# Questions tagged [elliptic-curves]

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### What is an absolutely irreducible smooth curve and how to determine if a curve is as such over a finite field F_q?

I'm new to this cite but I will try my best to convey my question in the best way possible. I was learning about Hasse-Weil Bound from Terence Tao's website and I had many questions from the beginning....
1 vote
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### Analog of chord-and-tangent for elliptic curves in 3 or higher dimensions

For an elliptic curve defined by a Weierstrass equation (or any degree-3 polynomial equation) in $\mathbb{P}^2$, the group law has a geometric interpretation by the chord-and-tangent method. Suppose ...
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1 vote
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### reference for Lemma 1.5 of Fermat's dream book

I am looking for another reference to the following Lemma in the elliptic curve chapter 1 of the book "Number Theory 1: Fermat’s Dream" by Kato, Kurokawa, Saito. The proof was left out as ...
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### Torsion subgroup of abelian variety

While studying elliptic curves I came over the question, whether for an elliptic curve $E$ and an integer $n$, we have $$(E[n])^k = E^k[n],$$ where $E^k$ is the $k$-...
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1 vote
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### Why does the continuous homomorphism group equal the homomorphism group?

I have the following question on Silverman's book The Arithmetic of Elliptic Curves, 2nd edition. First, let $K$ be a perfect field, $G_{\overline{K}/K}$ be the absolute Galois group of $K$, and $M$ ...
1 vote
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### Order of Picard groups of non-hyperelliptic algebraic curves

Let $q$ be prime. When $E/\overline{\mathbb{F}_q}$ is an elliptic curve, it is well-known that the group of $\mathbb{F}_q$-points of $E$ is isomorphic to the Picard group of degree $0$ divisors on $E$ ...
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### Uniformization Theorem and Non Existence of Family of Elliptic Curves over Riemann Sphere

A question concerning a statement from these notes introducing/motivating period map. On first page, left column, last sentence states: Since $\Bbb{H}$ (=complex upper half plane) is biholomorphic to ...
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1 vote
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### If the analytic rank is one then the sign in the functional equation is -1?

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $L(E, s)$ denote its $L$-function over $\mathbf{Q}$. Also $f$ denotes the weight two cusp form associated to $E$, but this shouldn't be ...
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### Reference Request: Studying elliptic curves in algebraic geometry

I am an undergraduate student who has studied some algebraic geometry. I want to read about elliptic curves from an algebraic-geometric perspective. I have read Hartshorne Chapter 1, and I know a ...
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1 vote
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### Why is $E$ an elliptic curve, and how to calculate its $j$-invariant?

I am reading an example of Springer fiber written by Bernstein and Kazhdan. Let $V$ and $V'$ be two $\mathbb C$-vector spaces, both of dimension $3$. Let $\lambda_1,\lambda_2,\lambda_3 \in \mathbb C$ ...
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### Why restricted product $\prod'$ is $\varinjlim_{S\subset I \text{ runs finite subset of} I} (\prod_{i\in S} X_{i}\times \prod_{v\in I-S}Y_i)$

This is a question related to this page. https://ncatlab.org/nlab/show/restricted+product . Let $I$ be a directed set. Let $X_i(i\in I)$ be a group. Let $\prod'_{i\in I}(X_i,Y_i)$ be a restricted ...
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### Can associativity of elliptic curve group law be verified by hand?

On p.15 of Silverman's Rational Points on Elliptic Curves, after sketching the proof of the associativity law of point addition on an elliptic curve $y^2=x^3+ax^2+bx+c$ via Bezout's theorem, the ...
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### How should the conjecture associated with Taniyama, Shimura, etc. be referred to?

I'm writing an article in which I need to refer to the Modularity theorem before it was a theorem. I always knew it as the first on the list below, but I'm aware there are alternatives, so I need to ...
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1 vote
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### Degree of an elliptic function

I'm attending a course about elliptic functions (over complex) The professor defined the degree of an elliptic function as the number of poles (counted with multiplicity) in the fundamental domain (...
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### Degree of $[\rho]^2-[\rho]$ with $\rho^3=1$

I have the elliptic curve $E:y^2=x^3+B$ where $B\in K^\times$ and $K$ is a field of characteristic distinct than $2,3$. I have the map $\mu:=[\rho]^2-[\rho]$ and I want to compute its degree. My ...
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### Chords and Tangents on Elliptic Curves Over Arbitrary Fields

I've been reading some books (e.g. Definition 3 in this chapter on elliptic curve cryptography or Group Law Algorithm 2.3 from The Arithmetic of Elliptic Curves) which define the group operation on ...
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### Are these elliptic curves over $\mathbb{Q}(\sqrt{2})$ isogenous?
I have the following elliptic curves: $$E:y^2=x^3-\sqrt{2}x \quad \quad E':y^2=x^3-2x$$ over $\mathbb{Q}(\sqrt{2})$. I want to determine whether they are isogenous. I have a few strategies to do this ...