It's no longer a conjecture, it's been proved so now it's a theorem.

I've put the original, extremely high-level overview in a more accurately named page, and the bickering about definitions, meanings, explanations and general arguments on an even better named page.

See
* TaniyamaShimuraWeilTheorem
* TaniyamaShimuraWeilArgument

If ever there is light instead of heat generated from the latter, perhaps it can be distilled into the former.

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CategoryMath