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May 20th, 2007, 08:19 PM   #1
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Supersingular elliptic curve question

I need help in interpreting the definition of a supersingular elliptic curve. Namely, what a "cubic equation of the form f(w,x,y)=0" means.

A definition can be found here: http://planetmath.org/encyclopedia/Supersingular.html

What I am having trouble is what the variable w is. In context of my problem, I have a curve in Weierstrass form and want to know what the cubic equation as referred to above corresponds to. I'm assuming that x and y are just the variables found in an elliptic curve.

Thanks in advance for the help. I'll post back here if I find out an answer before there are any replies.
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May 20th, 2007, 10:40 PM   #2
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f(w,x,y)=0 is the elliptic curve in normal form.
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May 30th, 2007, 04:00 AM   #3
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Re: Supersingular elliptic curve question

Quote:
Originally Posted by fathwad
I need help in interpreting the definition of a supersingular elliptic curve. Namely, what a "cubic equation of the form f(w,x,y)=0" means.

A definition can be found here: http://planetmath.org/encyclopedia/Supersingular.html

What I am having trouble is what the variable w is. In context of my problem, I have a curve in Weierstrass form and want to know what the cubic equation as referred to above corresponds to. I'm assuming that x and y are just the variables found in an elliptic curve.

Thanks in advance for the help. I'll post back here if I find out an answer before there are any replies.
*********************************************
I wrote my thesis on singular and supersingular curves. There I show that a curve over a perfect field of char. p is supersingular iff it fulfills several equivalent conditions. One of the nicest ones is that its invariant differential is exact, but others are:
An elliptic curve E over a perfect field of char. p is supersingular iff:
1) Its invariant differential is exact, iff:
2) E[p] = 0, where E[p] = { x in E / px = 0}, iff:
3) The Froebenius map P_p is purely inseparable, iff:
4) The map [p]: E --> E is purely inseparable, iff:
5) the coefficient of x^(p-1) in f(x)^(p-1)/2, when E: y^2 = f(x) and f(x) is a cubic in x with different roots.

You may want to check Silverman's book on elliptic cureves...
Reagrds
Tonio
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June 2nd, 2007, 12:19 PM   #4
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Thanks, but the ultimate criterion I used was a theorem from Lang's Elliptic functions which basically states that if a prime p stays inert from a rational to a quadratic field and an elliptic curve E has complex multiplication by an order of the quadratic field, then E mod p is supersingular. This fit perfectly with what I was doing and I didn't want to have to define things like invariant differential.

I appreciate the help though.

Frank
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