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Zbl 0631.14024
Elkies, Noam D.
The existence of infinitely many supersingular primes for every elliptic curve over $\Bbb Q$. (English)
[J] Invent. Math. 89, No. 3, 561-567 (1987). ISSN 0020-9910; ISSN 1432-1297/e

For an elliptic curve $E$ over $\Bbb Q$, a prime $p$ of good reduction of $E$ is said to be {\it supersingular} with respect to $E$ if the reduced elliptic curve $E_p$ has no points of order $p$ over the algebraic closure $\Bbb F_p$ of the prime field $\Bbb F_p=\Bbb Z/p\Bbb Z$; this is the case if and only if the ring of multiplicators of $E_p$ is a (noncommutative) maximal order in the quaternion algebra $\Bbb Q_{\infty,p}$. \par The author, ``thinking quaternionically'', establishes the existence of infinitely many supersingular primes with respect to a given elliptic curve $E$ over $\Bbb Q$, a fact not previously known for non-CM curves. He extends this result to elliptic curves over any algebraic number field $K$ of odd degree over $\Bbb Q$. The method of proof essentially depends on work of {\it M. Deuring} [Abh. Math. Semin. Hansische Univ. 14, 197--272 (1941; Zbl 0025.02003)].
[Horst G. Zimmer (Saarbrücken)]

MSC 2000:: *11G05 Elliptic curves over global fields
14G25 Global ground fields

Keywords: complex multiplication; existence of infinitely many supersingular primes; elliptic curve

Citations: Zbl 0025.02003

Cited in: Zbl 1234.11072 Zbl 1159.14015 Zbl 0898.11024 Zbl 0767.11028 Zbl 0733.14013 Zbl 0708.14020

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