#jsDisabledContent { display:none; } My Account | Register | Help

# Supersingular prime (for an elliptic curve)

Article Id: WHEBN0022965231
Reproduction Date:

 Title: Supersingular prime (for an elliptic curve) Author: World Heritage Encyclopedia Language: English Subject: Supersingular prime Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Supersingular prime (for an elliptic curve)

In algebraic number theory, a supersingular prime is a prime number with a certain relationship to a given elliptic curve. If the curve E 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 Fp.

Elkies (1987) showed that any elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero. Lang & Trotter (1976) conjectured that the number of supersingular primes less than a bound X is within a constant multiple of $\frac\left\{\sqrt\left\{X\right\}\right\}\left\{ln X\right\}$, using heuristics involving the distribution of Frobenius eigenvalues. As of 2012, this conjecture is open.

More generally, if K is any global field—i.e., a finite extension either of Q or of Fp(t)—and A is an abelian variety defined over K, then a supersingular prime $\mathfrak\left\{p\right\}$ for A is a finite place of K such that the reduction of A modulo $\mathfrak\left\{p\right\}$ is a supersingular abelian variety.

## References

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.