Fix a non-square integer $$k\ne 0$$ . We show that the number of curves $$E_B:y^2=x^3+kB^2$$ containing an integral point, where B ranges over positive integers less than N, is bounded by $$\ll _k N(\log N)^{-\frac{1}{2}+\epsilon }$$ In particular, this implies $$B\le N$$ such $$-3kB^2$$ discriminant elliptic curve $$\mathbb {Q}$$ o(N). The proof involves discriminant-lowering procedure on bina...

If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P ∈ E(Q), nP is integral for at most one value of n > C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ ⊆ E′(Q), Γ contains at most 6 integral ...

In (??), a is an SUn-connection on the trivial bundle over H, Fa is its curvature, iφ (the Higgs field) is a section of the adjoint bundle, and ∗ is the Hodge ∗-operator on H. We regard two monopoles as the same if they are gauge-equivalent. (The reason for our apparently eccentric notation for the Higgs field will become clear in §3.) We shall develop the twistor description of solutions of (?...