I’ve asked this question to quite a few people in person and so far haven’t seen a good answer... but I believe one should exist, so here goes!
Ok, we all know how to (roughly) prove Fermat’s Last Theorem:
Each solution to $a^p + b^p = c^p$ gives a Frey curve, an elliptic curve that is cleverly designed to encode the difference between trivial/non-trivial solutions geometrically (i.e. singular/non-singular Frey curve) and has a very special conductor.
One then invokes modularity in the non-trivial solution case to produce a modular form of a special level... which then is proved to not exist by level lowering arguments. Hence FLT is proved.
One of the current goals for many is to prove paramodularity, which in a nutshell should see certain abelian surfaces correspond to certain genus 2 Siegel modular forms of paramodular level.
The conjecture bas been made precise, for example see Conjecture 1.1.1 of this paper.
But anyway the details don’t matter too much. My question is the following naive one: if modularity is to FLT then paramodularity is to what?
Now I am not claiming that paramodularity is useless without Diophantine applications. However I feel it would be nice to know if it can be used to show there is a certain Diophantine equation (or even a family) that has no non-trivial solutions.
Perhaps there is a simple way to generalise the Frey curve construction to give an abelian surface which encodes the triviality/non-triviality of solutions for some Diophantine problem geometrically?
