Benjamin Rosenfield
The solvability or unsolvability of Diophantine equations (like those appearing in Fermat's Last Theorem or the Congruent Number Problem) is basically unknowable (Hilbert's Tenth problem to find a general algorithm for solving Diophantine equations was proven impossible). So it is, almost provably, an ad hoc field. The method of infinite descent is oldest, and the local-global principle is aesthetically pleasing, but both fail in general. The "fanciest" ad hoc method is to translate the question into a question about an elliptic curve and then to try to find an answer using modular forms. We will mainly discuss the (mostly conjectural) interrelationship between arithmetic geometry and automorphic forms of which this is a special case.
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