The files in this folder belong to the paper 

THE CAYLEY-OGUISO AUTOMORPHISM OF POSITIVE ENTROPY ON A K3 SURFACE

by 

DINO FESTI, ALICE GARBAGNATI, BERT VAN GEEMEN, AND RONALD VAN LUIJK

In particular, we check several computations from Sections 4 and 5. 

The files *.mg can be loaded in magma by typing the command 

load "filename.mg";

The file picard.mg counts the points on the surface given by det M0 = 0; the result is in the file picard.out. These numbers are used in the proof of 5.4.

The file makeorbits.mg prints the first 1000 points in the orbits under g of each of the three points on S with coordinates 0,1,-1. In other words, for each of these three points x, there is a file orbitj (j=1,2,3), which contains the list of the 1000 points g^j(x) with 0 <= j < 1000. 

The files oguiso.mg, oguiso.out, oguiso_polys.mg, and oguiso_check.mg belong to Section 4.4. The file oguiso_polys.mg contains a magma function oguiso_polys() that takes as input a polynomial ring in four variables and returns four quadruples of homogeneous polynomials, each describing the automorphism g on an open subset of S, as described in Section 4.4. The file oguiso.mg shows how these four quadruples were computed by recomputing them and checking that they are equal to the ones given by oguiso_polys(R). The file oguiso.mg has many explanatory comments. The output of running the file oguiso.mg is in oguiso.out, which also shows that these computations take a few hours. The file oguiso_check.mg checks that the four quadruples given by oguiso_polys(R) actually define g on some open subset of S. 

The files oguiso_inv* do the same as oguiso*, except for the inverse of g instead of g itself. 

The file three_quadruples.mg checks the last remark of 4.4, namely that three quadruples of those given by oguiso_polys(R) suffice to define g everywhere. This is also checked for the quadruples given by oguiso_inv_polys(R) describing the inverse of g. 

The file periodic.mg constructs the scheme Xi as in 5.5, or rather its reduction over F_p as used in the proof of 5.6, and checks that this reduction has dimension 0, degree 344, and is reduced for p=17 and p=101. The output of running periodic.mg is in periodic.out. 

The file algpoints.mg reconstructs the scheme Xi over F_p as in periodic.mg (see periodic.mg for some explanatory comments) and counts the number of points over extension fields F_{p^m} for m<= 150.  It also rechecks the dimension, degree and reducedness. The prime can be set in the third line of the file. The results for p=2, p=17 and p=101 are in the files algpointsp.out.