Fermat-type equations and modular curves

Wiles' groundbreaking proof of Fermat's Last Theorem initiated a new approach to tackle Diophantine equations called the modular method. In this talk we will discuss how modular curves and their rational points appear naturally when we try to apply generalizations of the modular method to solve new Diophantine problems, for example, the Fermat-type equation x^2 + y^3 = z^p.

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