The Catalan equation over function fields

Let K be the function field of a projective variety. Fix a, b, c E K*. We show that if max{m, n} is sufficiently large, then the Catalan equation axm + by c has no nonconstant solutions x, y E K. The Cassels-Catalan conjecture states that for fixed nonzero integers a, b, c, the equation axm + by = c has only finitely many solutions in integers x, y, m, n satisfying m , 3 and n, I x I 1 2. At present, the only known case of this conjecture is a = -b= c= 1, due to Tijdeman [6]. In this paper we prove a strengthened version of the Cassels-Catalan conjecture in the case that the number field Q is replaced by an arbitrary function field. The proof uses only elementary algebraic geometry, the principal tool being the Riemann-Hurwitz formula. THEOREM. Let k be a field of characteristic p (possibly with p = 0), and let K/k be the function field of a nonsingular projective variety. Fix a, b, c E K*. Then there are only finitely many pairs of integers m, n : 2 (prime to p ifp #& 0) for which the Cassels-Catalan equation axm + byn c has even a single nonconstant solution x,y E K, x,y X k. Further, for any particular pair m, n as above, there will be only finitely many solutions x, y E K unless either: (i) a/c is an mth power and b/c is an nth power in K, in which case there may be infinitely many solutions (x, y) = (a(a/c)l/m, f(b/c)'/n) with a, 1 Ez k satisfying a m + /n = 1; or (ii) (m, n) E {(2,2), (2,3), (3,2), (2,4), (4,2), (3,3)), in which case the CasselsCatalan equation defines a curve of genus 0 or 1 over K. PROOF. We first note that taking "generic" hyperplane sections of the variety whose function field is K, we are reduced by Bertini's theorem to the case that K = k(C) is the function field of a nonsingular projective curve C. (See, e.g., [5] for the details of this standard reduction.) Second, we may replace k by its algebraic closure, since at worst this will create extra solutions. Third, dividing the equation by c and replacing a and b by a/c and b/c, we may assume that our equation is (*) axm + byn = 1. Let D = max{deg(a), deg(b)}. (Note a and b are now functions on the curve C, so have degrees.) By symmetry, we may assume that m > n. Received by the editors May 28, 1981 and, in revised form, July 24, 1981. 1980 Mathematics Subject Classification. Primary I OB 15; Secondary 1 OJ06, 14J25. ?1982 American Mathematical Society 0002-9947/81/0000-0631 /$02.25 201 This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 06:00:20 UTC All use subject to http://about.jstor.org/terms