Boustrophedon transform

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 $\bar{K}$ and E an elliptic curve over K. Then the $\bar{K}$-valued points $E(\bar{K})$ 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

$E[p^r](\bar{K})\cong \{\begin{array}{ll}0& \text{or}\\ \mathbb{\mathbb{Z}}/p^r\mathbb{\mathbb{Z}}\end{array}$

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

Equivalent conditions

There are 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 $\bar{K}$) 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 $y^2=x(x-1)(x-\lambda )$. Then E is supersingular if and only if the sum

$\sum _{i=0}^k{(\genfrac{}{}{0ex}{}{k}{i})}^2{\lambda }^i$

vanishes, where $k=\frac{1}{2}(p-1)$. 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

$y^2+a_{3}y=x^3+a_{4}x+a_6$

is supersingular (see Washington2003, p. 122).

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

$y^2=x^3+a_{4}x+a_6$

is supersingular (see Washington2003, p. 122).

• For ${\mathbb{𝔽}}_p$ with p>3 we have that the elliptic curve defined by $y^2=x^3+1$ is supersingular if and only if $p\equiv 2\text{(mod 3)}$ and the elliptic curve defined by $y^2=x^3+x$ 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 $y^2=x(x-1)(x+2)$ is nonsingular over ${\mathbb{𝔽}}_p$ for $p\ne 2,3$. It is supersingular for p = 23 and ordinary for every other $p\le 73$ (see Hartshorne1977, 4.23.6).

• Template:Harvtxt showed that any elliptic curve defined over the rationals is supersingular for an infinite number of primes.
• Template:Harvtxt 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

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Robin Hartshorne (1977), Algebraic Geometry, Springer. ISBN 1-4419-2807-3
• Joseph H. Silverman (2009), The Arithmetic of Elliptic Curves, Springer. ISBN 0-387-09493-8
• Lawrence C. Washington (2003), Elliptic Curves, Chapman&Hall. ISBN 1-58488-365-0