Supersingular prime (algebraic number theory)

{{Short description|Prime number with a certain relationship to an elliptic curve}}

In algebraic number theory, a supersingular prime for a given elliptic curve is a prime number with a certain relationship to that curve. If the curve $E$ is defined over the rational numbers, then a prime $p$ is supersingular for E if the reduction of $E$ modulo $p$ is a supersingular elliptic curve over the residue field $\mathbb{F}_p$ .

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). {{harvtxt|Lang|Trotter|1976}} conjectured that the number of supersingular primes less than a bound $X$ is within a constant multiple of $\sqrt{X}/\log X$ , 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—i.e., a finite extension either of $\Q$ or of $\mathbb{F}_p(t)$ —and $A$ is an abelian variety defined over $K$ , then a supersingular prime $\mathfrak{p}$ for A is a finite place of $K$ such that the reduction of $A$ modulo $\mathfrak{p}$ is a supersingular abelian variety.

References

{{cite journal |authorlink=Noam Elkies |first=Noam D. |last=Elkies |title=The existence of infinitely many supersingular primes for every elliptic curve over Q |journal=Invent. Math. |volume=89 |year=1987 |issue=3 |pages=561–567 |doi=10.1007/BF01388985 |bibcode=1987InMat..89..561E | mr=0903384|s2cid=123646933 }}
{{cite book |last1=Lang |first1=Serge |authorlink1=Serge Lang|last2=Trotter |first2=Hale F. |authorlink2=Hale F. Trotter|year=1976 |title=Frobenius distributions in GL₂-extensions |location=New York |publisher=Springer-Verlag |isbn=0-387-07550-X | zbl=0329.12015 | series=Lecture Notes in Mathematics | volume=504 }}
{{cite book |authorlink=Andrew Ogg |last=Ogg |first=A. P. |chapter=Modular Functions |title=The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, Calif., June 25–July 20, 1979 |editor1-first=Bruce |editor1-last=Cooperstein |editor2-first=Geoffrey |editor2-last=Mason |location=Providence, RI |publisher=American Mathematical Society |pages=521–532 |year=1980 |isbn=0-8218-1440-0 | zbl=0448.10021 | series=Proc. Symp. Pure Math. | volume=37 }}
{{cite book | authorlink = Joseph H. Silverman |first=Joseph H. |last=Silverman | year = 1986 | title = The Arithmetic of Elliptic Curves | publisher = Springer-Verlag |location=New York |isbn=0-387-96203-4 | zbl=0585.14026 | series=Graduate Texts in Mathematics | volume=106 }}

Category:Classes of prime numbers

Category:Algebraic number theory

Category:Unsolved problems in number theory

Supersingular prime (algebraic number theory)

See also

References