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arithmetic progressions Assume average B(uo canonical height conjugacy class Corollary 3.3 corresponding solution curve dy2 d'Cd dCr(a deduce denote the corresponding divisor divisor functions E(uo E(uq elliptic curve error term finite function with x(t gcd(x gcd(x,y given Hence higher-power moments holds homogeneous polynomial hx(P hyperelliptic curve implied constant depends inequality irreducible polynomial Ivic J. W. S. Cassels K/Q is Galois lattice Lemma Lemma A.2 Let f Let f(x log(uy lower bound Math Mathematics Subject Classification measurable function mod pn multiplicative functions number field polynomial of degree positive integer positive real numbers progressions of modulus proof of Theorem Proposition 3.11 rational point rational prime result follows sampling datum satisfy solution to 2.1 square square-free integer square-free polynomial square-free sieve statement follows subset Suppose trivial up(DiscP up(x up(x)dx upper bound write S(N