zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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)].

MSC:
11G05Elliptic curves over global fields
14G25Global ground fields
WorldCat.org
Full Text: DOI EuDML
References:
[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