Isometries of Lattices and Automorphisms of K3 Surfaces

The paper’s Theorem 13.1 states that an even unimodular lattice with a semisimple isometry of characteristic polynomial F and Milnor index τ exists iff ε_τ=0, under (C1)–(C2), and proves it by assembling local data into a global Q[Γ]-form and then an even unimodular Z-lattice; this matches the candidate’s construction in outline and in key ingredients (local primary decompositions, Hasse–Witt bookkeeping via C0(I)→XF, trace forms from σ-hermitian spaces, and adelic intersection). The paper’s proof uses the equivariant Witt-group formalism and specific 2-adic input from [BT20], while the candidate appeals to classical Landherr/Hasse–Minkowski language; minor technicalities (notably the p=2 evenness and stability under t) are glossed in the candidate write-up but covered in the paper. Overall, both are correct and follow the same strategy, with the paper providing the missing technical details (e.g., the role of [BT20] and the construction of local lattices at p=2) that the model sketches. See Theorem 13.1 and the construction in the proof (hermitian forms q_f and the final intersection L=⋂L_p) ; definitions of XF and ε_τ are as in §§11–12 .