Supersingular prime (moonshine theory) (original) (raw)
En la rama matemática de la teoría monstrous moonshine, un primo supersingular es un número primo que divide el orden del grupo monstruo M, que es el grupo esporádico más grande.
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| dbo:abstract | En la rama matemática de la teoría monstrous moonshine, un primo supersingular es un número primo que divide el orden del grupo monstruo M, que es el grupo esporádico más grande. (es) In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order 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. (sequence in the OEIS) 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: 1. * The modular curve X0+(p) = X0(p) / wp, where wp is the Fricke involution of X0(p), has genus zero. 2. * Every supersingular elliptic curve in characteristic p can be defined over the prime subfield Fp. 3. * 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 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. (en) 在月光理論的數學分支中，超奇異質數（Supersingular prime）是怪兽群（Monster group，最大的簡單散在群）“M”階數的質因數 。超奇異質數只有15個：包括前11個質數（2、3、5、7、11、13、17、19、23、29、31）、41、47、59和71。（OEIS數列） 不是超奇異質數的質數有37、43、53、61、67，以及任何大於或等於73的質數。所有超奇異質數都是陳素數，但是不是不是超奇異質數的37、53和67也是陳素數，並且有無數個大於73的陳素數。 超奇異質數與以下所述的概念有關。對於素數“ p”，以下等價： 1. * 模曲線 X0+(p) = X0(p)/wp，其中 wp 是的X0(p)，其特征為零。 2. * 可以在素子場（prime subfield）上定義特徵“p”中的所有超奇異橢圓曲線 Fp. 3. * 怪物組的階數可被“p”整除。 上述等價敘述是由提出。奧格在1975年證明滿足第一個條件的素數，恰好是上面列出的15個超奇數素數，此後不久就得知（當時是猜想的）零星簡單群的存在，這些群正好把這些素數作為素數除數。這種奇怪的巧合即為怪兽月光理论的基礎。 三個非超奇異素數以另外兩個簡單散在群的階數出現：37和67是里昂群階數的因數，以及37和43是階數的因數。可以立即得出結論，這兩個不是魔群組的子商（它們是六個中的兩個）。其他的散在群（包括其他四個低群，若可計入散在群的話，也包括提次群）也具有僅超奇質數的階。 實際上，除了以外，它們都只能被小於或等於31的素數整除的階，不過只有怪獸群的因數包括所有的超奇異質數。超奇異質數47是小怪獸群階數的因數，而其他三個超奇異質數（41、59和71）只是怪獸群的因數，不是其他散在群的因數。 (zh) |
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| dbp:title | Sporadic group (en) Supersingular Prime (en) |
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| rdfs:comment | En la rama matemática de la teoría monstrous moonshine, un primo supersingular es un número primo que divide el orden del grupo monstruo M, que es el grupo esporádico más grande. (es) In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order 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. (sequence in the OEIS) The non-supersingular primes are 37, 43, 53, 61, 67, and any prime number greater than or equal to 73. (en) 在月光理論的數學分支中，超奇異質數（Supersingular prime）是怪兽群（Monster group，最大的簡單散在群）“M”階數的質因數 。超奇異質數只有15個：包括前11個質數（2、3、5、7、11、13、17、19、23、29、31）、41、47、59和71。（OEIS數列） 不是超奇異質數的質數有37、43、53、61、67，以及任何大於或等於73的質數。所有超奇異質數都是陳素數，但是不是不是超奇異質數的37、53和67也是陳素數，並且有無數個大於73的陳素數。 超奇異質數與以下所述的概念有關。對於素數“ p”，以下等價： 1. * 模曲線 X0+(p) = X0(p)/wp，其中 wp 是的X0(p)，其特征為零。 2. * 可以在素子場（prime subfield）上定義特徵“p”中的所有超奇異橢圓曲線 Fp. 3. * 怪物組的階數可被“p”整除。 上述等價敘述是由提出。奧格在1975年證明滿足第一個條件的素數，恰好是上面列出的15個超奇數素數，此後不久就得知（當時是猜想的）零星簡單群的存在，這些群正好把這些素數作為素數除數。這種奇怪的巧合即為怪兽月光理论的基礎。 (zh) |
| rdfs:label | Número primo supersingular (teoría moonshine) (es) Supersingular prime (moonshine theory) (en) 超奇異質數 (zh) |
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