For an abelian variety $A$ over a global field $K$, $\mathcal{A}$ its Néron model on $C$, either a smooth projective geometrically irreducible curve over a finite field, or the spectrum of the ring of integers of a number field, is it true that:

$$\text{H}^2(A_{ét},\mathbf{G}_m)/\text{H}^2(K,\mathbf{G}_m) = \text{H}^2(\mathcal{A}_{et},\mathbf{G}_m)/\text{H}^2(C_{et},\mathbf{G}_m)\ ?$$

In other words, is the following identity between cohomological Brauer groups true?

$$\text{Br}(A)/\text{Br}(K) = \text{Br}(\mathcal{A})/\text{Br}(C).$$

  • 1
    $\begingroup$ Trivial remark: In the function field case, $\mathrm{Br}(C) = 0$ and it is determined by the real places in the number field case, so $(\mathbf{Z}/2)^{s-1}$, both by Albert-Brauer-Hasse-Noether, which also calulates $\mathrm{Br}(K)$. I don't know anything about $\mathrm{Br}(A)$ and $\mathrm{Br}(\mathcal{A})$. $\endgroup$ – TKe Jan 13 '18 at 6:28
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    $\begingroup$ Yes, but this would help only upon analyzing the edge maps for the Leray spectral sequence with $E_2^{ij} = H^i(C_{ét},R^jf_*\mathbf{G}_m)$ and its generic counterpart. In other words, the question asks if the image of $H^2(\mathcal{A}_{ét},\mathbf{G}_m)\to H^1(C_{ét}, \text{Pic}_{\mathcal{A}/C}^0)$ (which is sometimes an isomorphism) agrees with the image of $H^2(A_{ét},\mathbf{G}_m)\to H^1(K, \text{Pic}_{A/K}^0)$. $\endgroup$ – user95222 Jan 13 '18 at 6:32

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