bio | website | |
---|---|---|
location | Nottingham, UK | |
age | ||
visits | member for | 4 years, 11 months |
seen | 13 hours ago | |
stats | profile views | 1,424 |
Mar 11 |
comment |
integral basis for the Lie algebra of the Neron model of an abelian variety
Lie$(A^{\vee})$ is defined to be the $O_K$-dual of the differentials on the Néron model. So if that is not free, neither will Lie$(A^{\vee})$. But a basis of Lie${}_K(A^{\vee})$ is all that you need. |
Mar 10 |
answered | integral basis for the Lie algebra of the Neron model of an abelian variety |
Mar 3 |
comment |
If $E(K)=E(L)$ for an elliptic curve $E$ and an algebraic extension $L/K$, what can we say about $Sel(E/K), Sel(E/L), L/K$?
I assume that Sel$(E/K)$ is the group that fits into a short exact sequence between $E(K)$ and Sha$(E/K)$. Otherwise you need to fix your $n$ in $n$-Selmer group (or your isogeny). So your question is equivalent to asking about the growth of the Tate-Shafarevich group in the tower. Basically anything can happen. Iwasaw theory gives examples of exploding growth as well as frequent stabilisation. |
Feb 13 |
comment |
Is it possible the division polynomials evaluated at fixed point to be perfect powers unbounded number of times?
Sure, sorry, if $x(P)$ and $y(P)$ are integers and the equation has integer coefficients then $a_n$ is an integer. - Non-square-free, yes, that is very easy, but perfect squares or even perfect powers are hard. But why do you want them perfect powers ? |
Feb 13 |
comment |
Is it possible the division polynomials evaluated at fixed point to be perfect powers unbounded number of times?
For a given $(E,P)$ it will only finitely many times give an integer, I would think. Why are you interested in this question ? It seems hard that one can say anything interesting about it other than that it is very unlikely that there could be infinitely many perfect powers in any such sequence. |
Feb 12 |
comment |
Is it possible the division polynomials evaluated at fixed point to be perfect powers unbounded number of times?
So $a_n$ is a rational number. |
Feb 12 |
comment |
Is it possible the division polynomials evaluated at fixed point to be perfect powers unbounded number of times?
Do you assume $E/\mathbb{Q}$ ? Do you assume $P$ to have integer coordinates in a given Weierstrass equation ? Do you assume that $P$ has good reduction at all primes ? |
Feb 8 |
comment |
Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling?
No. But you can place your cube such that it appears as two unit squares and the cube of twice the volume appears as a larger square with sides $\sqrt[3]{2}$. So in the two planes of descriptive geometry you face the same problem of constructing that number, no ? |
Feb 8 |
comment |
Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling?
In descriptive geometry it is quite clear that doubling the cube is the same as constructing $\sqrt[3]{2}$. But it may well be that your students did not see any descriptive geometry. |
Feb 5 |
comment |
Heegner points on elliptic curves
I might be wrong, but my guess is that we do not expect anything particular to happen when the elliptic curve you map the Heegner points to has complex multiplication. |
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 $n=7$. |