**Elliptic curves over Q(i) are potentially automorphic**

This is potentially big news! Announced by Richard Taylor (yes,

*that*Richard Taylor) at the Joint Mathematics Meeting earlier this month.

https://totallydisconnected.wordpress.com/2017/01/09/elliptic-curves-over-qi-are-potentially-automorphic/

I'm interested in knowing what the property of being automorphic means, in terms similar to what it means to be modular, if that's possible.

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- I'd indeed call this "nothing will happen" - for now (just like in that update - "Potential für neue Verfahren" because news like this raise interest in finding other constructs, and not more than that).

Cryptographers will probably start finding working alternate group constructions soon (and not "just" try to find practical quantum-safe cryptography as before) - not that there's any urgency there right now. I simply expect ECs to be replaced by a "better" construct long before EC are broken (so no "Cryptocalypse" at all).46w - Is it possible to explain in that case what is the big deal about this result? I don't mean a technical explanation -- just something like "This was the next stage in such-and-such a programme and people had been stuck on it for fifteen years," or whatever actually applies in this case.46w
- +Timothy Gowers The reply in Fefe's post that Lutz referenced explains something there. So EC have "some regularity" over Q, Q[sqrt(something)], and there were some partial results for EC over R, and now there's this for Q[i] (which isn't surprising, given this is almost a special case of Q[sqrt(something)]).

It's not much for itself - but the more such regularities are found, the more fragile/the less trustworthy EC over Z[p] appear. Note that Z[p] is the only one here that's finite, so there's that.46w - I'd like to see an explanation analogous to the case of when (a class of) elliptic curves are modular. Does this new result mean that there is an automorphic representation such that [something to do with the curves]? Much like when there is a Galois rep such that [blah] for the modularity theorem.46w
- @Timothy Gowers: "This was the next stage in such-and-such a programme and people had been stuck on it for fifteen years" - this is an excellent short summary of why people like me are excited about this result. This result looked completely hopeless even 5-6 years ago, but then Calegari-Geraghty and Caraiani-Scholze introduced some wonderful new ideas into the picture, and the ten authors mentioned on my blog assembled them with some other ideas to prove this great theorem.46w
- +Rudolf Polzer Yes, the fact that Z[p] is finite is crucial for cryptography. Not only because that way we can actually implement it (still no way to store infinite precision ;-), it also changes the situation for proofs.

A lot of things become easy to do, once you have infinite information or infinite amount of time/computation power. In reality, we never have that, and that includes real quantum computers, which is why post-quantum doesn't frighten me that much, either.46w - +Bernd Paysan Indeed. I don't see post-quantum as frightening either - there are already existing post-quantum cryptosystems, their only issue is that they take large processing time on commodity hardware and/or use very long keys (in the 10 to 100 kilobits range). Certainly not so bad that we can't use it at all if we have to.46w

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