Difference equations with semisimple Galois groups in positive characteristic

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.

Grantor dc:publisher

Publikationsserver der RWTH Aachen University

Year dc:date

2011

Author dc:creator

• Maier, Annette

Contributors dc:contributor

• Hartmann, Julia

• 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

Statement dc:rights

• info:eu-repo/semantics/openAccess

Language dc:language

eng

Identifier

https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114564%22

OAI record oai:identifier

https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A52331