2,456 reputation
716
bio website
location Nottingham, UK
age
visits member for 4 years
seen 19 hours ago

Apr
11
comment $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2
Theorem 11.3.11
Apr
11
comment $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2
$H^1(\mu_p)$ is not just the class group is also contains the $p$-units modulo $p$-th powers and they get a $\mu=1$. Should be some where in cohomology of number fields but I don't have it in front of me.
Mar
31
comment Kernel of a 3-isogeny between two elliptic curves
sage and probably magma have implemented the algorithm to find the dual. Often a look at the isogeny class (e.g. in Cremona's tables) is enough, say because there are only two curves there.
Mar
30
comment Kernel of a 3-isogeny between two elliptic curves
It is easy to check that $\ker \varphi =\mu_3$ by checking that the dual isogeny has a rational $3$-torsion point in its kernel.
Mar
30
awarded  Yearling
Mar
20
comment Ratio of periods for elliptic curves in an isogeny class
Sorry for the format, but I hope you see what code of functions you have to look up to see how it is done.
Mar
20
comment Ratio of periods for elliptic curves in an isogeny class
sage: E1 = EllipticCurve("19a1") sage: E2 = EllipticCurve("19a2") sage: phi = E1.isogeny(None, codomain=E2,degree=3) sage: phi.formal() sage: phihat = phi.dual() sage: phihat.formal() sage: E1.period_lattice().basis()[0]/E2.period_lattice().basis()[0]
Mar
19
comment Ratio of periods for elliptic curves in an isogeny class
I fear your question lacks precision, which makes it very difficult to reply to. My wild guess is the following: Let $E\to E'$ be an isogeny of elliptic curves defined over $\mathbb{Q}$. Then there is a positive real Néron period associated to both, obtained by integrating a Néron differential. These can be computed and the quotient of them is an interesting value associated to the isogeny - appearing for instance in the change of the $p$-adic $L$-function. The ratio can also be obtained by looking at the first coefficient of the formal expansion of the isogeny. Is that it ?
Mar
19
comment What is exceptional about the prime numbers 2 and 3?
Define an odd number $n>1$ to be prime when for all odd $d$ with $1<d\leq \sqrt{n}$ the remainder is non-zero. Then 5 and 7 become "exceptional", too.
Mar
15
answered Anomalous elliptic curves over finite rings
Mar
14
comment On $x^3-y^2=1728 \text{ unit}$ in number fields
$\mathcal{O}^*_K$ modulo $6$-th powers is a finite group.
Mar
14
comment On $x^3-y^2=1728 \text{ unit}$ in number fields
The equation $y^2 = x^3 -1726\,u$ for a fixed unit $u$ is an elliptic curve and so it has finitely many integral points by Siegel's theorem. So I read the question as asking to find solutions when $u$ is allowed to vary in all units.
Feb
2
awarded  Custodian
Feb
2
reviewed Approve suggested edit on Cartesian closed category
Jan
27
comment $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2
ok. i am sure you will find my email address.
Jan
26
comment $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2
$\mu_p$is the $p$-th roots of unity - as a Galois module. To get this look at the dual isogeny: it is made up by two isogenies of degree $3$ having a $\mathbb{Q}$-rational $3$-torsion point in the kernel.
Jan
24
answered $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2
Jan
23
comment $S$-Tate-Shafarevich groups of elliptic curves
Sure, sorry. I corrected this.
Jan
23
revised $S$-Tate-Shafarevich groups of elliptic curves
deleted 2 characters in body
Jan
23
answered $S$-Tate-Shafarevich groups of elliptic curves