TY - JOUR
T1 - Klein forms and the generalized superelliptic equation
AU - Bennett, Michael A.
AU - Dahmen, Sander R.
PY - 2013
Y1 - 2013
N2 - If F(x; y) 2 Z[x; y] is an irreducible binary form of degree k ≥ 3, then a theorem of Darmon and Granville implies that the generalized superelliptic equation F(x, y) = zl has, given an integer l ≥ max {2, 7 - k}, at most finitely many solutions in coprime integers x; y and z. In this paper, for large classes of forms of degree k = 3; 4; 6 and 12 (including, heuristically, "most" cubic forms), we extend this to prove a like result, where the parameter l is now taken to be variable. In the case of irreducible cubic forms, this provides the first examples where such a conclusion has been proven. The method of proof combines classical invariant theory, modular Galois representations, and properties of elliptic curves with isomorphic mod-n Galois representations.
AB - If F(x; y) 2 Z[x; y] is an irreducible binary form of degree k ≥ 3, then a theorem of Darmon and Granville implies that the generalized superelliptic equation F(x, y) = zl has, given an integer l ≥ max {2, 7 - k}, at most finitely many solutions in coprime integers x; y and z. In this paper, for large classes of forms of degree k = 3; 4; 6 and 12 (including, heuristically, "most" cubic forms), we extend this to prove a like result, where the parameter l is now taken to be variable. In the case of irreducible cubic forms, this provides the first examples where such a conclusion has been proven. The method of proof combines classical invariant theory, modular Galois representations, and properties of elliptic curves with isomorphic mod-n Galois representations.
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U2 - 10.4007/annals.2013.177.1.4
DO - 10.4007/annals.2013.177.1.4
M3 - Article
AN - SCOPUS:84872668672
SN - 0003-486X
VL - 177
SP - 171
EP - 239
JO - Annals of Mathematics
JF - Annals of Mathematics
IS - 1
ER -