Stark-Heegner points on elliptic curves over number fields

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My research focuses on equations a special kind called elliptic curves. These equations arise in many branches of pure and applied mathematics. Due to their special structure, elliptic curves have recently found numerous important applications in the field of number theoretic cryptography. I am currently attempting to develop systematic methods for solving the equations defining elliptic curves. The key to these methods are a family of results (and conjectures) called modularity theorems, the prototype of which was proved in the 1990s by Andrew Wiles in his landmark proof of Fermat's Last Theorem.