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Questions tagged [elliptic-curves]

For questions about elliptic curves.

An elliptic curve is a smooth, nonsingular projective curve of genus 1 with a specified point $\mathcal{O}$, defined over any field $K$. They form abelian groups under point addition. They are much studied in number theory, for example in cryptography and integer factorization.

An elliptic curve can be defined by an equation of the form: $$E:y^2=x^3+ax+b$$ with the discriminant $\triangle_E=-16(4a^3+27b^2)\ne 0$ so the curve is nonsingular, i.e. its graph has no cusps or intersections.

The elliptic curves with $a=0$ are .

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Why is an elliptic curve over a finite field symmetric about $y=p/2$?

I was reading this tutorial about elliptic curves. There are some example curves with $p=19$, $97$, $127$, $487$ etc, and they are all symmetric about $y=p/2$. Why is it the case?
frt132
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When is an elliptic curve of $j$-invariant $j=0,1728$ supersingular over $\mathbb{F}_p$?

Sorry for my bad English. Let $p$ be a prime, $E$ be an elliptic curve over $\mathbb{F}_p$ of $j$-invariant is 0 or 1728. Now I want to know if there is a criterion of when $E$ is supersingular. In Wikipedia,there is a table for small $p$, but I…
Yos
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Points of order 3 and inflection points of elliptic curves

Why is it true that on an elliptic curve a point on it $P$ is of order 3 iff $P$ is an inflection point on the curve? I'm taking a course on elliptic curves and this is something that I see in proofs everywhere, but I can't seem to find a proof of…
djurgen
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Elliptic Curve has no $p$-torsion over $K_n$

I'm currently studying Gross' paper about modular elliptic curves. Lemma 4.3 is quite difficult for me to understand. It states that an elliptic $E$ has no $p$-torsion rational over $K_n$, the ring class field of order $n$. The only thing I seem to…
defacto
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What is the canonical height of an elliptic curve?

Aside from the math involved, curious if there is a good layman's explanation for the notion of canonical height for an elliptic curve? I.e. if there is a geometric intepretation? Or perhaps if anyone can help link to a brief history of why it was…
gtr32x
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Complex conjugation of Heegner point

In Kolyvagin's work on modular elliptic curves, proposition 5.3, there is the following statement: let $\tau\in\text{Gal}(K/\mathbb{Q})$ be complex conjugation, then $$\tau y_n=\epsilon \sigma'y_n+\text{ (torsion) in }E(K_n), \text{for some…
defacto
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Supersingular curve on $\mathbb{F}_p$ is cyclic

I'm trying to solve the following problem, but I feel like I'm missing some key fact. Problem: Let $p\equiv 2\pmod{3}$ be an odd prime and consider the elliptic curve $E(\mathbb{F}_p)$ defined by $$y^2=x^3+B$$ where $B\not\equiv 0\pmod{p}$. Prove…
Federico Clerici
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Number of points on an elliptic curve over $F_{p^n}$

$E$ is an elliptic curve with non-split multiplicative reduction at prime $p$. I'm trying to find the number of points $E$ over $F_{p^n}$. I know that when I remove the singularity, the rest is a group isomorphic to the kernel of the norm map from…
Kartan12
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Torsion subgroup of elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$: $y^2+y=x^3-x$ I'm trying to find its torsion subgroup $E_{tors}$. Actually, I know that it is trivial. How to prove this? By using change of variable I've got: $y^2=x^3-x+\frac{1}{4}$ or…
Kartan12
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Mordell-Weil rank growth in Iwasawa tower

This is more of a reference request in case anyone can direct me to the right literature. If you have an elliptic curve $E/\mathbb Q$, and you consider the $\mathbb Z_p$ extension, $\mathbb Q_{\infty}$, then we know that the rank over $\mathbb…
fhn
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Clarification in Silverman's proof of the descent theorem

In the book by Silverman called The Arithmetic of Elliptic Curves, there is the Descent Theorem (Theorem 3.1). He proves the theorem and allong the way he writes $$h(P_{n}) \leq \left( \frac{2}{m}\right)^n h(P) + \left( \frac{1}{m^2} + \frac{2}{m^2}…
Bo Tielman
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For supersingular Elliptic Curves, how does [p] factor through the Frobenius endomorphism?

After having a look at Silverman's "Arithmetic of Elliptic Curves", I mostly understand the notion of a supersingular Elliptic Curve and its characterizations. However, some subtleties still confuse me. Could someone point out the error in the…
Feanor
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Is the order of elliptic curve the same as the order of point on it in finite field?

The question is the same as: is elliptic curve cyclic? how to prove it? update Seems the above answer is no. But I've a further question(maybe should post another thread?). Is there a bounding for the order of a random point on an elliptic curve?…
omg
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An example of adding points on an elliptic curve in Elliptic Curves, Number Theory and Cryptography by Lawrence Washington.

This question addresses an example that appears in Lawrence Washington's book, Elliptic Curves, Number Theory and Cryptography on page 16. Example 2.1. The author asserts that the following statements are true on the elliptic curve $$y^2…
student
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Show the equality in the sequence

Let $$\begin{align} a_0 &=0, & a_1&=1, & a_{n+1}+a_{n-1}&=\frac{8a_n}{16-5a_n^2}\end{align}$$ Show that $$a_{2n}^2 = \frac{64(5a_n^4-20a_n^2+16)a_n^2}{(5a_n^4-16)^2}$$ My thoughts so far: Since $a_n$ is rational number, it implies that…
Zhaohui Du
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