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Kummer surfaces and the computation of the Picard group. (English) Zbl 1297.14043

The authors test the method from R. van Luijk [Algebra Number Theory 1, No. 1, 1–17 (2007; Zbl 1123.14022)] to determine the geometric Picard number of surface \(X\) defined over a number field.
Van Luijk’s method uses the following observations: If \(\mathfrak{p}\) is a prime of good reduction for \(X\) then the geometric Picard number of the reduction \(X_{\mathfrak{p}}\) of \(X\) is at least the geometric Picard number of \(X\). The geometric Picard number of \(X_{\mathfrak{p}}\) can be bounded in terms of the zeta function of \(X_{\mathfrak{p}}\). If the Tate conjecture holds for the reduction of \(X\) then the geometric Picard number of \(X_{\mathfrak{p}}\) equals the above mentiond upper bound and has the same parity as \(h^2(X)\). However, the Picard number of \(X\) does not satisfy such parity statement in general. By considering the intersection pairing on the Néron-Severi group of reductions modulo different primes one may be able to deduce that the Picard number of \(X\) is strictly less then the Picard number of \(X_{\mathfrak{p}}\).
Hence in order to assure that Van Luijk’s method yields the Picard number of \(X\) we need to find at least two primes of good reduction such that \(\rho(X_{\mathfrak{p}})-\rho(X)\leq 1\).
In this paper the authors verify numerically that Van Luijk’s method works for many Kummer surfaces.
F. Charles [Algebra Number Theory 8, No. 1, 1–17 (2014; Zbl 1316.14069)] describes for which \(K3\) surface there exists at least two primes of good reduction such that \(\rho(X_{\mathfrak{p}})-\rho(X)\leq 1\).

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14F20 Étale and other Grothendieck topologies and (co)homologies
14G05 Rational points
11G35 Varieties over global fields
14C22 Picard groups
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