2-adic point counting on \(K3\) surfaces
The paper studies the problem of counting rational points of a class of \(K3\)-surfaces \(S\) defined over the finite field \(\mathbb F_p\) for an odd primt \(p\). By applying the Lefschetz trace formula, this is reduced to a statement on \(2\)-adic étale cohomology of said surfaces. Using the estimate \[ \lvert \# S(\mathbb F_p) - (p^2 +16p+1)\rvert \leq 6p, \] the authors conclude that it suffices to calculate \(\# S(\mathbb F_p)\) modulo a sufficiently large integer. Previous works due to Harvey and others give an algorithm to compute \(\# S(\mathbb F_p)\) modulo prime powers \(p^j\). The authors explain that while this algorithm is fairly efficient for computing this number modulo \(p^1\), it is already one order of magnitude slower for the modulus \(p^2\). They introduce a novel method to compute \(\# S(\mathbb F_p)\) modulo \(16.\). Combined with the above algorithm, this yields the number \(\# S(\mathbb F_p)\) modulo \(16p\), more than enough in light of the above estimate. Their algorithm relies on a very remarkable divisibility property for traces in the \(2\)-adic orthogonal group. The computational aspects of the paper seem to be done very diligently, implementing not only the paper's main algorithms but also methods from the background literature for comparison and validation. The data obtained for the primes up to \(10^8\) should be a treasure to search for new patterns. It is to be expected that the paper's code and data will appear shortly on its authors' websites, publicly available to the interested reader.
scientific article; zbMATH DE number 7608398
scientific article