BSD for a Large Class of Elliptic Curves

28 Jul

A Mind for Madness

I’m giving up on the p-divisible group posts for awhile. I would have to be too technical and tedious to write anything interesting about enlarging the base. It is pretty fascinating stuff, but not blog material at the moment.

I’ve been playing around with counting fibration structures on K3 surfaces, and I just noticed something I probably should have been aware of for a long time. This is totally well-known, but I’ll give a slightly anachronistic presentation so that we can use results from 2013 to prove the Birch and Swinnerton-Dyer conjecture!! … Well, only in a case that has been known since 1973 when it was published by Artin and Swinnerton-Dyer.

Let’s recall the Tate conjecture for surfaces. Let $latex {k}&fg=000000$ be a finite field and $latex {X/k}&fg=000000$ a smooth, projective surface. We’ve written this down many times now, but the long exact sequence associate to the Kummer sequence

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