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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 number]]s, then a prime ''p'' is '''supersingular for ''E''''' if the [[Good reduction#G|reduction]] of ''E'' modulo&nbsp;''p'' is a [[supersingular elliptic curve]] over the [[residue field]]&nbsp;'''F'''<sub>p</sub>.
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{{Harvtxt|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. {{harvtxt|Lang|Trotter|1976}} conjectured that the number of supersingular primes less than a bound ''X'' is within a constant multiple of <math>\frac{\sqrt{X}}{ln X}</math>, using heuristics involving the distribution of [[Frobenius eigenvalue]]s. {{As of|2012}}, this conjecture is open.
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More generally, if ''K'' is any [[global field]]—i.e., a [[finite extension]] either of '''Q''' or of '''F'''<sub>p</sub>(''t'')—and ''A'' is an [[abelian variety]] defined over ''K'', then a '''supersingular prime <math>\mathfrak{p}</math> for ''A''''' is a [[finite place]] of ''K'' such that the reduction of ''A'' modulo <math>\mathfrak{p}</math> 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=[[Inventiones Mathematicae|Invent. Math.]] |volume=89 |year=1987 |issue=3 |pages=561–567 |doi=10.1007/BF01388985 |ref=harv }}
*{{cite book |last=Lang |first=Serge |authorlink1=Serge Lang|last2=Trotter |first2=Hale F. |authorlink2=Hale F. Trotter|year=1976 |title=Frobenius distributions in GL<sub>2</sub>-extensions |location=New York |publisher=[[Springer-Verlag]] |isbn=0-387-07550-X | zbl=0329.12015 | series=Lecture Notes in Mathematics | volume=504 | ref=harv }}
*{{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 }}
 
{{Prime number classes}}
 
[[Category:Classes of prime numbers]]
[[Category:Algebraic number theory]]
[[Category:Unsolved problems in mathematics]]

Latest revision as of 17:25, 2 December 2014

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