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Results tagged with galois-cohomology
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user 11926
9
votes
Accepted
A cup product in Galois cohomology of Elliptic curve
One can use the exact sequence
$$ 0 \to E(K)/mE(K) \to H^1(K,E[m]) \to H^1(K,E)[m] \to 0 $$
to define a pairing
$$ E(K)/mE(K) \times H^1(K,E)[m] \to H^2(K,\mu_m) $$
by taking $(Q,\xi)$ to $\phi(Q'\cup …
- 45.3k
11
votes
Proof of $V\cong \overline{K} \otimes_{K} V_K$ using $H^1(G_{\overline{K}/K},\operatorname{G...
Here's a sketch of the proof. I encourage you to fill in the details yourself. The definition of $V_K$ is $V_K=H^0(G_{\overline K/K},V)$. The key part of the proof is to show that $V$ has a $\overline …
- 45.3k
5
votes
Forms of algebraic varieties
In case you are interested in seeing a situation where the map is not surjective, consider the set of rational maps of degree $f:\mathbb{P}^1\to\mathbb{P}^1$, modulo the conjugation action of $\phi\in …
- 45.3k
4
votes
Accepted
Triviality of torsors after a field extension of bounded degree
This is not even true for elliptic curves over $\mathbb Q$ or $\mathbb Q_p$. For example, by Tate duality the group $H^1(\mathbb Q_p,E)$ is dual to $E(\mathbb Q_p)$, which is an infinite group. This s …
- 45.3k
4
votes
Accepted
A question on the cohomology of elliptic curves over local fields
By the Kummer sequence, your kernel is isomorphic to $E(K_\nu)/p^nE(K_\nu)$ via the connecting homomorphism. If you also assume that your elliptic curve has good reduction ath $\nu$, then the cocycle …
- 45.3k
3
votes
Accepted
Notation for the restriction map in Galois cohomology
I've seen both $\text{Res}_k^K$ and $\text{Res}_{K/k}$, and similarly for the inflation map if $K/k$ is Galois. More generally, if $\Gamma$ is a group acting on a group $G$ and if $\Lambda\subseteq\Ga …
- 45.3k
15
votes
Galois cohomologies of an elliptic curve
$H^2$ appears quite prominently in the duality theory of elliptic curves. For a local field such as a finite extension of $\mathbb Q_p$, one has $H^2(\text{Gal}(\bar K/K),\bar K^*)=\mathbb Q/\mathbb Z …
- 45.3k