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

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4
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3answers
138 views

Example of an elliptic curve with trivial torsion subgroup and rank 0

What is an example of an elliptic curve over $\mathbb{Q}$ with trivial torsion subgroup and rank 0?
0
votes
0answers
112 views

Some basic questions about Jacobians of curves

Let $C$ be a curve defined over $\mathbb{Q}$, of positive genus. Let $J$ denote its Jacobian. I would like to ask a couple of basic (I presume) questions: 0) Why is $J$ an algebraic variety? 1) For ...
0
votes
1answer
47 views

Elliptic curve and restriction

Let $E$ be an elliptic curve. Let $\xi$ be a class of $H^{1}(\mathbb{Q}, E[m])$ unramified at the prime $\ell$. Then $\xi$ restricted to $H^{1}(I_{\ell}, E[m])$ where $I_{\ell}\subset ...
4
votes
1answer
152 views

Mathematics for Pleasure of a Beginner

I've just read "The Music of the Primes" by Marcus du Sautoy, it is worth a read. I'm not from a maths background, but I'd like to develop a deeper understanding of the concepts. The poetry of math is ...
1
vote
0answers
69 views

Do points on elliptic curves exist where the denominators of point multiples grows more slowly than normal?

Looking at prime multiples of $P=[1,1]$ on the curve $y^2=x^3+x-1$ the size of the denominator grows quite rapidly. So ...
2
votes
1answer
116 views

Given a real number, how do I produce an elliptic curve with j-invariant equal to that number?

I have formula for computing the j-invariant but I was wondering if given number $j$, is there a formula for getting a curve $y^2=x^3+a_2x^2+a_4x+a_6$ with j-invariant j?
2
votes
1answer
82 views

$H^{0}$ cohomology group and elliptic curve

Let $E$ be an elliptic curve with good reduction at $\ell$. Is it possible that one can have $H^{0}(\mathbb{Q}_{\ell}, E[\ell]) = E[\ell]$?
3
votes
1answer
104 views

projective cubic curve to complex projectie space

Suppose we are given the equation $$ y^2z = x(x - z)(x - 2z) $$ I would like to define a degree two map $g$ on this curve into complex projective space. I hate to say I am already lost here - how do I ...
10
votes
2answers
130 views

Solve : $ab(a+b)(a-b)=c^2-1$

As we know that $ab(a+b)(a-b)=c^2$ has no integer solution in $Z^+$.However, it seems that $$ab(a+b)(a-b)=c^2-1$$ has infinite positive integer solutions,could you prove it? Here are some of them: ...
3
votes
0answers
142 views

Good source of problems for Knapp's Elliptic Curves?

I'm studying elliptic curves (and eventually modular forms) out of Knapp's book because of the softer algebraic geometry prereqs. It's incredibly accessible but the problem is that I don't know I can ...
9
votes
2answers
257 views

Intuitively, what is the height of a point on an abelian variety?

I have been reading through Silverman's classic text on elliptic curves and I just can't seem to wrap my head around the height functions. It just kind of shows up. What exactly does the height ...
2
votes
0answers
219 views

A strong form of the Nagell-Lutz theorem

The motivation of this question can be found in Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order? Given the elliptic curve: $$C:y²=x³+ax+b$$ ...
1
vote
0answers
56 views

Is it possible to say that there is a curve $C$ such that its rank exactly $r$?

The motivation of this question can be found in Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order? Given the elliptic curve: $$C:y²=x³+ax+b$$ ...
3
votes
1answer
120 views

Finding number of solutions to an equation in $\mathbb F_p$

$p=3 \pmod 4$ is a prime. Let $b\in \mathbb F_p^*$. Show that the equation $v^2=u^4-4b$ has $p-1$ solutions $(u,v)$ with $u, v \in \mathbb F_p$. If we write the given equation as $v+u^2=x$ and ...
2
votes
1answer
272 views

Elliptic curve as an intersection of quadrics

Let $E$ be an elliptic curve. If one starts with embedding associated with invertible sheaf $\mathcal{O}(3x)$ where $x$ is some point on $E$ then one gets cubic in $\mathbb{P}^2$ and this embedding is ...
0
votes
0answers
68 views

Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not?

The motivation to this question can be found in How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$ We know that there is an isomorphism ...
4
votes
1answer
81 views

How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$

Let $C$ be an 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 (is the ...
1
vote
0answers
52 views

Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order?

Let $C$ an elliptic curve over $\mathbb Q$. Assume that the rank of $C(ℚ)$ is equal to $r$. Then the cardinality of a maximal independent set in $C(ℚ)$ is $r$, thus there exists $r$ independent points ...
0
votes
1answer
68 views

What is the point $\{∞\}$?

The set of all rational points in an elliptic curve $C$ over $ℚ$ is denoted by $C(ℚ)$ and called the Mordell-Weil group, i.e., $C(ℚ)=\{\text{points on } $C$ \text{ with coordinates in } ℚ\}∪\{∞\}$. ...
6
votes
0answers
142 views

What is the complex *algebraic* moduli of elliptic curves?

It's well-known that $SL_2(\mathbb{Z}) \backslash \mathfrak{h}$ is a coarse moduli space for complex elliptic curves. Thus, I would expect this to be related to the pullback of $\mathcal{M}_{ell} ...
3
votes
1answer
141 views

Stark's formula for the j-invariant

In his paper On the "gap" in Heegner's proof (which you can find : here) Stark gives the following formula for the $j$-invariant (for some $\tau \in \mathcal{H}$ and $q=e^{2i\pi\tau}$) $$ j(\tau) = ...
3
votes
1answer
236 views

Inverse Scalar Multiplication of a point over elliptic curve

I was implementing point arithmetic operation, and was exploring the properties of point arithmetic, and I am unable to conclude whether $$ k^{-1}(kP) = P $$ where P is a point over elliptic curve $ ...
7
votes
3answers
318 views

Diophantine equation $x^2 + 32x = y^3$

I am trying to find all solutions to the Diophantine equation $x^2 + 32x = y^3$. I think that the first step is to factorise: $x(x+32)=y^3$. If $x$ is odd, then $x+32$ is also odd. The common ...
10
votes
3answers
447 views

Integer solutions for $x^3+2=y^2$?

I've heard a famous result that $26$ is the only integer, such that $26-1=25$ is a square number and $26+1=27$ is a cubic number.In other words, $(x,y)=(5,3)$ is the only solution for $x^2+2=y^3$. ...
2
votes
0answers
143 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
2
votes
0answers
81 views

Two quartic polynomials to be made a square?

Given two generally non-square quartic polynomials that are to be simultaneously made squares for particular values of $x$, $$c_1x^4+c_2x^3+c_3x^2+c_4x+c_5 = y_1^2$$ ...
1
vote
1answer
55 views

What is the upper bound of order of $ E(F_{599})$

Can you please help me to slove this problem: Let E be the elliptic curve $y^{2}=x^{3}+1$ over the finite field $F_{599}$. Using Hasse's theorem find What is the upper bound of order of $ E(F_{599})$ ...
4
votes
2answers
217 views

Rank of the elliptic curve $y^2=x^3+px$

I need to prove that the rank of the curve $y^2=x^3+px$ is $0$, if $p\equiv 7 \pmod {16}$ is a prime. Using the standard technique, we need to show that none of the following two equations admits an ...
7
votes
1answer
102 views

Why is $H^1(\text{Gal}(\overline{K}/K),A)=\lim_{\rightarrow}H^1\left(L/K, A^{\text{Gal}(\overline{K}/L)}\right)$

I'm doing a course on elliptic curves, and I'm stuck on this. Here are the definitions: Let $G$ be a group and let $A$ be a $G$-module (that is, a $\mathbb{Z}[G]$-module). We have the "cochains" ...
4
votes
1answer
147 views

Elliptic curve question

Let $P$ be a point on an elliptic curve over $\mathbb{R}$. Give a geometric condition that is equivalent to P being a point of order (a) $2$ , (b) $3 $ , (c) $ 4$ . Could someone explain this to ...
4
votes
3answers
270 views

The rational points on the curve: $y^2=ax^4+bx^2+c$.

I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$. Is there infinite rational points on this curve? For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
4
votes
1answer
557 views

Computing the divisors of a meromorphic function defined by a hyperelliptic curve.

Let $X$ be a hyperelliptic curve defined by $y^2=h(x).$ Let $\pi : X\to \mathbb{P}^1$ be the double covering map sending $(x,y)$ to $x$. Let $\omega=\pi^{*}(dx/h(x)).$ Compute div$(\omega)$. I ...
1
vote
1answer
2k views

Is it possible to compute order of a point over Elliptic curve?

In the elliptic Curve cryptography, it is said that the order of base point should be a prime number, and order of a point $P$ is defined as $k$, where $kP = \mathcal{O}$. And to compute the order we ...
12
votes
2answers
222 views

Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube

Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube. I found $x=0$, any others ?
2
votes
2answers
347 views

order of elliptic curve $y^2 = x^3 - x$ defined over $F_p$, where $p \equiv 3 \mod{4}$

It is said that the elliptic curve $y^2 = x^3 - x$ defined over a prime field $\mathbb{F}_p$, where $p \equiv 3 \mod{4}$ has an order $p + 1$. When I tried to get the elements of $E = \{(x,y) \in ...
0
votes
1answer
70 views

Find lift($E_{p^2}$) of an elliptic curve $E_p$ defined in field $F_p$ where $p$ is a prime

How to find $E_{p^2}$ of an elliptic curve $E_p$ defined over finite field $F_p$ where $p$ is a prime number?
1
vote
3answers
110 views

Combine two given Elliptic Curves

I want to combine two Elliptic curves such $E_p$ (defined in the field $F_p$) and $E_q$ (defined in the field $F_q$) i.e to find $E_n$ where $n=pq$. Is there any method to do it?
3
votes
1answer
109 views

Torsors under elliptic curves splitting over the same fields

I have a question somewhat related to my last question. Suppose $C$ and $C'$ are two genus $1$ curves (smooth, projective, geom conn.) over a perfect field $k$ with no $k$-rational points and that $C$ ...
8
votes
1answer
130 views

Proof in Kummer Theory - why is this subgroup finite?

I'm doing a course on elliptic curves. We're working with a field $K$ with $\mu_n \subset K$ ($K$ contains all $n$th roots of unity) and $\mbox{char}(K)\not|\;n$. I'm trying to understand the proof ...
2
votes
1answer
78 views

Testing to see if $\ell$ is of split or nonsplit multiplicative reduction

Suppose an elliptic curve $E/\mathbb{Q}$ has multiplicative reduction at $\ell$. Are there any other ways of seeing if $\ell$ is of split or nonsplit reduction aside from computing ...
10
votes
1answer
115 views

If $E/\mathbf Q$ is an elliptic curve and $n$ is odd, then the $n$-torsion $E(\mathbf Q)[n]$ is cyclic; elementary proof?

I know that this follows from the existence and non-degeneracy of the Weil pairing. A consequence of the existence of the Weil pairing is that, if the whole $n$-torsion is defined over $\mathbf Q$, ...
35
votes
3answers
757 views

The resemblance between Mordell's theorem and Dirichlet's unit theorem

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other ...
2
votes
0answers
103 views

Examples of non-minimal Weierstrass equations that make $[E(K):E_0(K)]$ arbitrarily large.

I'm taking a course on elliptic curves, in which the lecturer briefly mentioned that the Tamagawa number $c_K(E)=[E(K):E_0(K)]$ satisfies $c_K(E)=\mbox{ord}(\Delta)$ or $c_K(E) \leq 4$. He said that ...
6
votes
2answers
495 views

Turning an elliptic curve over C into a complex torus

I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would ...
7
votes
1answer
117 views

Short exact sequence of modules and elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve with a 3-torsion point $P$. Let $E_{d}$ denote the quadratic twist of $E$ by $d$. Then the action of $\sigma \in ...
1
vote
1answer
226 views

The Group of points on the Elliptic curve $y^2=x^3+1$ over $\mathbb{F}_5$

So I'm trying to understand the group of points of $y^2=x^3+1$ over $\mathbb{F}_5$ and for some reason I seem to be getting nonsense answers and I'm not sure what I'm doing wrong. So basically my ...
2
votes
1answer
44 views

Using the power series expansion for $w(t)$ to construct a subgroup of the rational points on an elliptic curve.

I'm doing a course on elliptic curves, and I'm stuck on a line in a proof which is supposedly using the uniqueness in Hensel's lemma. Starting with an elliptic curve ...
4
votes
1answer
236 views

Proving that the differential on an elliptic curve $E$ given by $\omega=\frac{dx}{y}$ is translation invariant

I'm taking a course on elliptic curves and I'm stuck on a line in a proof. We're assuming we're in an algebraically closed field $K$ and char($K)\not=2$. We have our elliptic curve ...
5
votes
1answer
136 views

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
1
vote
1answer
95 views

Like Diophantine equation

The equation $x^n - ny^x-nxy$ = $0$ has solution set $(n, x, y) = (1, 1, \frac12), (2, 1, \frac14), (3, 1, \frac16), \ldots$ I would like to know/learn the following (Kindly discuss) 1) If we ...