Let E be a supersingular curve over a finite field. Why is the j-invariant always in F_p^2?

Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$: (i) $p$ is inert in End($E$) (ii) $E_p$ is supersingular (iii) The trace of the Frobenius …

I am currently doing a self study on Algebraic geometry but my ultimate goal is to study more on elliptic curves. Which are the most recommended textbooks I can use to study? I nee …

Let $E$ be your favorite elliptic curve, and let $Tor^m$ be the moduli stack of torsion sheaves of degree $m$ on $E$. This sounds horrible, but it's not so bad; it's a global quot …

(Warning: a student asking) Let $E$ be an elliptic curve over $\mathbf Q$. Let $P(a,b)$ be a (nontrivial) torsion point on $E$. Is there an easy description of the ring of algebrai …

I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised. In this question, in Charles Rezk's answer, I erroneously clai …

This is maybe the first question I actually need to know the answer to! Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of …

Consider the Weierstrass cubic $$y^2z = x^3 + A\, xz^2+B\,z^3.$$ This defines a curve $E$ in $\mathbb{P}^2$, which if smooth is an elliptic curve with basepoint at $[0,1,0]$. I'm …

Notation. Let $p$ be a prime number, $K$ a finite extension of $\mathbb{Q}_p$ and $E|K$ an elliptic curve which has good reduction. The discriminant $d_{E|K}$ of $E|K$ is an elem …

Why do elliptic curves have bad reduction at some point if they are defined over Q, but not necessarily over arbitrary number fields?

My question points in a direction similar to Qiaochu's, but it's not the same (or so I think). Let me provide you with a little bit of background first. Let E be an elliptic curve …

Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really …

The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look …

What is the quadratic twist of an elliptic curve in Legendre Form? How do you show an elliptic curve and its quadratic twist is isomorphic when they are in Legendre Form?

If we have an elliptic curve E over a field k and we pick a non-square d in k-{0}. Suppose E is isomorphic to E^(d). (E^(d) is the quadratic twist) Why must j(E) = 1728 and why is …