Linear differential equations with finite differential Galois group

For a finite irreducible subgroup H⊂PSL(Cn) and an irreducible, H-invariant curve Z⊂P(Cn) such that C(Z)H=C(t), a standard differential operator Lst∈C(t)[d/dt] is constructed. For n=2 this is essentially Klein's work. For n>2 an actual calculation of Lst is done by computing an evaluation of invariants C[X1,…,Xn]H→C(t) and applying a scalar form of a theorem of E. Compoint in a “Procedure”. Also in some cases where Z is unknown evaluations are produced. This new method is tested for n=2 and for three irreducible subgroups of SL3. This supplements [18]. The theory developed here relates to and continues classical work of H.A. Schwarz, G. Fano, F. Klein and A. Hurwitz

Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page