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Jan
21
awarded  Nice Answer
Dec
24
comment $\zeta$ function for ambiguous class group
It seems to me this is just the Riemann zeta function at $[K:\mathbb{Q}]s$ with finitely many factors modified, no ?
Dec
21
comment Can the pre-image of the real points in the complex upper-half plane of a modular elliptic curve under the modular parametrization be identified?
Coordinates on the affine model over $\mathbb{Q}$ are $j(\tau)$ and $j(N\tau)$. These are power-series in $q=e^{2\pi i \tau}$ with real coefficients. Now $\bar{q} = e^{2\pi i(-\bar\tau)}$ confirms that complex conjugation is the reflection on the imaginary axis.
Dec
21
comment Can the pre-image of the real points in the complex upper-half plane of a modular elliptic curve under the modular parametrization be identified?
I guess you answer what the preimage of $X_0(N)(\mathbb{R})$ in the upper half plane looks like. Surely the imaginary axis belongs to it as complex conjugation on the curve comes from the reflection through the imaginary axis $\tau\mapsto -\bar\tau$.
Nov
25
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 ?
Nov
25
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 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?
added 108 characters in body
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