On the equation
On the equation a³ + b³ⁿ = c²
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.
On the Euler-Poincaré characteristics of finite dimensional -adic Galois representations
On the existence of abelian varieties with good reduction.
On the existence of supersingular curves of given genus.
On the ...-factor attached to motives.
On the Finite Field Nullstellensatz for the intersection of two quadric hypersurfaces.
On the Fourier coefficients of modular forms. II.
On the Gauss maps of space curves in characteristic p
On the Gauss maps of space curves in characteristic p, II
On the generic injectivity of the Gauss map in positive characteristic.
On the Geometry of a Siegel Modular Threefold.
On the group of components of a Néron model.
On the group orders of elliptic curves over finite fields
On the Hasse principle for homogeneous spaces with finite stabilizers
On the height constant for curves of genus two
On the height constant for curves of genus two, II
On the Hodge-Tage decomposition in the imperfect residue field case.
On the Holomorphy on Certain Non-Abelian L-Series.
On the index theorem for -adic differential equations. III. (Sur le théorème de l'indice des équations différentielles -adiques. III.)