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2d
comment Fermat's last theorem over larger fields
Could a Russian speaker add a summary of the results in this paper to this answer, please ?
2d
answered On a minimal algebraic number field which satisfies the principal ideal theorem
Nov
22
awarded  Nice Question
Nov
20
comment Is elliptic curve point division defined over the field of real numbers?
If you are looking for "procedures" to compute it for general fields, then look at sage's function P.division_points(n). It starts by finding roots of the division polynomial in the field.
Nov
20
comment elliptic curves and tower of finite fields
What do you mean by "reduce"? To me it seems only when $m$ divides $n$, there is a natural map, the trace map, which sums the Galois conjugate points.
Nov
20
comment How to explicitly compute lifting of points from an elliptic curve to a modular curve?
If increasing the conductor is computationally hard, maybe you could enlarge the base field to get to rank 2 examples. Find a small conductor curve with large rank over a small quadratic field.
Nov
17
comment Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)
The Galois group of the equation for $p=3/5$ and $n=7$ is $S_7$, which is not soluble. So don't expect a solution by radicals for $n=7$.
Nov
12
comment For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
Yes that is possible for $p=5$. It turns out to be equivalent to be a quadratic twist by 5 of a curve with a rational 5-torsion point.
Nov
11
revised For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
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Nov
11
comment For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
You are right. More generally, If $G$ is the group of all matrices of the form $(\begin{smallmatrix} 1 & * \\ 0 & * \end{smallmatrix})$, then $H^1(G,E[p])= \mathbb{F}_p$ if $p =3$ and it is zero if $p>3$. Your example, and many other curves with a 3-torsion point rational over $\mathbb{Q}$, has indeed this group.
Nov
11
revised For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
deleted 4 characters in body
Nov
11
answered For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
Nov
11
comment For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
I don't know of an example with $p>2$. Maybe the group theorists can tell us examples of $H^1(H, V)\neq 0$ for subgroups $H$ of $\operatorname{SL}_2(\mathbb{F}_p)$ with $V$ the 2-dimensional vector space over $\mathbb{F}_p$. Say with $p=3$ or $p=5$.
Nov
11
comment For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
Over $\mathbb{Q}$, the determinant $G\to \mathbb{F}_p^{\times}$ must be surjective. So your $G$ won't appear as a group for an elliptic curve over $\mathbb{Q}$.
Nov
9
comment A number theoretic identity
it is true at least up to $2n+1<1000$.
Nov
9
awarded  Enlightened
Nov
9
awarded  Nice Answer
Oct
24
comment $j$-invariants of elliptic curves over finite fields
For the second, look up "twists" in chapter X of Silverman's book. Together with the knowledge of what the endomorphism ring is yo ushould get the answer.
Oct
16
comment Is the Cassels-Tate pairing defined for elliptic curves over function fields?
There are though known problems with the proof that the kernels are the maximal divisible subgroup. See Harari-Szamuely and Gonzalez-Aviles for the corrections, both in Crelle 632 in 2009 with a title containing "arithmetic duality theorems".
Sep
19
comment What are all positive integers n for which the congruence $a^{n+1} \equiv a (mod n)$ holds?
Moreover this is a duplicate from mathoverflow.net/questions/37097/…