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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
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References:
[1] 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
[2] C.D. Gonzalez-Aviles, Brauer groups and Tate-Shafarevich groups , J. Math. Sci. Univ. Tokyo 10 (2003), 391-419. · Zbl 1029.11026
[3] S. Lichtenbaum, The period-index problem for elliptic curves , Amer. J. of Math. 90 (1968), 1209-1223. · Zbl 0187.18602 · doi:10.2307/2373297
[4] 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
[5] 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
[6] J. Silverman, The arithmetic of elliptic curves , GTM 106 , Springer-Verlag. · Zbl 0585.14026
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