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Supersingular primes for elliptic curves over real number fields. (English) Zbl 0708.14020

Let \(E\) be an elliptic curve defined over a number field \(K\). The author proved in an earlier paper [Invent. Math. 89, No. 3, 561–567 (1987; Zbl 0631.14024)] that if \([K:\mathbb Q]\) is odd then \(E\) has infinitely many primes of supersingular reduction. This is generalized in the present article by showing that the same result holds if \(K\) has at least one real embedding.

MSC:

11G05 Elliptic curves over global fields
14G25 Global ground fields in algebraic geometry

Citations:

Zbl 0631.14024
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References:

[1] H. Davenport , Multiplicative Number Theory , 2nd ed. New York-Heidelberg -Berlin: Springer-Verlag 1980. · Zbl 0453.10002
[2] M. Deuring , Die Typen der Multiplikatorenringe elliptischer Funktionenk√∂rper . Abh. Math. Sem. Hansischen Univ. 14, (1941) 197-272. · Zbl 0025.02003
[3] N.D. Elkies , The existence of infinitely many supersingular primes for every elliptic curve over Q . Invent. Math. 89, (1987) 561-567. · Zbl 0631.14024
[4] B.H. Gross and D. Zagier , On singular moduli . J. Reine Angew. Math. 335, (1985) 191-220. · Zbl 0545.10015
[5] S. Lang and H. Trotter , Frobenius distributions in GL2-extensions . Lect. Notes in Math., vol. 504. Berlin-Heidelberg-New York: Springer 1976. · Zbl 0329.12015
[6] J.-P. Serre , A Course in Arithmetic . New York- Heidelberg-Berlin: Springer-Verlag 1973. · Zbl 0256.12001
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