Abstract:We use recent advances in the local evaluation of Brauer elements to study the role played by odd torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over arbitrary number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check,… Show more
“…Theorem 5.5). This theorem proves what whose already claimed by Ieronymous after some computational evidences, see [18,Remark 2.6].…”
Section: Ordinary Case: Examplessupporting
confidence: 85%
“…Let p be a prime above 2 and O p be the valuation ring of k p . We have that 2 • α -1 ∈ O p if and only if α 2 ≡ 0 mod 8; we can define X α to be the O p -scheme defined by equation (18). If α 2 ≡ 0 mod 8, then X α is smooth and hence X α has good reduction at p. Theorem 7.2.…”
Following Bright and Newton, we construct an explicit K3 surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.
“…Theorem 5.5). This theorem proves what whose already claimed by Ieronymous after some computational evidences, see [18,Remark 2.6].…”
Section: Ordinary Case: Examplessupporting
confidence: 85%
“…Let p be a prime above 2 and O p be the valuation ring of k p . We have that 2 • α -1 ∈ O p if and only if α 2 ≡ 0 mod 8; we can define X α to be the O p -scheme defined by equation (18). If α 2 ≡ 0 mod 8, then X α is smooth and hence X α has good reduction at p. Theorem 7.2.…”
Following Bright and Newton, we construct an explicit K3 surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.