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A relation between the Brauer group and the Tate-Shafarevich group. (English) Zbl 1302.11039
Let $$K$$ be a number field or a function field, and $$C$$ a smooth geometrically connected curve over $$K$$. For each place $$v$$ of $$K$$ let $$\delta_v$$ denote the local index of $$C_{K_v}$$, and $$\delta$$ the index of $$C$$. Let $$A$$ denote the Jacobian variety of $$C$$, $$\text Ш(A)$$ the Tate-Shafarevich group of $$A$$, and $$\mathrm{Br}(C)$$ the Brauer group of $$C$$. Define $\mathrm{Br}(C)':=\mathrm {ker}\big[ \mathrm{Br}(C) \to \oplus_v \mathrm{Br}(C_{K_v})\big].$ Motivated by the works of Grothendieck and of Artin, J. S. Milne [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 735–743 (1981; Zbl 0503.14010)] conditionally extended Artin’s result and refined Grothendieck’s exact sequence as follows: suppose that $$\delta_v=1$$ for all $$v$$. If $$\text Ш(A)$$ has no nonzero infinitely divisible elements, then we have an exact sequence $0 \to \mathrm{Br}(C)' \to \text Ш(A)/T_1 \to T_2 \to 0$ where $$\# T_1 = \delta$$ and $$\#T_2 = \delta/p^r$$ where $$p$$ is the characteristic of $$K$$ and $$p^r=1$$ if $$K$$ is a number field. If one of $$\text Ш(A)$$ or $$\mathrm{Br}(C)'$$ is finite, then so is the other, and $$\delta^2\cdot\#\mathrm{Br}(C)' = p^r\cdot\#\text Ш(A)$$.
Motivated by P. L. Clark [J. Reine Angew. Math. 594, 201–206 (2006; Zbl 1097.14024)], here the author considers a number field $$K$$, a place $$w$$ and, a curve $$C$$ of genus $$1$$ that corresponds to an element in $$\ker\big[ \mathsf{H}^1( K,E) \to \oplus_{v\neq w}\mathsf{H}^1(K_v,E)\big]$$ where $$E$$ is the Jacobian of $$C$$, which is an elliptic curve. Hence, $$\delta_v=1$$ for all $$v$$ except for $$v=w$$. Let $$d$$ and $$d_w$$ be the indices of $$C$$ and $$C_{K_w}$$. The author obtains an exact sequence for $$C$$ same as Milne’s, and proves
$(d/d_w)^2\cdot\#\mathrm{Br}(C)' = \#\text Ш(E)$ provided that one of $$\text Ш(E)$$ or $$\mathrm{Br}(C)'$$ is finite. In fact, C. D. Gonzalez-Avilez obtains a more general result in [J. Math. Sci., Tokyo 10, No. 2, 391–419 (2003; Zbl 1029.11026)] where the conditions on $$\delta_v$$’s are (1) the local indices are equal to the local periods and (2) the local indices are pairwise relatively prime.
Let $$B$$ denote the subgroup of $$\mathrm{Br}(C)$$, $\mathrm{ker}\big[ \mathrm{Br}(C) \to \oplus_v \mathsf{H}^1(K_v,\mathrm{Pic}(C_{\bar K_v}))\big].$ Additionally, the author obtains a result on Brauer-Manin obstruction: If $$\text Ш(E)$$ is finite, then $\left( \prod_{v\neq w} C(K_v) \right)^B \neq \emptyset\quad \text{ if and only if} \quad d/d_w = 1.$
##### MSC:
 11G05 Elliptic curves over global fields 11G35 Varieties over global fields
##### Keywords:
Brauer group; Tate-Shafarevich group
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##### References:
  P. Clark, There are genus one curves of every index over every number field. J. Reine Angew. Math. 594 (2006), 201-206. · Zbl 1097.14024 · doi:10.1515/CRELLE.2006.040  C.D. Gonzalez-Aviles, Brauer groups and Tate-Shafarevich groups , J. Math. Sci. Univ. Tokyo 10 (2003), 391-419. · Zbl 1029.11026  S. Lichtenbaum, The period-index problem for elliptic curves , Amer. J. of Math. 90 (1968), 1209-1223. · Zbl 0187.18602 · doi:10.2307/2373297  S. Lichtenbaum, Duality theorems for curves over p-adic fields , Inventiones Math. 7 (1969), 120-136. · Zbl 0186.26402 · doi:10.1007/BF01389795 · eudml:141956  J. Milne, Comparison of the Brauer group with the Tate-Shafarevich group , J. Fac. Science, Univ. Tokyo, Sec. IA V. Scharaschkin, The Brauer-Manin obstruction for curves . · Zbl 0503.14010  J. Silverman, The arithmetic of elliptic curves , GTM 106 , Springer-Verlag. · Zbl 0585.14026
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