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Stream: theory: mathematics

Topic: Categorifying the modularity theorem

John Baez (Apr 20 2024 at 18:25):

The Modularity Theorem was the hard part of proving Fermat's Last Theorem - though Wiles and Taylor only needed to prove a special case to deduce Fermat's Last Theorem, and the rest was done later. I certainly don't understand it. But now at least I understand the statement of it, because Bruce Bartlett just wrote a nice post on the n-Category Cafe explaining this:

• Bruce Bartlett, The Modularity Theorem as a bijection of sets.

I find his statement of it clearer than most, in part because most people don't phrase it as a bijection of sets and he does. (Also he very efficiently reviews the concepts required to understand what's going on.)

But one of these two sets he's describing a bijection between is manifestly a set of equivalence classes of objects in a category. Not isomorphism classes, interestingly: morphism classes. Still, this suggests that maybe at a deeper level the Modularity Theorem expresses an equivalence of categories, or at least a functor between categories.

And indeed, pondering it a bit, this seems to be true! I explained my idea in the comments.

John Baez (Apr 20 2024 at 18:31):

(An isogeny between elliptic curves is a surjective holomorphic homomorphism between them, where you regard them as abelian groups in the category of complex 1-manifolds. A funny fact about isogenies is that if there exists an isogeny $f: E \to E'$ then there exists an isogeny $g: E' \to E$. There is thus an equivalence relation on elliptic curves where $E \sim E'$ iff there exists an isogeny $f: E \to E'$. In this case we say $E$ and $E'$ are isogenous. I would like to understand this a bit more category-theoretically than people usually seem to bother to. E.g. I don't even know if the category of elliptic curves and isogenies is a [[dagger-category]].)

Julius Hamilton (Apr 21 2024 at 11:50):

Copied and pasted into my Obsidian knowledge base for further study :+1: