# finiteness of the Brauer group for flat proper schemes over $\mathrm{Spec}\,\mathbf{Z}$

One fundamental conjecture on the Brauer group is that $\mathrm{Br}(X)$ is finite for $X/\mathrm{Spec}\,\mathbf{Z}$ proper. By class field theory (the theorem of Albert-Brauer-Hasse-Noether), this is true for $\dim{X} = 1$. For varieties $X$ over finite fields, the finiteness of $\mathrm{Br}(X)[\ell^\infty]$ is equivalent to the surjectivity of the cycle class map $\mathrm{Pic}(X) \otimes \mathbf{Z}_\ell \to \mathrm{H}^2(X,\mathrm{Z}_\ell(1))$ (the Tate conjecture in dimension $1$). The Tate conjecture in dimension $1$ over finite fields is known for smooth projective curves (trivial), Abelian varieties (Tate) and K3 surfaces (at least for characteristic $> 3$), and hence for products of such varieties.

Are there results on the finiteness (of an $\ell$-primary part) of the Brauer group of schemes flat and proper over $\mathrm{Spec}\,\mathbf{Z}$ of dimension $> 1$?

• For an elliptic curve, isn't this equivalent to finiteness of Sha, and thus aren't there partial results coming from the various partial results on BSD? – Will Sawin Nov 7 '17 at 15:05
• @WillSawin: This is at least equivalent for function fields. – user19475 Nov 7 '17 at 15:07
• Doesn't the same cohomological argument work? – Will Sawin Nov 7 '17 at 15:17
• @WillSawin: You are right, I have posted this as an answer. – user19475 Nov 7 '17 at 15:21
• @Timo Keller What motivates your question? The question is wide open in the flat proper case over $\text{Spec}(\mathbf{Z})$ as much as in the proper case over $\text{Spec}(\mathbf{Z})$, so you must have something specific in mind. – user95222 Nov 25 '17 at 4:51

Let $\mathscr{C}/X$ be a relative curve with a section, e.g. a relative elliptic curve. Then there is a short exact sequence $$0 \to \mathrm{Br}(X) \to \mathrm{Br}(\mathscr{C}) \to Ш(\mathbf{Pic}^0_{\mathscr{C}/X}/X) \to 0$$ (see https://www.timokeller.name/TateShafarevich.pdf, Theorem 4.27; for $\dim{X}=1$, this is a theorem of Grothendieck).
So let $X$ be the spectrum of the ring of integers of a number field and $\mathscr{C}/X$ a relative elliptic curve with finite Ш.
• This is true when $X$ is a smooth projective curve over a finite field by a Thm of Grothendieck, and if $X$ is smooth projective geometrically irreducible over a finite field by fairly elementary arguments, but it's not quite true if $X$ is the spectrum of the ring of integers of an algebraic number field $K$. $\text{Br}(\mathscr{C})$ should surject onto $H^1(X_{\rm\acute{e}t}, \text{Pic}^0_{\mathscr{C}/X})$ (actually onto its image into $H^1(X_{\rm\acute{e}t}, \text{Pic}_{\mathscr{C}/X})$, which is off from the Tate-Shafarevich group by a finite group of exponent $2$ if $K$ has a real place.) – user87684 Nov 25 '17 at 5:04