By Cohn P.M.

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Faltings and G. , Vieweg, 1992). Zimmer, Computing the Mordell-Weil Group of an Elliptic Curve over Q, in Elliptic Curves and Related Topics (ed. Kisilevsky and M. Ram Murthy), CRM Proceedings and Lecture Notes, vol 4, pp 61-83 (Amer. Math. , 1994). Gross, Kolyvagin’s work on modular elliptic curves, in L-functions and Arithmetic ed. Taylor (Cambridge, 1991). Zagier, Heegner points and derivatives of L-series II, Math. Ann. 278(1987), 497-562. Zagier, Heegner points and derivatives of L-series, Invent.

For suppose that u0 , v0 have been so chosen that there is only one good prime p0 which divides c(u0 , v0 ); then the product in (7) reduces to the left hand side of (6), and so (6) holds for this prime. Now choose an open set N ⊂ S0 such that (7) holds throughout N for each c(u, v); by a slightly modified version of Schinzel’s Hypothesis we can choose (u0 , v0 ) in N so that every c(u0 , v0 ) is the product of one good prime and possibly some factors in B. As c runs through all irreducible factors of a1 a2 a3 , p0 runs through all those primes for 21 which we have to verify (6).

David, (Cambridge, 1996). Heath-Brown, The Density of Rational Points on Cayley’s Cubic Surface, (unpublished). Hooley, On nonary cubic forms, J. Reine Angew. , 386(1988), 32-98 and 415(1991), 95-165 and 456(1994), 53-63. Hulsbergen, Conjectures in Arithmetic Algebraic Geometry, (Vieweg, 1992). Kleiman, Algebraic cycles and the Weil conjectures, in Dix expos´es sur la cohomologie des sch´emas, ed. Kuiper (NorthHolland, 1968). Kolyvagin, Finiteness of E(Q) and X(E/Q) for a class of Weil curves, Izv.