Supersingular prime - Citizendia

If E is an elliptic curve 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 $\mathbb{F}_p$ . In Mathematics, the Hasse-Witt matrix H of a Non-singular Algebraic curve C over a Finite field F is the More generally, if K is any global field -- i. e. , a finite extension either of $\mathbb{Q}$ 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.

Alternately, in some contexts the term supersingular prime is used for a prime divisor of the order of the Monster group M, the largest of the sporadic simple groups. In the Mathematical field of Group theory, the Monster group M or F 1 (also known as the Fischer-Griess Monster or the Friendly Giant In this sense there are precisely 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71. In mathematics Two has many properties in Mathematics. An Integer is called Even if it is divisible by 2 ---- In mathematics Three is the first odd Prime number, and the second smallest prime This article discusses the number five. For the year 5 AD see 5. In mathematics Seven is the fourth Prime number. It is not only a Mersenne prime (since 23 &minus 1 = 7 but also a 17 ( seventeen) is the Natural number following 16 and preceding 18. 19 ( nineteen) is the Natural number following 18 and preceding 20. This article is about the number 23 For the year see 23. For the movies see 23 (film and The Number 23. 29 ( twenty-nine) is the Natural number following 28 and preceding 30. 31 ( thirty-one) is the Natural number following 30 and preceding 32. 41 ( forty-one) is the Natural number following 40 and preceding 42. 47 ( forty-seven) is the Natural number following 46 and preceding 48. 59 ( fifty-nine) is the Natural number following 58 and preceding 60. 71 ( seventy-one) is the Natural number following 70 and preceding 72.

Although these two usages are certainly distinct (the first is relative to a particular elliptic curve, whereas the second is not), they are related. Indeed, for a prime number p, the following are equivalent:

(i) The modular curve $X_0^+(p)$ has genus zero. In Number theory and Algebraic geometry, a modular curve is a Riemann surface, or the corresponding Algebraic curve, constructed as a quotient

(Let H denote the upper half-plane. In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \} For a natural number n, let Γ₀(n) denote the modular group Γ₀, and let w_n be the Fricke involution defined by the block matrix [[0, −1], [n, 0]]. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a congruence subgroup of a Matrix group with Integer entries is a Subgroup defined by congruence conditions on the entries Furthermore, let the modular curve X₀(n) be the compactification (with added cusps) of

$Y_0(n) = \Gamma_0(n)\setminus H,$

and for any prime p, define

$X_0^+(p) = X_0(p)/w_p.$ )

(ii) Every supersingular elliptic curve of characteristic p can be defined over the prime subfield $\mathbb{F}_p$ . In Number theory and Algebraic geometry, a modular curve is a Riemann surface, or the corresponding Algebraic curve, constructed as a quotient

(iii) The order of the Monster group is divisible by p.

The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed that the primes satisfying (i) are exactly the 15 primes 2,. . . ,71 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. In Mathematics, monstrous moonshine is a term devised by John Horton Conway and Simon P

References

Eric W. Weisstein, Supersingular Prime at MathWorld. In Mathematics, the Hasse-Witt matrix H of a Non-singular Algebraic curve C over a Finite field F is the Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

Joseph H. Silverman (1986). Joseph H Silverman (born 1955 is currently a professor of mathematics at Brown University. The Arithmetic of Elliptic Curves. Springer.

Ogg, A. P. "Modular Functions. " In The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, Calif. , June 25-July 20, 1979 (Ed. B. Cooperstein and G. Mason). Providence, RI: Amer. Math. Soc. , pp. 521-532, 1980.

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