1 O ct 2 00 8 DESCENT ON ELLIPTIC CURVES AND HILBERT ’ S TENTH PROBLEM

Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecid-ability results for some large subrings of the rational numbers. In 1970, Matijasevič [10], building upon earlier work of Davis, Put-nam and Robinson [5], resolved negatively Hilbert's Tenth Problem for the ring Z, of rational integers. This means there is no general algorithm which will decide if a polynomial equation, in several variables, with integer coefficients has an integral solution. Equivalently, one says Hilbert's Tenth Problem is undecidable for the integers. See [14, Chapter 1] for a full overview and background reading. The same problem, except now over the rational field Q, has not been resolved. In other words, it is not known if there is an algorithm which will decide if a polynomial equation with integer coefficients (or rational coefficients, it doesn't matter) has a rational solution. Recently, Poonen [11] took a giant leap in this direction by proving the same negative result for some large subrings of Q. To make this precise, given a prime p of Z, let |.| p denote the usual p-adic absolute value. Let S denote a set of rational primes. Write Z S = Z[1/S] = {x ∈ Q : |x| p ≤ 1 for all p / ∈ S},