For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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4
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1answer
104 views

motivation for talking about torsion points on an elliptic curve

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points ...
1
vote
1answer
39 views

Find coordinate $y$ of an elliptic curve point

If I have an elliptic curve over a finite filed $F_p$ ($p$ is prime) defined as $$ y^2 \equiv x^3 + ax + b\pmod p,$$ such that $4a^2 + 27b^2 \neq 0$ and suppose I have only given the coordinate $x$, ...
1
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0answers
21 views

moduli of lattices

Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$. Does M identify in some way with Siegel upper half space $H_g$? Let's say a lattice has CM if it has ...
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0answers
34 views

Function field of a projective variety

I am reading Silverman's "The Arithmetic of Elliptic Curves". On page 10 he defines the function field of a projective variety $V$ over a field $K$ to be the function field of $V\cap\mathbb{A}^n$, ...
1
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1answer
43 views

Lattices and Elliptic curves and number fields

Let $K$ be a number field with ring of integers $O_K$. If $K$ is totally real, then $O_K$ is a lattice in $\mathbb R$. If $K$ is imaginary quadratic, then $O_K$ is a lattice in $\mathbb C$. If $K$ ...
3
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0answers
70 views

Finding the prime element of a place of a function field of a elliptic curve.

Let $p$ be an elliptic curve in $\mathbb{C}[X,Y] $. Consider the quotient ring $A = \mathbb{C}[X,Y]/(p) $ and its field of fractions $F = frac(A) $. For all $f + (p) \in A$, define $deg_A(f + (p))= ...
3
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0answers
30 views

does every elliptic curve E/S have infinitely many sections after passing to an etale extension of S?

Let E/S be an elliptic curve, where S is any scheme. Must there exist a scheme $S'$, etale and surjective over $S$, such that the pullback $E' := E\times_S S'$ has infinitely (or even > 1) many ...
11
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2answers
197 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
3
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1answer
51 views

Reduction map on torsion of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at a prime $p$. It is well-known that the map $$E[N]\to E_p[N]$$ is injective when $p\nmid N$. It is even a bijection since both ...
0
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0answers
17 views

What is an embedding degree of elliptic curve?

I am dealing with MOV algorithm to transform ECDLP to DLP in $GF(p^k)$, but at the first step I have to determine embedding degree k. I have read the definitions of embedding degree, but still I am ...
1
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1answer
36 views

Silverman AEC Corollary 6.4

Quick question about Chapter 3 Corollary 6.4 [p. 86] in Silverman's Arithmetic of Elliptic Curves. I feel like I'm misreading it and would like clarification. He claims that for an elliptic curve E ...
3
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1answer
85 views

$L$-function of an elliptic curve and isomorphism class

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We have a $L$-function $$L(E,s)$$ built from the local parameters $a_p(E)$. If two elliptic curves are isomorphic, they clearly have the same ...
2
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1answer
70 views

History of the Coefficients of Elliptic Curves — Why $a_6$? [duplicate]

I would like to know what is the motivation behind the naming convention of the Weierstrass form of elliptic curves given as $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ I can see that $a_1,a_2,a_3,a_4$ ...
0
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2answers
42 views

Derivative of Integral of (g) with g in the limit

I would like to evaluate the following: $$\frac{\partial }{\partial \beta }\int _0^{\cos ^{-1}(\beta )}\text{dx} \sqrt{\beta +\cos (x)}$$ given that $0\leq\beta\leq1$ basically I'd like to find ...
4
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1answer
111 views

Group law for an elliptic curve using schemes

I was trying to understand better the definition of the group law for an elliptic curve given in Katz and Mazur's book ...
1
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1answer
61 views

associativity on elliptic curves — Milne's proof

In the proof that the group law on an Elliptic curve is associative, Milne (http://www.jmilne.org/math/Books/ectext5.pdf, page 28) sets up 3 cubics, and claims that they all contain the $8$ points ...
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0answers
36 views

The cardinality of the preimage of a point under a nonzero isogeny equals the separable degree of the isogeny

Let $f:E_1\rightarrow E_2$ be a nonzero isogeny between elliptic curves. Take a point $Q \in E_2$. I am looking for a reference to a proof, or a proof, of the following fact: ...
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0answers
119 views

Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$. Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
0
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0answers
43 views

the torsion subgroup of E(Q) (eliptic curves)

if $E$ is an elliptic curve over $Q$, then why $E(Q)_{\rm tor}$ is group and finite set ?
5
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1answer
67 views

Exact rank of Elkies curve

A naïve question. We definitely know an elliptic curve of rank $28$ or more exists by Elkies but no one knows exactly what the rank is for this curve (and for similar examples given previously). ...
7
votes
1answer
61 views

Torsion on $y^2=x^3+d$

A question that I am stuck on is: prove that the $\mathbb{Q}$-torsion subgroup of the elliptic curve $y^2=x^3+d$ has order dividing 6. Any hints on how to start would be nice. I tried saying ...
6
votes
1answer
75 views

Intersection of two quadrics

How to understand (maybe, informally) why the intersection of two quadrics in general position in $\mathbb{CP}^3$ is an elliptic curve? It is obvious that it is a compact 2-manifold, i.e. a sphere ...
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0answers
79 views

Intuition behind using projective geometry for defining the addition on an elliptic curve

We already had the chord-and-tangent construction that can be used to define a way of "adding" points on an elliptic curve. Also this addition satisfies all the group laws. Still why one needs to ...
3
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1answer
69 views

Endomorphism Ring of an Elliptic Curve over Finite Field

Let $~E:y^2=x^3+x~$ be an elliptic curve over finite field $\mathbb{F}_{5},$ I compute the trace of Frobenius is $2$($E/\mathbb{F}_{5}$ obvious is ordinary). (By the theory of CM, I know (when $E$ ...
0
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1answer
53 views

Existence of certain homogenous forms

Let $D(X,Y), E(X,Y)\in\mathbb{Z}[X,Y]$ forms of the same degree $n$ and suppose that the resultant $R=Res(D,E)$ of $D$ and $E$ is not $0$. Show that there are homogenous forms $L_0(X,Y),M_0(X,Y), ...
1
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2answers
57 views

Sage usage to calculate a cardinality

I would like to compute the cardinality of an elliptic curve group over the finite field $\mathbb{F}_{991}$. I'm trying to use sage but I still have an error in the syntax (I never used it before and ...
1
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1answer
64 views

Selmer and Shafarevich-Tate Groups

I'm currently trying to under the Selmer and Shafarevich-Tate Groups from Silverman's Arithmetic of Elliptic Curves (2nd edition), pg. 331 onwards. I have a couple of questions I think is derived from ...
3
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1answer
115 views

Why Elliptic Curves have so many nice properties

As the definition referred from Silverman's book: An elliptic curve is a pair $(E,O)$, where $E$ is a nonsingular curve of genus one and $O\in E$. (We generally denote the elliptic curve by $E$, the ...
1
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2answers
66 views

When Frobenius map equal to multiplication-by-m map

Here is a homework, the result brought me some trouble. Let $p = 7$, and consider the finite field ${ \mathbb{F}}_{p^{2}}$ , which we may represent explicitly as $${ \mathbb{F}}_{p^{2}}\simeq ...
0
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1answer
65 views

an Example of Elliptic Curve over finite field has no CM

I have known this property (from Silverman's The Arithmetic of Elliptic Curves): Let $\operatorname{char}(K)=p>0,$ and let $E/K$ be an elliptic curve with $j(E)~ \overline{\in}~ \overline{ ...
1
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1answer
32 views

Elliptic Curves Nagell-Lutz question

Let $y^2 = x^3 + ax + b$ be an elliptic curve defi ned over $\mathbb{Z}$. If $b=a^2$, find a point of infinite order on $\mathcal{E}(\mathbb{Q})$. The previous part of the question implies that I ...
4
votes
2answers
84 views

How does $\text{Gal}(L/K)$ act on the automorphism group of an elliptic curve?

Let $L/K$ be a finite Galois extension of number fields; I'm interested mainly in the case $K = \Bbb{Q}$ and $L= \Bbb{Q}(\sqrt{d})$. Let $X$ be an elliptic curve over $K$ and $\text{Aut}(X_L)$ the ...
7
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0answers
78 views

Motivation for Weil pairing

The Weil pairing $$e_\phi:E[\phi]\times E'[\hat{\phi}]\to \mu_n$$ for an elliptic curve is defined as follows. Let $\phi:E\to E'$ be an isogeny of degree $n$ and $\hat\phi:E'\to E$ be the dual ...
3
votes
2answers
67 views

The torsion of an elliptic curve over a finite field

There is a result for $p$ prime, $E$ an elliptic curve over $\mathbb F_p$, then $E(\overline{\mathbb{F}_p})[m]\cong (\mathbb{Z}/m\mathbb{Z})^2$ for $m \nmid p$. The book on cryptography I am using ...
2
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1answer
57 views

elliptic curve group law

Let $C$ be an elliptic curve over a field $k \supset \mathbb{Q}$. Then given $P$ and $Q$, we can draw the line between $P$ and $Q$ (call this line $L$) and then "find the third intersection point", ...
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0answers
45 views

How to find the subgroups of the group $C(ℚ)$?

Let $C$ be a fixed elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $$C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$$ where $C(ℚ)^\mathrm{tors}$ is a finite abelian group ...
0
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1answer
72 views

Public Key Scheme decryption. [closed]

You have been sent a message based on the following Public Key Scheme. 1) Bob chooses two large primes $\ p,q $ with $ p \equiv q \equiv 2 \pmod 3$ and computes $ n=pq. $ 2) Bob chooses integers $ e,d ...
0
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2answers
65 views

Elliptic Curves and “roots”

Given elliptic curve $\omega$ in $\mathbb{R}^2$ such that $y^2 = x^3 + ax + b$, how can you find how many solutions (and what they are) of $x^3+ax+b$ have a $y$ value of $0$; or as they call it, a ...
4
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1answer
95 views

seeing the differential dx/y on an elliptic curve as an element of the sheaf of differentials

$\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\Spec}{\text{Spec }}$ It's a well known fact that every elliptic curve (say, over a field $k$) has a global holomorphic nowhere vanishing differential. If ...
2
votes
2answers
63 views

Does the conductor of an elliptic curve always divide the minimal discriminant?

Of course, the primes dividing the conductor are precisely those dividing the minimal discriminant. But I cannot find any source that addresses the possibility of a prime appearing to the first power ...
4
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1answer
70 views

Fundamental period of the WeierstrassP elliptic function?

Consider the WeierstrassP elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia when ...
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0answers
92 views

From inverse Weierstrass function to Jacobi elliptic/inverse elliptic functions?

As a conclusion to a previous question on integrals, I get an answer in terms of inverse Weierstrass elliptic function : $$ f\left(x\right)=\wp^{-1}\left( \beta + \frac{9\beta^2-1}{3(x-\beta)} \right) ...
4
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0answers
50 views

There are no elliptic curves over $\mathbb{F}_8$ satisfying either $\#E(\mathbb{F}_8)=7$ or $\#E(\mathbb{F}_8)=11$

This is taken from The Arithmetic of Elliptic Curves by Silverman on page 154, Q5.10(f). One way of directly solving this problem is to work out on sage all 8^5 possibilities of elliptic curves and ...
1
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1answer
32 views

$\forall p\geq 3, E:y^2=x^3+x$ satisfies $\#E(\mathbb{F}_p)=0\mod4$

The question is taking from Arithmetic of Elliptic Curves by Silverman, Q5.12 on page 154. I've managed to show the supersingular case when $p=3 \mod 4$, which was done more generally for elliptic ...
2
votes
1answer
43 views

What is the argument used to dsitinguish the cases (a) and (b)

We know from [B. Mazur, Modular curves and the Eisenstein ideal, Publ. math. IHES 47 (1977), 33-186] that if $C$ is an elliptic curve of the form ($C:y²=x³+ax+b$ with $a,b∈ℤ$), then $C(ℚ)^{tors}$ (the ...
2
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1answer
77 views

Question about divisors

Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
4
votes
1answer
233 views

Where does a CM elliptic curve have bad reduction?

Let $d>1$ be square-free, and $K=\mathbf Q(\sqrt{-d})$. Choose an embedding of $K$ in $\mathbf C$, and let $E = \mathbf C/\mathcal O_K$. It is known that $E$ admits a model over the Hilbert class ...
2
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1answer
50 views

An isogeny of elliptic curves induces a $\mathbb{Z}_l$-linear map

On page 89 of The Arithmetic of Elliptic Curves (second edition), Silverman says: Let $\phi:E_1\rightarrow E_2$ be an isogeny of elliptic curves. Then $\phi$ induces maps $\phi:E_1[l^n]\rightarrow ...
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2answers
91 views

Constructing a meromorphic function

I need help with the following problem. "Let $C : y^2 = x^3 − 5x^2 + 6x$ be a cubic curve with the standard group law. Find a meromorphic function on $C$ having the pole of order two at ...
5
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1answer
84 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...