On pg. 1 of the slides of a talk, Henri Darmon wrote:

Question:What is an interesting Diophantine equation?

A “working definition”. A Diophantine equation is interesting if it reveals or suggests a rich underlying mathematical structure.

One can adopt the perspective that the interesting elliptic curves over $\mathbb{Q}$ are those that are modular and view the fact that *all* elliptic curves over $\mathbb{Q}$ are modular as being of minor significance (in the sense that if some weren't, they wouldn't be so interesting).

I realize that my raising this perspective may come across as an affront to some of the celebrated research of recent times, and would hasten to emphasize that I'm asking my questions here in good faith with a view toward learning more.

What would we lose if we decided to focus only on those Galois representations that are attached to automorphic forms and ignore the possibility that some do not?

One thing that one would lose is Wiles' proof of Fermat's last theorem. Until recently, my attitude had been that the Frey Curve construction is a curiosity and that a "morally right" proof would come from the $abc$-conjecture applied to sufficiently large exponents together with the theory of arithmetic of cyclotomic fields to rule out the possibility of nontrivial solutions to the equations with smaller exponents. However, recently I came across slides from a talk by Minhyong Kim in which Kim wrote (pg. 29):

The idea of encoding points into 'larger' geometry is a common one in Diophantine geometry, as when solutions

$a^n + b^n = c^n$

to the Fermat equation are encoded into the elliptic curves

$y^2 = x(x - a^n)(x + b^n)$.

The geometry of the path torsor $\pi_1(X(\mathbb{C}); b, x)$ is an extremely canonical version of this idea.

I find the idea that the Frey curve construction is canonical to be fascinating! It raises the possibility that the proof of Fermat's last theorem coming from the study of Frey equations is morally right. [**Edit:** As KristianJS aptly points out, I misread Kim's quote. So I'd recur to my remark above about my impression on what a "morally right" proof of Fermat's Last Theorem would look like.]

Anyway, I'd be very interested in further examples concerning the significance of *all* L-functions attached to Galois representations (of suitable type) being modular.

[**Added:** If I remember correctly, In "The Map of My Life" Shimura wrote that he was more interested in the fact that suitable cusp forms correspond to elliptic curves than in the converse. This seems relevant. However, I cannot find the quotation and may be misremembering. I would welcome a reference from anybody who remembers this.]

[**Added**: I just asked another question that touches on material that may provide a partial answer to this question.]

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