Galois extension with Galois group $G$. Let $Aut(K)$ be the group of $\mathbb F_q$-automorphisms of $K$.
Obviously, $G\subseteq Aut(K)$. It is well known that
$H^1(G,...
- 3,996
8
votes
2
answers
900
views
Forms of algebraic varieties
Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...
- 7,722
4
votes
1
answer
265
views
What is the interpretation of this galois cohomology set?
Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$
The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are ...
- 605
12
votes
1
answer
706
views
Is there a canonical height on the Weil-Chatelet group?
Jon Hanke and I were just chatting and realized we didn't know the answer to the following question. If E is an elliptic curve over a number field, is there in any sense a "canonical height" on the ...
- 19.2k
1
vote
1
answer
811
views
Is the direct limit of Weil restriction of an elliptic curve a scheme?
In a discussion today on the Shafarevich-Tate group of an elliptic curve, the following structure and question came up. I will abuse many notations and be very vague about some things, but am very ...
- 4,593