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The existence of infinitely many supersingular primes for every elliptic curve over . (English) Zbl 0631.14024

For an elliptic curve E over , a prime p of good reduction of E is said to be supersingular with respect to E if the reduced elliptic curve E p has no points of order p over the algebraic closure 𝔽 p of the prime field 𝔽 p =/p; this is the case if and only if the ring of multiplicators of E p is a (noncommutative) maximal order in the quaternion algebra ,p .

The author, “thinking quaternionically”, establishes the existence of infinitely many supersingular primes with respect to a given elliptic curve E over , 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 . The method of proof essentially depends on work of M. Deuring [Abh. Math. Semin. Hansische Univ. 14, 197–272 (1941; Zbl 0025.02003)].


MSC:
11G05Elliptic curves over global fields
14G25Global ground fields
References:
[1]Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hansischen Univ.14, 197-272 (1941) · 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
[3]Gross, B.H., Zagier, D.: On singular moduli. J. Reine Angew. math.335, 191-220 (1985)
[4]Lang, S., Trotter, H.: Frobenius distributions in GL2-extensions. Lect. Notes in Math., vol. 504. Berlin-Heidelberg-New York: Springer 1976
[5]Schoof, R.: Elliptic curves over finite fields and the computation of square roots modp. Math. Comput.44, 483-494 (1985)
[6]Silverman, J.: The arithmetic of elliptic curves. New York: Springer 1985