Take an abelian surface with an involution with \(16\) isolated fixed points, then the desingularization of the quotient is a \(K3\) surface which is called Kummer surface associated to the abelian surface, with Picard rank 17, due to the 16 \((-2)\)-curves obtained as the resolution of the 16 fixed isolated points. V. V. Nikulin [Math. USSR, Izv. 9, 261–275 (1976; Zbl 0325.14015)] proved that a \(K3\) surface containing 16 disjoint smooth rational curves is a Kummer surface associated to an abelian surface. If \(X\) is a \(K3\) surface, a Kummer structure on \(X\) is an abelian surface \(A\) (up to isomorphism) such that \(X \cong \mathrm{Kum}(A)\). A Nikulin configuration is a set of \(16\) disjoint rational curves on \(X\). By Nikulin we have a bijecition between Kummer structures and Nikulin configurations up to automorphisms.
A motivational question is the one asked by T. Shioda [J. Fac. Sci., Univ. Tokyo, Sect. I A 24, 11–21 (1977; Zbl 0406.14023)]: is it possible to have two non-isomorphic abelian surfaces \(A\) and \(B\) such that Kum\((A)\) and Kum\((B)\) are isomorphic? T. Shioda and N. Mitani [Lect. Notes Math. 412, 259–287 (1974; Zbl 0302.14011)] give a negative answer to this question, later V. Gritsenko and K. Hulek [Math. Proc. Camb. Philos. Soc. 123, No. 3, 461–485 (1998; Zbl 0930.11028)] answer positively to this question taking into consideration an abelian surface \(A\) and its dual \(\hat{A}\). In [Math. Ann. 373, No. 1–2, 597–623 (2019; Zbl 1411.14044)] the authors exhibit the first geometric construction of two distinct Kummer structures, more precisely they give explicit examples of Nikulin configurations \(C\) and \(C'\) on some \(K3\) surface \(X\) such that the abelian surfaces \(A\) and \(A'\) associated to these configurations are not isomorphic. This construction was done for generic Kummer surfaces, such that the orthogonal complement of the \(16\) rational curves in \(C\) was generated by a class \(L\) such that \(L^{2}=2k(k+1)\) for some integer \(k\). The purpose of this paper is to extend this result for other Kummer surfaces. Let \(t \in \mathbb{N}^{*}\) and let \(X\) be a general Kummer surface with Nikulin configuration \(C\) such that the orthogonal complement of the \(16\) \((-2)\)-curves in \(C\) is generated by \(L\) with \(L^{2}=4t\). A class \(C\) of the form \(C=\beta L - \alpha A_1\) with \(\beta \in \mathbb{N}^{*}\) has self intersection \(-2\) if and only if the coefficients \((\alpha, \beta)\) satisfy the Pell-Fermat equation \(\alpha^{2}-2t \beta^{2}=1\). There is a non-trivial solution if and only if \(2t\) is not a square. In this case there is a so called fundamental solution \((\alpha_0, \beta_0)\) and the main result of the paper (Theorem 1) gives a necessary and sufficient condition, in the case \(\beta_0\) is even, to have the same Kummer structure on the Kummer surface \(X\). This condition involves the solution of the Pell-Fermat equation and the authors also find a geometric construction for \(X\) in the case in which the Pell-Fermat equation has some solutions. As a consequence finding values of the parameters such that the Pell-Fermat equation has no solutions gives examples of two distinct Kummer structures. Finally the authors give a proof as a consequence of a result of P. Stellari [Math. Z. 256, No. 2, 425–441 (2007; Zbl 1138.18006)] of the fact that if \(\mathrm{Kum}(A) \cong\mathrm{Kum}(B)\) then \(A\) and \(B\) are isogenous. Moreover they dedicate part of the paper showing why \(\beta_0\) odd does not work in Theorem 1.