# Questions tagged [elliptic-curves]

Elliptic curves are objects of algebraic geometry met in somewhat advanced parts of number theory. They also appear in applications to cryptography. Use the tag, if this applies. Questions on ellipses should be tagged [conic-sections] instead.

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### Elliptic curves with complex multiplication by an order $\mathcal{O}$

I am reading a paper about elliptic curves with CM, and since I'm new to this I'm having troubles understanding the essence of some things. Let $E$ be an elliptic curve with complex multiplication by ...
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### Homogenous space of elliptic curve E/$\Bbb Q$

Homogenous space of elliptic curve E/$\Bbb Q$ always has $\Bbb Qp$ rational points for every prime $p$? And why? I know homegenous space has $\Bbb Q$ rational points only if it is not a trivial class. ...
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### Elliptic curve: Type of reduction mod 2, how can I show the curve has a cusp?

I want to know what type of reduction the curve $E : y^2 = x^3 + 7x$ has at $p=2$. From online search, I obtain that it has additive/cuspidal reduction. But this disagrees with my own computation, ...
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### Do the all points in subgroup of an elliptic curve with prime order have the same order? [closed]

A subgroup $G$ of elliptic curve can constructed with point $P$ with order $q$ by $G=\langle P\rangle$. Now, if $q$ is prime, do the all points in subgroup $G$ (except infinity point) have same order ...
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### Tate-Shafarevich group and Hasse principle

I' m looking for a proof of the fact that the Hasse local-global principle holds for an homogeneous space of elliptic curve $E$ defined over $\Bbb Q$ if and only if the Tate-Shafarevich group of $E$ ...
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### Elliptic Curve over $\mathbb Z_p$ isomorphic to a subgroup of that Elliptic Curve over reals?

Is every Elliptic Curve group over finite field $\mathbb F_p$ isomorphic to some subgroup of the Elliptic Curve group over the reals having the same equation? Update: this is at last sometime the ...
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### Computing the genus of $y^2=x(x^2-1)$ using 1-forms

I'm trying to compute the genus of the projective curve $C:=V(Y^2Z-X(X^2-Z^2))\subset\Bbb{P}^2_\Bbb{C}$ explicitly using differential forms. I know beforehand that this is an elliptic curve, so the ...
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### Smallest solutions to show that a number is congruent

On wikipedia, under the 'smallest solutions' section, there is a table of congruent numbers, where for each congruent number $n$ they put a triple $(a, b, c)$ with $a^2 + b^2 = c^2$ and $n = ab/2$. It ...
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### Naive question about computational cost of doubling and points addition over elliptic curve

When we talk about the computational cost of doubling and points addition we usually give a result in terms of field multiplication M and field squaring S. For example, here we can read points ...
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### Group Structure of Rational Points a Supersingular Elliptic Curve over $\mathbb{F}_{p^2}$ with $p=2^a3^b-1$?

I'm looking at this Jao-De Feo-Plut paper and this expository paper by Craig Costello. Both claim that for a prime $p$ of the form $p=2^a3^b-1$, the entire $(p+1)$-torsion of a supersingular elliptic ...
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### Elliptic Curve Check [closed]

I am working with a 256 bit elliptic curve. I have data which contains the following: Scalar integer, (x,y) pair. I have been able to check the scalar vs the total number of points (N) and the Y-...
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### Doubly periodic meromorphic function with prescribed poles and zeros

The field of the meromorphic functions on a complex torus $\mathbb{C} \mathbin{/} \Lambda$ is $\mathbb{C}(\wp, \wp')$, where $\wp$ is the weierstrass p-function to the lattice $\Lambda$. Furthermore, ...
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### Is there an analogous of quadratic residue over elliptic curves?

I've faced this question some days ago looking at this paper Kleptography: Using Cryptography Against Cryptography In this article, the writers discuss about a kleptographic attack against Diffie-...
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### Rational functions on elliptic curves

Recall that an elliptic curve over a field $k$ i.e a proper smooth connected curve of genus $1$ equipped with a distinguished $k$-rational point, I'll be really grateful for any help in understanding ...
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### Proofs of abelian group axioms for elliptic curve over finite fields + tangent-chord group law

I'm looking for the proofs of the abelian group axioms for elliptic curve over finite fields (e.g. integers mod p) with the tangent-chord group law (i.e. the "standard" group law for ...
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### Defining the action of elliptic curve $E$ on a principal homogeneous space of $E$

I recently learnt (using Milne's book about elliptic curves) about principal homogeneous space for elliptic curves. For completeness I state the definition here: Let $E$ be an elliptic curve over a ...
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### Elliptic curve confusion

I am confused by some lecture I heard. It was stated that the elliptic curve over a finite field $\mathbb{F}_{q^k}$ would be a subset of the elliptic curve over the finite field $\mathbb{F}_{q}$. But ...
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### Elliptic curve over finite non-prime fields

I have a problem understanding elliptic curves over finite non-prime fields, i.e., $E / {\Bbb F}_{p^k}$, where $p$ is prime and $k > 1$. Let's say we have a defining Weierstrass equation of the ...
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### Understanding the compactification of elliptic curves

I'm having trouble understanding the following example from my lecture notes: Example: Let $E$ be an elliptic curve over a field $k$ given by the compactification of the affine equation $y^2=x^3+ax+b$...
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### Function field for projective variety

Let $V$ be a projective variety over a field $k$. For any affine patch we can define the function field of $V$ to be $K(V)=K(V\cap\mathbb{A}^n)$, and these are all canonically isomorphic. In Silverman'...
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$L-$function of elliptic curves is Dirichlet series and defined to be $$L(E,s) = \sum_{n\ge 1}\frac{a_n}{n^s} = \prod_p L_p(E,s),$$where the Euler factor at $p$ is $$L_p(E,s) = \begin{cases}(1-a_pp^... 1answer 56 views ### Determinant of Frobenius action on the exterior power on cohomology Given an elliptic curve E, the Frobenius action \mathrm{Frob}, and \mathrm{det}(1-\mathrm{Frob}_E T | H^1(E)), how do we find an expression for:$$\mathrm{det}(1-\mathrm{Frob}_X T | \wedge^2 (H^...
So I'm reading Milne's book on elliptic curves, which is freely available here. On page 38, he defines a regular map: Let $k$ be a perfect field. A regular map $\varphi : C_{g_1} \to C_{g_2}$ of ...