The Brauer group of Kummer surfaces and torsion of elliptic curves (Q2890362)

The main object of the paper under review is a Kummer surface \(X\) constructed from an abelian surface \(A\). The authors' goal is to compute the Brauer group of \(X\), with an eye towards using it, in the case where the ground field \(k\) is a number field, for the study of arithmetic properties of \(X\), such as the Hasse principle and weak approximation, one of obstructions to which, the Brauer-Manin obstruction, lives in \(\mathrm{Br}(X)\). Here is a list of some of their general results:NEWLINENEWLINE-- over an algebraic closure, there is an isomorphism of Galois modules NEWLINE\[NEWLINE\mathrm{Br}(\bar X)\cong \mathrm{Br}(\bar A);NEWLINE\]NEWLINENEWLINENEWLINE-- when \(A\) is a product of elliptic curves, the algebraic part \(\mathrm{Br}_1(X)\) of \(\mathrm{Br}(X)\) is often trivial (equals \(\mathrm{Br}(k)\));NEWLINENEWLINE-- for \(n\)-torsion parts, there is an inclusion of transcendental parts NEWLINE\[NEWLINE\mathrm{Br}(X)_n/\mathrm{Br}_1(X)_n\subset \mathrm{Br}(A)_n/\mathrm{Br}_1(A)_n,NEWLINE\]NEWLINE which is an equality for odd \(n\);NEWLINENEWLINE-- the latter group is expressed in more explicit terms in the case where \(A\) is a product of elliptic curves;NEWLINENEWLINE-- for some Kummer surfaces without rational points, arising from twists of abelian surfaces, it is shown that the algebraic Brauer group is zero.NEWLINENEWLINEA number of interesting results are produced for \(k=\mathbb Q\):NEWLINENEWLINE-- many pairs \(E,E'\) of elliptic curves are found for which the transcendental part of the Brauer group of \(A=E\times E'\) is small (0 or a finite abelian 2-group); for instance, the first case occurs if \(A=E\times E\) and for all primes \(\ell\) the image of the Galois representation on \(E_\ell\) is \(GL(2,\mathbb F_ \ell )\) (this happens for most elliptic curves over \(\mathbb Q\), according to \textit{W.~Duke} [C. R. Acad. Sci., Paris, Sér. I, Math. 325, No. 8, 813--818 (1997; Zbl 1002.11049)]);NEWLINENEWLINE-- many Kummer surfaces with trivial Brauer groups are produced; it is shown that most of the surfaces of the form \(Kum(E\times E')\) enjoy this property;NEWLINENEWLINE-- exhibited are Kummer surfaces whose Brauer group contains a transcendental element of prime order \(\ell\leq 13\).NEWLINENEWLINESome results on cohomology of abelian varieties, appearing in the course of the proofs, are interesting in their own rights.