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$\DeclareMathOperator{\gal}{Gal}$ Here's a comment which one can make to differential geometers which at least explains what etale cohomology "does". Given an algebraic variety over the reals, say a s …

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader kno …

Here is a completely different kind of answer to this question. A perfectoid space is a term of type PerfectoidSpace in the Lean theorem prover. Here's a quote from the source code: structure perfe …

Hey Bjorn. Let me try for a counterexample. Consider a hypersurface in projective $N$-space, defined by one degree 2 equation with integral coefficients. When is such a gadget smooth? Well the partial …

OK so let's see if I can use MO to explicitly compute an example of something, by getting other people to join in. Sort of "one level up"---often people answer questions here but I'm going to see if I …

This is still way open, I should think. "Elementary" methods won't even solve the analogous problem over $\mathbf{Z}$, so you need to use "modular form" methods. The problem is that even if the result …

This question came up recently on the NMBRTHRY mailing list and I can't resist paraphrasing a comment I made there. So, perhaps surprisingly, this question has links to automorphic forms! For if $x$ i …

Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details). IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has …

No there are not any mistakes in these papers of any interest. In the 1990s there were a bazillion study groups and seminars across the world devoted to these papers; I personally read all three of th …

Let $F$ be a real quadratic field and let $E/F$ be an elliptic curve with conductor 1 (i.e. with good reduction everywhere; these things can and do exist) (perhaps also I should assume E has no CM, ev …

For what it's worth, here are some historical comments. Both Birch and S-D spoke in Cambridge a few weeks ago, on the history of their conjecture. To my surprise, both of them emphasized the role not …

EDIT: Hendrik Lenstra emailed me a proof of Conjecture 2. I'll append it below. So Jagy's question is now solved. OK so I think that Jagy wants to make the following conjecture: CONJECTURE 1: an i …

I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt correc …

The Eichler-Shimura relation doesn't just prove the Hasse-Weil conjecture for modular curves. It e.g. attaches Galois representations to modular forms of weight 2. More delicate arguments (using etale …

Let me highlight some issues that Emerton doesn't: 1) you seem to hint that you don't know that modular forms can be viewed as a product of a bunch of local terms. So there is an adelic story, where …