On quartic Thue equations with trivial solutions

Abstract

Let K be a quartic number field with negative absolute discriminant and let L = Q ( d ) {\mathbf {L}} = {\mathbf {Q}}(\sqrt d ) be its real quadratic subfield, with d ≡ 3 ( mod 4 ) d \equiv 3\;\pmod 4 . Moreover, assume K to be embedded in the reals. Further, let ξ > 1 \xi > 1 generate the subgroup of units relative to L in the group of positive units of K. Under certain conditions, which can be explicitly checked, and for suitable linear forms X ( u , v ) X(u,v) and Y ( u , v ) Y(u,v) with coefficients in Z [ d ] {\mathbf {Z}}[\sqrt d ] , the diophantine equation \[ Norm K / Q ( X ( u , v ) + Y ( u , v ) ξ 2 ) = 1 , {\text {Norm}_{{\mathbf {K}}/{\mathbf {Q}}}}(X(u,v) + Y(u,v){\xi ^2}) = 1, \] which is a quartic Thue equation in the indeterminates u and v, has only trivial solutions, that is, solutions given by X Y = 0 XY = 0 . Information on a substantial number of equations of this type and their associated number fields is incorporated in a few tables.