Cubic forms, powers of primes and the Kraus method

• Autorzy: Andrzej Dąbrowski , Tomasz Jędrzejak , Karolina Krawciów
• Języki publikacji: EN

Abstrakty

EN

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).

Kategorie tematyczne

• 11F80: Galois representations
• 11D41: Higher degree equations; Fermat's equation
• 11G05: Elliptic curves over global fields

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2012
• Tom: 128
• Numer: 1
• Strony: 35-48

Daty

wydano

2012

Twórcy

autor

Andrzej Dąbrowski

• Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland

autor

Tomasz Jędrzejak

• Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland

autor

Karolina Krawciów

• Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland

Identyfikatory

DOI

10.4064/cm128-1-5