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Cm Elliptic Curves and Bicolorings of Steiner Triple Systems
Author(s) -
Dummit David S.
Publication year - 2002
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609301008785
Subject(s) - mathematics , combinatorics , elliptic curve , prime (order theory) , steiner system , order (exchange) , square (algebra) , prime number , supersingular elliptic curve , discrete mathematics , pure mathematics , geometry , finance , economics
It is shown that a necessary condition for the existence of a bicolored Steiner triple system of order n is that n can be written in the form A 2 +3 B 2 for integers A and B . In the case when n = q is either a prime congruent to 1 mod 3, or the square of a prime congruent to 2 mod 3, it is shown that the numbers of colored vertices in the triple system would be unique, and are given by the number of points on specific twists of the CM elliptic curve y 2 = x 3 −1 over the finite field F q . 2000 Mathematics Subject Classification 05B07, 11G20, 14G15 (primary); 11G15, 14K22 (secondary).