h1

% IMPORTANT: The following is UTF-8 encoded.  This means that in the presence
% of non-ASCII characters, it will not work with BibTeX 0.99 or older.
% Instead, you should use an up-to-date BibTeX implementation like “bibtex8” or
% “biber”.

@PHDTHESIS{Maier:52331,
      author       = {Maier, Annette},
      othercontributors = {Hartmann, Julia},
      title        = {{D}ifference equations with semisimple {G}alois groups in
                      positive characteristic},
      address      = {Aachen},
      publisher    = {Publikationsserver der RWTH Aachen University},
      reportid     = {RWTH-CONV-114564},
      pages        = {135 S.},
      year         = {2011},
      note         = {Prüfungsjahr: 2011. - Publikationsjahr: 2012; Aachen,
                      Techn. Hochsch., Diss., 2011},
      abstract     = {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 "smallest"
                      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.},
      keywords     = {Lineare Differenzengleichung (SWD) /
                      Frobenius-Endomorphismus (SWD) / Galois-Theorie (SWD) /
                      Galois-Gruppe (SWD)},
      cin          = {115210 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)115210_20140620$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      urn          = {urn:nbn:de:hbz:82-opus-39092},
      url          = {https://publications.rwth-aachen.de/record/52331},
}