I would like to know the relation between the BSD conjecture for abelian schemes (as stated for example in T Keller's thesis) and the clasical BSD conjecture.
In particular, can one state the classical version from the general one..?
1 Answer
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The BSD conjecture for an Abelian variety $A$ over, say, a global field $K$ with model $X$ (the spectrum of the ring of integers of $K$ if $K$ is a number field and the smooth projective model of $K$ in the function field case) is (up to some modifications for the infinite places in the number field case) the BSD conjecture for the Néron model $\mathscr{A}/X$.
In https://www.timokeller.name/TateShafarevich.pdf, Theorem 4.4 and 4.5, is the comparison of $Ш(\mathscr{A}/X) := \mathrm{H}^1_\mathrm{et}(X,\mathscr{A})$ with the classical $Ш(A/K)$ defined as a kernel of Galois cohomology groups.
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$\begingroup$ You asked in your answer (before I edited it) how to create Ш in math formulas on this site. Just putting the Cyrillic letter Ш within a pair of dollar signs does the trick, producing $Ш$. $\endgroup$– KConradCommented Aug 21, 2017 at 20:29
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$\begingroup$ @KConrad: How does one "put" that character within a pair of dollar signs if one doesn't have a Cyrillic keyboard? Is there a more robust method? $\endgroup$– nfdc23Commented Aug 22, 2017 at 2:04
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$\begingroup$ @nfdc23, do it by the universal answer to lots of questions: use Wikipedia. Copy and paste a Ш from the Wikipedia page about Shafarevich (his name in Russian appears on the first line), the page about the Tate-Shafarevich group, or the page about the letter Ш itself (google "letter sha" and it should be the first hit). $\endgroup$– KConradCommented Aug 22, 2017 at 3:42