For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.
0
votes
0answers
15 views
About fibers of an elliptic fibration.
Consider the pencil of cubics
$\lambda C_1+ \mu C_2$ where $C_1=y^2z$ and $C_2=x(x^2+2xz+z^2)$ and the elliptic fibration $\tilde X \rightarrow \mathbb P^1$ induced by the blow-up of $\mathbb P^2$ ...
6
votes
0answers
45 views
Surfaces ruled over elliptic curves
Ground field $\Bbb{C}$. Algebraic category.
Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve.
Suppose $E$ is an elliptic ...
1
vote
0answers
25 views
What is the exact mathematical formulation of a claim
The motivation to this question can be found in
http://mathoverflow.net/questions/103846/why-are-galois-representations-so-important-in-number-theory
My question is concerned with this sentence: ...
0
votes
0answers
31 views
Why is [-2,0] not listed as a generator of the elliptic curve 389a1
The elliptic curve $y^2 + y = x^3 + x^2 - 2*x$ clearly has $x=-2, y=0$ as a point and its multiples are
$[-2, 0]$
$[39, -247]$
$[-2869/1681, 52850/68921]$
$[2319450/243049, -3736242957/119823157]$
...
5
votes
0answers
44 views
Abelian Elliptic Surfaces
By abelian surface we mean a 2-dimensional algebraic complex torus. Thus
$$ S=\Bbb{C}^2/\Gamma$$
where $\Gamma$ is a rank $4$ lattice in $\Bbb{C}^2$ and such that $S$ is algebraic. It has trivial ...
1
vote
0answers
37 views
Question about paper on Selmer groups
Let $\textrm{Sel}_{n}E$ denote the $n$-Selmer group and $\textrm{Sel}_{p^{\infty}}E = \varinjlim_{n}\textrm{Sel}_{p^{n}}E$.
Proposition 5.10 of this paper http://arxiv.org/abs/1304.3971 states that
...
1
vote
1answer
34 views
Modularity theorem and some results
Let $C$ be an elliptic curve over rationals. Then we can attach to $C$ an L-series $L(C,s)$. I read about the Modularity theorem
http://en.wikipedia.org/wiki/Modularity_theorem
In the section ...
2
votes
1answer
68 views
Infinity of Right Triangle Elliptic Curve
Translate the congruent number problem into elliptic curve, we conclude that an integer $n\in\mathbb Z^+$ is area of a right triangle with $a,b,c\in\mathbb{Q}$ if and only if the corresponding ...
4
votes
1answer
63 views
Computing cohomology groups of elliptic curves
I'm skimming through Silverman's text to recall some theory of elliptic curves that I've learned in undergrad. In practice however, I'm having trouble actually computing the cohomology groups. For ...
2
votes
1answer
58 views
Fields where all smooth projective curves of genus $1$ are elliptic curves
A common definition of an elliptic curve over a field $k$, is that it is a smooth projective curve of genus $1$ (defined over $k$) with a distinguished $k$-rational point. The distinguished point is ...
5
votes
1answer
32 views
deg functions and maps
For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman.
For algebraic functions on elliptic ...
1
vote
0answers
22 views
deg of composition on supersingular curve
Let we have supersingular curve $E(\bar{\mathbb{F}_q})$.
Let we have algebraic function $f \in \bar{\mathbb{F}_q}(E)$ with div($f) = \sum_{i=0}^{i=n}n_iP_i$.
Then div$(f) \circ [q] = ...
2
votes
0answers
31 views
number of solutions eqauation on supersingular elliptic curve
To Frobenius endomorphism on supersingular elliptic curve
I want to prove that equation $\pi_q(X) = A$ has 1 solution for any point
$A \in E(\bar{\mathbb{F}_q}))$ where $E$ is supersingular.
Is it ...
1
vote
0answers
34 views
Frobenius endomorphism on supersingular elliptic curve
Let we have supersingular curve $E(\bar{\mathbb{F}_q})$.
Is it true that for every point $P$ $q$-Frobenius endomorphism $\pi_q$ can be write as $A + [q]B$ where $P = A + B$? It is true if ...
2
votes
0answers
58 views
Pole of differential
Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$.
We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
3
votes
1answer
113 views
Find all integer solutions to $x^2+4=y^3$.
Find all integer solutions to $x^2+4=y^3$.
Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
3
votes
1answer
32 views
Question about quadratic twists of elliptic curves
Let $E$ be an elliptic curve and $d$ be a squarefree integer. If $E'$ and $E$ are isomorphic over $\mathbb{Q}(\sqrt{d})$, must $E'$ be a quadratic twist of $E$?
3
votes
2answers
67 views
Isomorphism of Elliptic Curves:
In Stinson's Cryptography Theory and Practice, a theorem is given without proof:
Theorem 6.1 Let $E$ be an elliptic curve defined over $Z_p$, where $p$ is prime and $p > 3$. Then there exist ...
3
votes
1answer
44 views
Explicitly computing finite subgroups on elliptic curves
I have a simple cubic curve, say
$y^2 = x^3 - x.$
Is there a simple way to find a small finite subgroup of points lying on this curve? (with respect to the elliptic curve group law.)
Otherwise, does ...
1
vote
0answers
33 views
Multiples of a point in a non-elliptic curve
Let $E:y^2-xy+y-x^3=0$ over a field $K$, $P=(0,0)$. If $\text{char}(K)\neq2$, $E$ is an elliptic curve for which I can easily get $n*P$. Now, something goes wrong with $\text{char}(K)=2$: Basically, I ...
-1
votes
0answers
34 views
elliptic curve in complex projective space is homeomorphic to real torus?
I need an answer to the following problem, but have been told to post on here instead of thinking about it!
Prove that if $\lambda\neq0,1,\infty$ then ...
1
vote
2answers
46 views
Relation between Galois representation and rational $p$-torsion
Let $E$ be an elliptic curve over $\mathbb{Q}$. Does the image of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under the mod $p$ Galois representation tell us whether or not $E$ has rational ...
6
votes
3answers
100 views
The genus of a curve with a group structure
I'm reading Milne's Elliptic Curves and came across this statement: If a nonsingular projective curve has a group structure defined by polynomial maps, then it has genus 1.
In this question a similar ...
5
votes
2answers
68 views
Other ways to compute the torsion subgroup of elliptic curves
Suppose I have a family of elliptic curves $E_{n}/\mathbb{Q}$. I would like to determine the torsion subgroup of $E_{n}(\mathbb{Q})$ denoted by $E_{n}(\mathbb{Q})_{\textrm{tors}}$. Two ways to do this ...
1
vote
2answers
68 views
How do you determine if an elliptic curve over a finite field is cyclic?
I know the group order and the points of the elliptic curve $y^2 = x^3 + Ax + B$, but I am confused on how to determine if they from a cyclic group
The curve $y^2 = x^3 + 2x +2$ in $\Bbb F_{11}$ ...
7
votes
2answers
75 views
Rank $2$ Elliptic Curves
I'm on a quest for some rank $2$ elliptic curves. My question is actually twofold:
Is there a way to easily construct a curve with this property?
Is there a database of elliptic curves with given ...
5
votes
2answers
50 views
Property of the $p^n$-Selmer group
Consider the $p^n$-Selmer group of an elliptic curve $E/\mathbb{Q}$, $\operatorname{Sel}_{p^n}(E)$. Must we always have $\operatorname{Sel}_{p^n}(E) \cong (\mathbb{Z}/p^n\mathbb{Z})^{s}$ for some $s$? ...
1
vote
0answers
69 views
On the elliptic curve $x^4+y^4 =193z^2$
Given the simultaneous Diophantine equations,
$$u^2+v^2=w^2\tag{1}$$
$$x^4+y^4 = (u^6+v^6)t^2\tag{2}$$
the only solutions seem to be for the first Pythagorean triple $u,v,w = 3,4,5$ which yield the ...
2
votes
0answers
54 views
Every smooth cubic curve has a flex point
I want to show that every smooth irreducible plane cubic $C$ has a flex point, i.e. a point $P$ with $i_P(C, T_C(P)) = 3)$ (where $T_C(P)$ is the tangent to $C$ at $P$). I know how to do this in ...
4
votes
3answers
61 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
30 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
40 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 ...
5
votes
1answer
93 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
51 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
28 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
68 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
59 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 ...
8
votes
2answers
93 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
37 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 ...
8
votes
2answers
128 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 ...
1
vote
0answers
48 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
38 views
Is it possible to say that there is a cuvre $C$ such that its rank exactely $r$?
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
76 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 ...
0
votes
0answers
17 views
Number of solutions to a conic in $Z_p$
$p\not=2$ is a prime. $a, b, c, d\in F_p$ and $acd\not =0$. C is the conic given by the homogeneous equation
$ax^2+bxy+cy^2=dz^2$.
If $b^2\not =4ac$, prove that #$C(F_p)=p+1$
Note: Solutions are to ...
2
votes
1answer
58 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
48 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 ...
3
votes
1answer
51 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 ...
0
votes
0answers
39 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
60 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\}∪\{∞\}$.
...
3
votes
0answers
88 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} ...
