In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
branch of
moonshine theory, a supersingular prime is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
that
divides the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of the
Monster group ''M'', which is the largest
sporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes (
2,
3,
5,
7,
11,
13,
17,
19,
23,
29, and
31), as well as
41,
47,
59, and
71.
The non-supersingular primes are
37,
43,
53,
61,
67, and any prime number greater than or equal to
73.
Supersingular primes are related to the notion of
supersingular elliptic curves as follows. For a prime number ''p'', the following are equivalent:
# The
modular curve ''X''
0+(''p'') = ''X''
0(''p'') / ''w''
p, where ''w''
p is the
Fricke involution of ''X''
0(''p''), has
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
zero.
# Every supersingular elliptic curve in characteristic ''p'' can be defined over the
prime subfield
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
F
''p''.
# The order of the Monster group is divisible by ''p''.
The equivalence is due to
Andrew Ogg
Andrew Pollard Ogg (born April 9, 1934, Bowling Green, Ohio) is an American mathematician, a professor emeritus of mathematics at the University of California, Berkeley.
Education
Ogg was a student at Bowling Green State University in the mid 195 ...
. More precisely, in 1975 Ogg showed that the primes satisfying the first condition are exactly the 15 supersingular primes listed above and shortly thereafter learned of the (then
conjectural) existence of a sporadic simple group having exactly these primes as prime divisors. This strange coincidence was the beginning of the theory of
monstrous moonshine.
Three non-supersingular primes occur in the orders of two other sporadic simple groups: 37 and 67 divide the order of the
Lyons group, and 37 and 43 divide the order of the
fourth Janko group. It immediately follows that these two are not
subquotients of the Monster group (they are two of the six
pariah groups). The rest of the sporadic groups (including the other four pariahs, and also the
Tits group, if that is counted among the sporadics) have orders with only supersingular prime divisors. In fact, other than the
Baby Monster group, they all have orders divisible only by primes less than or equal to 31, although no single sporadic group, other than the Monster itself, has all of them as prime divisors. The supersingular prime 47 also divides the order of the Baby Monster group, and the other three supersingular primes (41, 59, and 71) do not divide the order of any sporadic group other than the Monster itself.
All supersingular primes are
Chen primes, but 37, 53, and 67 are also Chen primes, and there are infinitely many Chen primes greater than 73.
References
*
*
*
{{Prime number classes
Classes of prime numbers
Moonshine theory
Sporadic groups
fr:Nombre premier super-singulier
it:Primi supersingolari