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f(Q1, Q2) are indeed linear in A, B, C. But it turns out that these two conditions completely determine f: there is up to scaling a unique PSL2-covariant bilinear map from V3 V3 to V3; equiv- alently, V3 occurs exactly once in the representation V3 " V3 of PSL2. In fact it is known (see e.g. [FH, 11.2]) that V3 " V3 decomposes as V1 " V3 " V5, where V1 is the trivial representation and V5 is the space of homogeneous polynomi- als of degree 4 in X, Y . The factor V3 is particularly easy to see, because it is 2 just the antisymmetric part V3 of V3 " V3. Now the next-to-highest exte- dim V -1 rior power V of any finite-dimensional vector space V is canonically dim V " isomorphic with (det V ) " V , where det V is the top exterior power V . Taking V = V3, we see that det V3 is the trivial representation of PSL2. More- over, thanks to the invariant quadric B2 - AC we know that V3 is self-dual as a 2 PSL2 representation. Unwinding the resulting identification V3 ! V3" ! V3, we find: 2 Proposition A. Let Qi = AiX2 + 2BiXY + CiY (i = 1, 2) be two polyno- mials in V3 without a common zero. Then the unique involution of P1 switching the roots of Q1 and also of Q2 is the involution whose fixed points are the roots of 2 (A1B2 - A2B1)X2 + (A1C2 - A2C1)XY + (B1C2 - B2C1)Y , (88) i.e. the fractional linear transformation (A1C2 - A2C1)t + 2(B1C2 - B2C1) t !! . (89) 2(B1A2 - B2A1)t + (C1A2 - C2A1) Shimura Curve Computations 45 " 2 Proof : The coordinates of Q1 '" Q2 for the basis of V3 dual to (X2, 2XY, Y ) " are (B1C2 -B2C1, A2C1 -A1C2, A1B2 -A2B1). To identify V3 with V3 we need "2 a PSL2-invariant element of V3 . We could get this invariant from the invariant ""2 quadric B2 - AC " V3 , but it is easy enough to exhibit it directly: it is 1 2 2 X2 " Y - 2XY " 2XY + Y " X2, (90) 2 the generator of the kernel of the multiplication map Sym2(V3) ! V5. The 2 resulting isomorphism from V3" to V3 takes the dual basis of (X2, 2XY, Y ) to 2 (Y , -XY, X2), and thus takes Q1 '" Q2 to (88) as claimed. Of course this is not the only way to obtain (89). A more geometrical approach (which ultimately amounts to the same thing) is to regard P1 as a conic in P2. Then involutions of P1 correspond to points p " P2 not on the conic: the involution associated with p takes any point q of the conic to the second point of intersection of the line pq with the conic. Of course the fixed points are then the points q such that pq is tangent to the conic at q. Given Q1, Q2 we obtain for i = 1, 2 the secant of the conic through the roots of Qi, and then p is the intersection of those secants. From either of the two approaches we readily deduce 2 Corollary B. Let Qi = AiX2 + 2BiXY + CiY (i = 1, 2, 3) be three polyno- mials in V3 without a common zero. Then there is an involution of P1 switching the roots of Qi for each i if and only if the determinant A1 B1 C1 A2 B2 C2 (91) A3 B3 C3 vanishes. As an additional check on the formula (88), we may compute that the dis- criminant of that quadratic polynomial is exactly the resolvent A1 2B1 C1 0 0 A1 2B1 C1 det (92) A2 2B2 C2 0 0 A2 2B2 C2 of Q1, Q2 which vanishes if and only if these two polynomials have a common zero. References A. Arno, S.: The imaginary quadratic fields of class number 4. Acta Arith. 60 (1992), 321 334. Be. Beckmann, S.: Ramified primes in the field of moduli of branches coverings of curves. J. of Algebra 125 (1989), 236 255. BK. Birch, B.J., Kuyk, W., ed.: Modular Functions of One Variable IV. Lect. Notes in Math. 476, 1975. 46 Noam D. Elkies C. Cremona, J.E.: Algorithms for modular elliptic curves. Cambridge University Press, 1992. D. Deuring, M.: Die Typen die Multiplikatorenringe elliptische Funktionkrper, Abh. Math. Sem. Hansischen Univ. 14, 197 272 (1941). E1. Elkies, N.D.: ABC implies Mordell, International Math. Research Notices 1991 #7, 99 109. E2. Elkies, N.D.: The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89 (1987), 561 568. E3. Elkies, N.D.: Supersingular primes for elliptic curves over real number fields, Compositio Math. 72 (1989), 165 172. E4. Elkies, N.D.: Heegner point computations. Pages 122 133 in Algorithmic number theory (Ithaca, NY, 1994; L.M. Adleman and M.-D. Huang, eds.; Lect. Notes in Computer Sci. #877; Berlin: Springer, 1994). E5. Elkies, N.D.: Elliptic and modular curves over finite fields and related compu- tational issues. Pages 21 76 in Computational Perspectives on Number Theory: Proceedings of a Conference in Honor of A.O.L. Atkin (D.A. Buell and J.T. Teit- elbaum, eds.; AMS/International Press, 1998). [ Pobierz całość w formacie PDF ] |