Cubic forms, powers of primes and the Kraus method

Andrzej Dąbrowski, Tomasz Jędrzejak, and Karolina Krawciów. "Cubic forms, powers of primes and the Kraus method." Colloquium Mathematicae 128.1 (2012): 35-48. <http://eudml.org/doc/283983>.

@article{AndrzejDąbrowski2012,
abstract = {We consider the Diophantine equation $(x+y)(x²+Bxy+y²) = Dz^\{p\}$, where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).},
author = {Andrzej Dąbrowski, Tomasz Jędrzejak, Karolina Krawciów},
journal = {Colloquium Mathematicae},
keywords = {Diophantine equations; modular forms; elliptic curves; Galois representations},
language = {eng},
number = {1},
pages = {35-48},
title = {Cubic forms, powers of primes and the Kraus method},
url = {http://eudml.org/doc/283983},
volume = {128},
year = {2012},
}

TY - JOUR
AU - Andrzej Dąbrowski
AU - Tomasz Jędrzejak
AU - Karolina Krawciów
TI - Cubic forms, powers of primes and the Kraus method
JO - Colloquium Mathematicae
PY - 2012
VL - 128
IS - 1
SP - 35
EP - 48
AB - We consider the Diophantine equation $(x+y)(x²+Bxy+y²) = Dz^{p}$, where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).
LA - eng
KW - Diophantine equations; modular forms; elliptic curves; Galois representations
UR - http://eudml.org/doc/283983
ER -