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The existence of infinitely many supersingular primes for every elliptic curve over $\Bbb Q$. (English) Zbl 0631.14024
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 $\Bbb Q_{\infty,p}$. 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)].

11G05Elliptic curves over global fields
14G25Global ground fields
Full Text: DOI EuDML
[1] Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hansischen Univ.14, 197-272 (1941) · Zbl 0025.02003 · doi:10.1007/BF02940746
[2] Gross, B.H.: Arithmetic on elliptic curves with complex multiplication. Lect. Notes in Math., vol. 776. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0433.14032
[3] Gross, B.H., Zagier, D.: On singular moduli. J. Reine Angew. math.335, 191-220 (1985) · Zbl 0545.10015
[4] Lang, S., Trotter, H.: Frobenius distributions in GL2-extensions. Lect. Notes in Math., vol. 504. Berlin-Heidelberg-New York: Springer 1976 · Zbl 0329.12015
[5] Schoof, R.: Elliptic curves over finite fields and the computation of square roots modp. Math. Comput.44, 483-494 (1985) · Zbl 0579.14025
[6] Silverman, J.: The arithmetic of elliptic curves. New York: Springer 1985 · Zbl 0613.14029