Tagged Questions

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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 ...
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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 ...
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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 $C$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 $K$ ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 $$E_1: y^2+y=x^3+x^2âˆ’9xâˆ’15 \enspace [Cremona:19A1]$$ Let $\varphi$ be the $3$-isogeny from $E_3$ to $E_1$. I want ...
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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 ...
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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 ...
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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?
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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 ...
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Elliptic curves on a K3 surface

Let $X$ be an elliptic K3 surface. Let $\alpha$ be a smooth curve of genus $\geq3$. Define $$d(\alpha)=\min\lbrace \epsilon\cdot \alpha \ | \ \epsilon \mbox{ is an elliptic curve on } X \rbrace,$$ ...
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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 ...
Let E be the elliptic curve $y^2 + y = x^3$ over $F_2$. Prove #E($F_{2^n})$$= \left\{ \begin{array}{ll} 2^n+1 & \quad n=odd \\ 2^n+1-2(-2)^{n/2} & \quad ... 2answers 110 views Moduli space of isogeny classes of elliptic curves The modular curve Y(1) classifies isomorphism classes of elliptic curves, namely its K-points for any field \mathbb Q\subseteq K\subseteq \mathbb C correspond via the j-invariant to \mathbb ... 1answer 138 views How to visualize projective plane I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ... 1answer 36 views Why is the answer set limited here? This question is based on pp 67 - 68 of Ash and Gross's "Elliptic Tales". Here the authors discuss points on a curve in the projective plane. We have an equation f(x,y) = x^2+y^2 We can ... 0answers 26 views Reduction of isogenies at bad primes Let L be a number field and E,E' two elliptic curves defined over L. Suppose \varphi\colon E\to E' is an isogeny defined over L. Let \mathfrak p be a prime of bad reduction for E,E'. ... 1answer 28 views How to determine the group structure of E(\mathbb{R}) for an elliptic curve E/\mathbb{R} Using Weierstrass' \wp function it can be proved that the group of complex points on an elliptic curve E /\mathbb{C}: y^2 = x^3 + ax + b satisfies E(\mathbb{C}) \cong \mathbb{R}/\mathbb{Z} \oplus ... 0answers 29 views Inflection points on elliptic curves over a field of characteristic 2 I'm looking at the elliptic curve C:={\cal Z}(XY^2+ZX^2+YZ^2) in the field k:=\overline{\mathbb{F}_2}. I want to prove that this curve has 9 inflection points. Since the characteristic of k is ... 1answer 71 views A special cubic curve How can I transfer following cubic curve to a Weierstrass normal form?$$2x^2y+4xy^2+2y^3-2axy-ay^2+a=0,$$where$a$is a fixed rational number. 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 ... 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 ... 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 ... 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 ...
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 ...