Supersingular elliptic curve

In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular are called ordinary and these two classes of elliptic curves behave fundamentally differently in many aspects. Hasse (1936) discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that in positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and Deuring (1941) developed their basic theory.

The term "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular. It comes from the phrase "singular values of the j-invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorphism ring has the maximal possible rank 2. In positive characteristic it is possible for the endomorphism ring to be even larger: it can be an order in a quaternion algebra of dimension 4, in which case the elliptic curve is supersingular.

Definition

There are many different but equivalent ways of defining supersingular elliptic curves that have been used. Some of the ways of defining them are given below. Let K be a field with algebraic closure \overline{K} and E an elliptic curve over K.

 E[p^r](\overline{K}) \cong \begin{cases} 0 & \mbox{or}\\ \mathbb{Z}/p^r\mathbb{Z} \end{cases}
for r = 1, 2, 3, ... In the first case, E is called supersingular. Otherwise it is called ordinary. In other words, an elliptic curve is supersingular if and only if the group of geometric points of order p is trivial.
F^*: H^1(E, \mathcal{O}_E) \to H^1(E,\mathcal{O}_E).
The elliptic curve E is supersingular if and only if F^* equals 0.
F^*: H^0(E, \Omega^1_E) \to H^0(E,\Omega^1_E).
The elliptic curve E is supersingular if and only if F^* equals 0.
\sum_{i=0}^n {n\choose{i}}^2\lambda^i
vanishes, where n = (p-1)/2. Using this formula, one can show that there are only finitely many supersingular elliptic curves over K (up to isomorphism).
When q=p is a prime greater than 3 this is equivalent to having the trace of Frobenius equal to zero (by the Hasse bound); this does not hold for p=2 or 3.

Examples

y^2+a_3y = x^3+a_4x+a_6
with a3 nonzero is a supersingular elliptic curve, and conversely every supersingular curve is isomorphic to one of this form (see Washington2003, p. 122).
y^2+y = x^3+x+1
y^2+y = x^3+1
y^2+y = x^3+x
with 1, 3, and 5 points. This gives examples of supersingular elliptic curves over a prime field with different numbers of points.
y^2+y=x^3,
with j-invariant 0. Its ring of endomorphisms is the ring of Hurwitz quaternions, generated by the two automorphisms x\rightarrow x+\omega and y\rightarrow y+x+\omega,x\rightarrow x+1 where \omega^2+\omega+1=0 is a primitive cube root of unity. Its group of automorphisms is the group of units of the Hurwitz quaternions, which has order 24 and contains a normal subgroup of order 8 isomorphic to the quaternion group.
y^2 = x^3+a_4x+a_6
with a4 nonzero is a supersingular elliptic curve, and conversely every supersingular curve is isomorphic to one of this form (see Washington2003, p. 122).
y^2 = x^3-x
y^2 = x^3-x+1
y^2 = x^3-x+2
y^2 = x^3+x
y^2=x^3-x,
with j-invariant 0. Its ring of endomorphisms is the ring of quaternions of the form a+bj with a and b Eisenstein integers. , generated by the two automorphisms x\rightarrow x+1 and y\rightarrow iy,x\rightarrow -x where i is a primitive fourth root of unity. Its group of automorphisms is the group of units of the these quaternions, which has order 12 and contains a normal subgroup of order 3 with quotient a cyclic group of order 4.
2, 19, 29, 199, 569, 809, 1289, 1439, 2539, 3319, 3559, 3919, 5519, 9419, 9539, 9929,... A006962

Classification

For each positive characteristic there are only a finite number of possible j-invariants of supersingular elliptic curves. Over an algebraically closed field K an elliptic curve is determined by its j-invariant, so there are only a finite number of supersingular elliptic curves. If each such curve is weighted by 1/|Aut(E)| then the total weight of the supersingular curves is (p–1)/24. Elliptic curves have automorphism groups of order 2 unless their j-invariant is 0 or 1728, so the supersingular elliptic curves are classified as follows. There are exactly ⌊p/12⌋ supersingular elliptic curves with automorphism groups of order 2. In addition if p≡3 mod 4 there is a supersingular elliptic curve (with j-invariant 1728) whose automorphism group is cyclic or order 4 unless p=3 in which case it has order 12, and if p≡2 mod 3 there is a supersingular elliptic curve (with j-invariant 0) whose automorphism group is cyclic of order 6 unless p=2 in which case it has order 24.

Birch & Kuyk (1975) give a table of all j-invariants of supersingular curves for primes up to 307. For the first few primes the supersingular elliptic curves are given as follows. The number of supersingular values of j other than 0 or 1728 is the integer part of (p−1)/12.

prime supersingular j invariants
2 0
3 1728
5 0
7 1728
11 0, 1728
13 5
17 0,8
19 7, 1728
23 0,19, 1728
29 0,2, 25
31 2, 4, 1728
37 8, 3±√15

See also

References

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