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Can the Selmer curve be viewed as an elliptic curve?

Yes, for example you can adjoin an algebraic element $\alpha$ which is a root of $3X^3 + 4$ to $\mathbb Q$ to get a number field $k$, upon which this curve obtains a rational point $P = [\alpha:1:0]$. ...
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Kummer sequence for "affine" elliptic curves

Multiplication by $n$ is never regular on $E-P_1,\ldots,P_r$ (unless $n=0$ or $n$ has trivial kernel, ie. $n=\pm 1$ or $char(k)=p$, $n=p^l$ and $E$ supersingular) because $\{ Q\in E, \exists m, [n^m]Q ...
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2 votes
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Question From Serre's "On the 2-Dimensional Modular Representations of $G(\overline{\mathbb Q}/\mathbb Q)$"

As pointed out in the comments, if the representation $G_{\Bbb Q} \to \mathrm{Aut}(E[p])$ is reducible, then there is some subgroup $X \subset E[p] \subset E(\overline{\Bbb Q})$ of order $p$ (1-...
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Isogenous Elliptic Curves and Reduction Type

The local factor of the $L$-function is defined as it is because of the Tate module: it is defined to be $$\det(1-p^{-s}\mathrm{Frob}_p | V_\ell^{I_p})^{-1},$$ where $V_\ell(E)^{I_p}$ is the subspace ...
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2 votes
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What is the relationship between these two versions of BSD?

The $L$-function reformulation of BSD was given by Goldfeld here: Goldfeld, Dorian. Sur les produits partiels eulériens attachés aux courbes elliptiques. (French) [On the partial Euler products ...
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Cardinality of elliptic curve using Hasse bound

The Hasse bound with given element with order $7$ indicates that one have either order 7 or 14. 14 means we have an element of order 2, i.e. $P + P = \mathcal{O}$ that is $P = - P$ and this implies $P....
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Torsion subgroup of elliptic curve

We can apply the strategy used in this post, which uses the following proposition (Proposition VII.3.1(b) of Silverman's The Arithmetic of Elliptic Curves.) Here $K$ is a local field with residue ...
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2 votes

The lattice $\displaystyle \Lambda $ associated to the pendulum equation is rectangular

For the first part, let $$v = -u-E/3$$ so that $$v'^2 =v^3-v(1+E^2/3)+2E/3-2 E^3/27 $$ which corresponds to the elliptic curve $$C:y^2=x^3-Ax-B, \qquad A = 1+\frac{E^2}3,\quad B=\frac{2E}3- \frac{2E^3}...
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1 vote
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curves on abelian surface $E\times E$, part II

There is an extra condition of $E$ in Kollar's book. Namely, the elliptic curve $E$ is very general in the moduli. Otherwise, for example, if $E=\mathbb C/\mathbb Z+\omega_3\mathbb Z$, where $\omega_3=...
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1 vote
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curves on abelian surface $E\times E$

Since the intersection form is bilinear. This reduces the knowing the intersection of any pair out of the 3 curves and there self intersections. First the self-intersections. Since $E_{1,0}$ and $E_{0,...
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Characterization of Isogenous Elliptic Curves over Finite Fields

Yes, this is true but not obvious! It was proved by John Tate (1966) in his paper "Endomorphisms of Abelian Varieties over Finite Fields", p. 139. You can find the proof also in the ...
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Graphical representation of quadratic forms in 2 or 3 dimentions

The quadratic form is actually $ a_{11} x^2 + a_{22} y^2 + a_{33} z^2 + 2 a_{12} x y + 2 a_{13} x z + 2 a_{23} yz $ and can be written compactly as $ p^T A p $ where $ p = [x, y, z]^T $ and $ A = \...
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Divisors: function composition with multiplication by n map on elliptic curve

The fact that the zeroes and poles of $f \circ [n]$ have the same orders as those of $f$ follows from the facts that: (i) ramification index is multiplicative with respect to composition; and (ii) $[n]...
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1 vote

Why weierstrass equation should not be used for elliptic curves over the field char=2?

Actually, the problem is the discriminant. By definition, an elliptic curve must have non-zero discriminant, otherwise it is degenerate. A clear explanation of this with examples of finite fields by ...
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1 vote
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Reductions of Isogenous Elliptic Curves

Let $f : E \to E'$ be an isogeny over $\mathbb{Q}$ and let $p$ be a prime of good reduction for $E, E'$. One way to see that the reductions (of global minimal integral Weierstrass models) $\tilde{E}, \...
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