{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:52331"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:52331","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Difference equations with semisimple Galois groups in positive characteristic","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Maier, Annette"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Hartmann, Julia"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011","date_published":"2011","updated_at":"2026-07-30T19:40:50Z","subjects":["info:eu-repo/classification/ddc/510","Lineare Differenzengleichung","Frobenius-Endomorphismus","Galois-Theorie","Galois-Gruppe","Mathematik","Differenzen Galoistheorie","Differenzengleichungen","Frobeniusmoduln","Algebraische Gruppen","Klassische Gruppen","difference Galois theory","difference equations","Frobenius modules","algebraic groups","classical groups"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114564%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114564%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114564%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/52331","outbound_label":"Repository record","outbound_source":"dc:identifier"},"source_record":{"url":"https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A52331","prefix":"oai_dc"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Hartmann, Julia"]},{"key":"dc:creator","label":"Author","values":["Maier, Annette"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2011"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-39092"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/510","Lineare Differenzengleichung","Frobenius-Endomorphismus","Galois-Theorie","Galois-Gruppe","Mathematik","Differenzen Galoistheorie","Differenzengleichungen","Frobeniusmoduln","Algebraische Gruppen","Klassische Gruppen","difference Galois theory","difference equations","Frobenius modules","algebraic groups","classical groups"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/52331","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114564%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University 135 S. (2011). = Aachen, Techn. Hochsch., Diss., 2011"]},{"key":"dc:title","label":"Title","values":["Difference equations with semisimple Galois groups in positive characteristic"]}]}],"canonical_facts":{"dc:contributor":["Hartmann, Julia"],"dc:coverage":["DE"],"dc:creator":["Maier, Annette"],"dc:date":["2011"],"dc:description":["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."],"dc:identifier":["https://publications.rwth-aachen.de/record/52331","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114564%22"],"dc:language":["eng"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-39092"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University 135 S. (2011). = Aachen, Techn. Hochsch., Diss., 2011"],"dc:subject":["info:eu-repo/classification/ddc/510","Lineare Differenzengleichung","Frobenius-Endomorphismus","Galois-Theorie","Galois-Gruppe","Mathematik","Differenzen Galoistheorie","Differenzengleichungen","Frobeniusmoduln","Algebraische Gruppen","Klassische Gruppen","difference Galois theory","difference equations","Frobenius modules","algebraic groups","classical groups"],"dc:title":["Difference equations with semisimple Galois groups in positive characteristic"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:40:50Z"}