On second order linear differential equations with few transcendental solutions (Q1081641)

Let k be a differential field of characteristic 0 and with algebraically closed field C, L a linear homogeneous differential equation of order 2 with coefficients in k, and G(L) the differential Galois group of ''the'' Picard-Vessiot extension generated by a fundamental system of solutions of L. We give necessary and sufficient conditions for G(L) to be a subgroup of the group SD(2,C) of diagonal \(2\times 2\)-matrices of determinant 1 (i.e. \(G_ m\) as an algebraic group over C), and derive the general 2nd order equation with this group. The conditions are exploited (i) to show that equations with polynomial coefficients have larger Galois groups except in trivial cases; (ii) to give an explicit and geometrical description of the set of those hypergeometric differential equations L having \(G(L)=SD(2,C)\).