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%0 Thesis
%A Maier, Annette
%T Difference equations with semisimple Galois groups in positive characteristic
%C Aachen
%I Publikationsserver der RWTH Aachen University
%M RWTH-CONV-114564
%P 135 S.
%D 2011
%Z Prüfungsjahr: 2011. - Publikationsjahr: 2012
%Z Aachen, Techn. Hochsch., Diss., 2011
%X Let F be a field with an automorphism sigma on F.  A (linear) difference equation over F is an equation of the form sigma(y)=Ay with A in GL_n(F) and y a vector consisting of n indeterminates. There is the notion of a Picard-Vessiot ring which is in some sense a ßmallest" difference ring extension R of F such that there exists a full set of solutions with entries in R to the given difference equation. If there exists a Picard-Vessiot ring, one can assign a difference Galois group to the Picard-Vessiot ring, which turns out to be a linear algebraic group (in the scheme theoretic sense). Let F = F_q(s,t) with sigma defined to be the automorphism that fixes F_q(t) pointwise and maps s to s^q. The main result of this thesis is that the following groups occur as difference Galois groups over F: the special linear groups SL_n, the symplectic groups Sp_2d, the special orthogonal groups SO_n (here we have to assume q odd), and the Dickson group G_2 (in both cases q odd and even). We give explicit difference equations for all of these groups. As another result, we show that every semisimple and simply-connected group G that is defined over F_q occurs as a difference Galois group over F_(q^i)(s,t) for some i, where now sigma(s)=s^(q^i). Let F_q(s)' denote an algebraic closure of F_q(s). We can lift our difference equations from F_q(s,t) to F_q(s)'(t) using the fact that all of our constructed Galois groups are connected. As a result we obtain rigid analytically trivial pre-t-motives with the same Galois groups. The category of rigid analytically trivial pre-t-motives contains the category of t-motives, which occurs in the arithmetic of function fields.
%K Lineare Differenzengleichung (SWD)
%K Frobenius-Endomorphismus (SWD)
%K Galois-Theorie (SWD)
%K Galois-Gruppe (SWD)
%F PUB:(DE-HGF)11
%9 Dissertation / PhD Thesis
%U https://publications.rwth-aachen.de/record/52331