Difference equations with semisimple Galois groups in positive characteristic

  • Differenzengleichungen mit halbeinfachen Galoisgruppen in positiver Charakteristik

Maier, Annette; Hartmann, Julia (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2011, 2012)
Dissertation / PhD Thesis

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.

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