bio | website | |
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location | Nottingham, UK | |
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visits | member for | 4 years |
seen | 19 hours ago | |
stats | profile views | 1,242 |
Apr 11 |
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$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2
Theorem 11.3.11 |
Apr 11 |
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$\mu$-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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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On $x^3-y^2=1728 \text{ unit}$ in number fields
$\mathcal{O}^*_K$ modulo $6$-th powers is a finite group. |
Mar 14 |
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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 |
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$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2
ok. i am sure you will find my email address. |
Jan 26 |
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$\mu$-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 |
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$S$-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 |