algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

, a supersingular prime for a given

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...

with a certain relationship to that curve. If the curve

E

is defined over the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s, then a prime

p

is supersingular for ''E'' if the reduction of

E

modulo

p

is a

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 ''ordi ...

over the

residue field In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ri ...

\mathbb_p

. Noam Elkies showed that every elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero (if

E

does not have complex multiplication). conjectured that the number of supersingular primes less than a bound

X

is within a constant multiple of

\sqrt/\log X

, using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2019, this conjecture is open. More generally, if

K

is any

global field In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global functio ...

—i.e., a

finite extension In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory—in ...

either of

\Q

or of

\mathbb_p(t)

—and

A

is an

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...

defined over

K

, then a supersingular prime

\mathfrak

for ''A'' is a finite place of

K

such that the reduction of

A

modulo

\mathfrak

is a supersingular

References

* * * * {{Prime number classes Classes of prime numbers Algebraic number theory Unsolved problems in number theory

See also

References