2
votes
0answers
31 views

Elliptic curves as $\mathbb{C}^*/\mathbb{Z}$

I apologize in advance if my question is rather trivial, but i have trouble understanding a basic fact about elliptic curves.. I have always wrote an elliptic curve $E$ as $\mathbb{C}/\Lambda$, where ...
0
votes
1answer
39 views

Compute principal divisor for a rational function on a curve

During the lecture we defined the principal divisor of a rational function on a smooth curve as it follows: Consider the smooth curve $C\subseteq\mathbb{P}^2$. Take $g\in{K(C)^*}$. Then the principal ...
0
votes
1answer
69 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
5
votes
1answer
57 views

The Moduli Stack of Elliptic curves - What is it?

I have often heard the words "Moduli Stack of Elliptic Curves", but I have nowhere found a from-scratch definition of this object. I do understand the motivation: There are cusps in the moduli space ...
3
votes
1answer
36 views

Can you have a nontrivial automorphism of an elliptic curve $E/S$ which when restricted to a geometric fiber is the identity?

Ie, let $E/S$ be an elliptic curve over some scheme $S$. Is it possible to have an automorphism $\alpha$ of $E$ over $S$ such that for some geometric point $s\in S$ its pullback to $E_s$ is the ...
1
vote
0answers
37 views

Flex point on an elliptic curve

I have just started working through Pete Clark's elliptic curve notes, which are available here: http://math.uga.edu/~pete/EllipticCurves.pdf Early on, in section 2.1 on page 6, it is shown that the ...
1
vote
1answer
59 views

Relating ramification index of a map of curves to degree of vanishing

I am little confused about explicitly computing ramification index and relating it to degree of vanishing a polynomial. In particular I have the following example (when trying to prove the genus ...
0
votes
0answers
53 views

How can we compute the order of 1-form on Riemann surfaces

Let X be a hyperellictic curve defined by $y^2=h(x)$. Let $\pi:X\rightarrow\mathbb{P}^1$ be the double covering map seding $(x,y)$ to $x$. Let $\omega=\pi^*(dx/h(x))$. How can we compute the orders of ...
1
vote
1answer
83 views

Number of points on an elliptic curve over $ \mathbb{F}_{q} $.

I have the following elliptic curve: $$ E: \quad Y^{2} = X^{3} + 1 ~ \text{over} ~ \mathbb{F}_{q}, ~ \text{where} ~ q \equiv 1 ~ (\text{mod} ~ 3). $$ I want to know the number of points on this curve. ...
1
vote
1answer
39 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 $K$ ...
2
votes
0answers
26 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 ...
10
votes
1answer
123 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 ...
4
votes
1answer
105 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
vote
0answers
35 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: ...
6
votes
0answers
111 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 ...
1
vote
0answers
73 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
votes
1answer
108 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 ...
0
votes
1answer
60 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{ ...
4
votes
2answers
81 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 ...
4
votes
1answer
78 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
1answer
71 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 ...
5
votes
1answer
76 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 $E$, ...
3
votes
1answer
47 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
4
votes
1answer
49 views

Confusion with computing kernel of an isogeny between two elliptic curves

Consider the two elliptic curves $$E_3: y^2+y=x^3+x^2+x \enspace [Cremona:19A3]$$ and $$E_1: y^2+y=x^3+x^2−9x−15 \enspace [Cremona:19A1]$$ Let $\varphi$ be the $3$-isogeny from $E_3$ to $E_1$. I want ...
9
votes
4answers
136 views

What are some applications of the Weil conjectures for algebraic curves?

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and ...
7
votes
2answers
70 views

How to show there exists no solution to a discrete logarithm problem on an Elliptic Curve?

The exact problem is to show that $\nexists$k such that $k(1,2) = (4,5)$ on the elliptic curve defined by $\widetilde{E}: y^2 = x^3 -14x + 17$ over $\mathbb Q$. Background: E: $y^2 = x^3 + 3$ over ...
4
votes
1answer
62 views

Automorphisms of elliptic curve

Consider an elliptic curve $y^2=x^3+b$ over $\mathbb{R}$. How to find all real automorphisms of this curve of order 3?
2
votes
1answer
72 views

Is $y^2 =$ quartic in $x$ smooth at infinity?

Let $q(x)\in K(x)$ be a quartic polynomial in x with distinct roots over the algebraically closed field $K$. Consider the curve $C\subset \Bbb P^2$ given by $y^2-q(x)$. Is $C$ smooth? Well, at least ...
4
votes
1answer
111 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
1
vote
0answers
60 views

Elliptic curves in $\Bbb P^3$

How can I check that a curve inside of $\Bbb P^3$ is an elliptic curve? Specifically, let $C$ be the plane cubic $$C:aX^3+bY^3+cZ^3=0$$ and $\phi:\Bbb P^2\to \Bbb P^3$ given by ...
4
votes
1answer
92 views

Computing the kernel of an isogeny between two elliptic curves

Consider the two rational elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650 $ $ [\text{Cremona}:35a2] $ $ E_{2}: y^{2}+y=x^{3}+x^{2}-x $ $ [\text{Cremona}:35a3] $ We know that ...
2
votes
1answer
124 views

Order of a point on an Elliptic Curve

I am currently struggling with the determination the order of a point on an elliptic curve. We had to do the following exercise: $C = V(y^2+x^3-1)$ and $P = (0,1)$. Now Wikipedia told me that I can ...
3
votes
1answer
74 views

Isogeny of an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime. Then what does it mean by "$E$ has a $\mathbb{Q}$-isogeny of degree $p$"?
6
votes
2answers
87 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
1
vote
1answer
36 views

How to prove $-P = (x, -a_1x - a_3 - y)$ for an Elliptic Curve of the General Weirstrass equation for P not the identity?

Let $P = (x,y) \ne \{\infty\}$. Then $-P$ is the other finite point of intersection of the curve and the vertical line through $P$. General Weirstrass equation: E: $a_1y^2+a_3xy+a_5y = ...
2
votes
1answer
67 views

Elliptic curves on a K3 surface

Let $X$ be an elliptic K3 surface. Let $\alpha$ be a smooth curve of genus $\geq3$. Define $$d(\alpha)=\min\lbrace \epsilon\cdot \alpha \ | \ \epsilon \mbox{ is an elliptic curve on } X \rbrace, $$ ...
3
votes
4answers
381 views

Two circles intersect in two points

Take for example two circles $$\begin{cases}x^2+y^2=1\\x^2+y^2-x-y=0\end{cases}$$ These two circles intersect in two points namely $(0,1)$ and $(1,0)$. But by Bezout's theorem they must intersect four ...
1
vote
1answer
80 views

E: $y^2+y=x^3$ an elliptic curve over $F_{2}$. How to prove the number of $E(F_{2^n})$ = $2^n+1$ if n is odd, …

Let E be the elliptic curve $y^2 + y = x^3$ over $F_2$. Prove $ $#E($F_{2^n})$$ = \left\{ \begin{array}{ll} 2^n+1 & \quad n=odd \\ 2^n+1-2(-2)^{n/2} & \quad ...
3
votes
2answers
110 views

Moduli space of isogeny classes of elliptic curves

The modular curve $Y(1)$ classifies isomorphism classes of elliptic curves, namely its $K$-points for any field $\mathbb Q\subseteq K\subseteq \mathbb C$ correspond via the $j$-invariant to $\mathbb ...
3
votes
1answer
137 views

How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ...
0
votes
1answer
36 views

Why is the answer set limited here?

This question is based on pp $67$ - $68$ of Ash and Gross's "Elliptic Tales". Here the authors discuss points on a curve in the projective plane. We have an equation $f(x,y) = x^2+y^2$ We can ...
1
vote
0answers
25 views

Reduction of isogenies at bad primes

Let $L$ be a number field and $E,E'$ two elliptic curves defined over $L$. Suppose $\varphi\colon E\to E'$ is an isogeny defined over $L$. Let $\mathfrak p$ be a prime of bad reduction for $E,E'$. ...
1
vote
1answer
28 views

How to determine the group structure of $E(\mathbb{R})$ for an elliptic curve $E/\mathbb{R}$

Using Weierstrass' $\wp$ function it can be proved that the group of complex points on an elliptic curve $E /\mathbb{C}: y^2 = x^3 + ax + b$ satisfies $E(\mathbb{C}) \cong \mathbb{R}/\mathbb{Z} \oplus ...
2
votes
0answers
29 views

Inflection points on elliptic curves over a field of characteristic 2

I'm looking at the elliptic curve $C:={\cal Z}(XY^2+ZX^2+YZ^2)$ in the field $k:=\overline{\mathbb{F}_2}$. I want to prove that this curve has 9 inflection points. Since the characteristic of $k$ is ...
1
vote
1answer
71 views

A special cubic curve

How can I transfer following cubic curve to a Weierstrass normal form? $$2x^2y+4xy^2+2y^3-2axy-ay^2+a=0,$$ where $a$ is a fixed rational number.
1
vote
2answers
76 views

Elliptic Curve and Conjugation

If I consider an elliptic curve $C$ as a Riemann surface cut out in $\mathbb{C}P^2$ by a homogenous cubic, and if that cubic is defined over $\mathbb{R}$, then I think we have a conjugation map ...
2
votes
1answer
96 views

Linear Equivalence of Divisors on Projective Plane Cubic

I'm self-studying Miranda's Algebraic Curves and Riemann Surfaces and am uncertain of how I'm supposed to solve problem V.2c on linearly equivalent divisors: Let $X$ be the projective plane cubic ...
3
votes
1answer
61 views

Embedding of elliptic curves into $\mathbb{P}^2$ by arbitrary line bundle of degree $3$

Let $E$ be a complex elliptic curve, with distinguished point $x_0 \in E$. Any divisor of degree three is equivalent to the divisor $D=x+2x_0$. If $x=x_0$, it is well known that $L(D)$ has an explicit ...
2
votes
0answers
52 views

Trivial divisor on elliptic curve

Suppose $E$ is an elliptic curve over $k$, and $(E,+)$ is an abelian group(suppose we fix some closed point as identity). Let $[p]$ denote the Weil divisor corresponding to the closed point $p \in ...
8
votes
0answers
142 views

Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...