Supersingular elliptic curve

In algebraic geometry, a branch of mathematics, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0. 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.

Definition

Let K be a field with algebraic closure $\overline{K}$ and E an elliptic curve over K. Then the $\overline{K}$-valued points $E(\overline{K})$ have the structure of an abelian group. For every n, we have a multiplication map $[n]: E\to E$. Its kernel is denoted by E[n]. Now assume that the characteristic of K is p > 0. Then one can show that either

$E[p^r](\overline{K}) \cong \begin{cases} 0 & \mbox{or}\\ \mathbb{Z}/p^r\mathbb{Z} \end{cases}$

for r = 1, 2, 3, ... In the first case, E is called supersingular. Otherwise it is called ordinary. The term 'supersingular' does not mean, of course, that E is singular since all elliptic curves are smooth.

Equivalent conditions

There is a number of equivalent conditions to supersingularity:

• Supersingular elliptic curves have many endomorphisms in the sense that an elliptic curve is supersingular if and only if its endomorphism algebra (over $\overline{K}$) is an order in a quaternion algebra. Thus, their endomorphism group has rank 4, while the endomorphism group of every other elliptic curve has only rank 1 or 2.

• Let G be the formal group associated to E. Since K is of positive characteristic, we can define its height ht(G), which is 2 if and only if E is supersingular and else is 1.

• We have a Frobenius morphism $F: E\to E$, which induces a map in cohomology

$F^*: H^1(E, \mathcal{O}_E) \to H^1(E,\mathcal{O}_E)$.

The elliptic curve E is supersingular if and only if F ^* equals 0.

• Suppose E is in Legendre form, defined by the equation y² = x(x − 1)(x − λ). Then E is supersingular if and only if the sum

$\sum_{i=0}^k {k\choose{i}}^2\lambda^i$

vanishes, where $k = \frac12(p-1)$. Using this formula, one can show that there are only finitely many supersingular elliptic curves for every K.

Examples

• If K is a field of characteristic 2, every elliptic curve defined by an equation of the form

y² + a₃y = x³ + a₄x + a₆

is supersingular (see Washington2003, p. 122).

• If K is a field of characteristic 3, every elliptic curve defined by an equation of the form

y² = x³ + a₄x + a₆

is supersingular (see Washington2003, p. 122).

• For $\mathbb{F}_p$ with p>3 we have that the elliptic curve defined by y² = x³ + 1 is supersingular if and only if $p\equiv 2 \text{(mod 3)}$ and the elliptic curve defined by y² = x³ + x is supersingular if and only if $p\equiv 3 \text{(mod 4)}$ (see Washington2003, 4.35).

• There are also more exotic examples: The elliptic curve given by y² = x(x − 1)(x + 2) is nonsingular over $\mathbb{F}_p$ for $p\neq 2,3$. It is supersingular for p = 23 and ordinary for every other $p\leq 73$ (see Hartshorne1977, 4.23.6).

• Elkies (1987) showed that any elliptic curve defined over the rationals is supersingular for an infinite number of primes.
• Birch & Kuyk (1975) give a table of all 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	0=1728
5	0
7	6=1728
11	0, 1=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

References

• Birch, B. J.; Kuyk, W., eds. (1975), "Table 6", Modular functions of one variable. IV, Lecture Notes in Mathematics, 476, Berlin, New York: Springer-Verlag, pp. 142–144, doi:10.1007/BFb0097591, ISBN 978-3-540-07392-5, MR0376533 
• Elkies, Noam D. (1987), "The existence of infinitely many supersingular primes for every elliptic curve over Q", Inventiones Mathematicae 89 (3): 561–567, doi:10.1007/BF01388985, ISSN 0020-9910, MR903384 
• Robin Hartshorne (1977), Algebraic Geometry, Springer. ISBN 1441928073
• Joseph H. Silverman (2009), The Arithmetic of Elliptic Curves, Springer. ISBN 0387094938
• Lawrence C. Washington (2003), Elliptic Curves, Chapman&Hall. ISBN 1584883650