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On the equation a³ + b³ⁿ = c²

Michael A. Bennett;  Imin Chen;  Sander R. Dahmen;  Soroosh Yazdani — 2014

Acta Arithmetica

We study coprime integer solutions to the equation a³ + b³ⁿ = c² using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from ℚ-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.

Relations between jacobians of modular curves of level p 2

Imin Chen;  Bart De Smit;  Martin Grabitz — 2004

Journal de Théorie des Nombres de Bordeaux

We derive a relation between induced representations on the group GL 2 ( ℤ / p 2 ℤ ) which implies a relation between the jacobians of certain modular curves of level p 2 . The motivation for the construction of this relation is the determination of the applicability of Mazur’s method to the modular curve associated to the normalizer of a non-split Cartan subgroup of GL 2 ( ℤ / p 2 ℤ ) .

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