Lectures on Shimura Curves 4.5: a Crash Course on Linear Algebraic Groups

Lectures on Shimura Curves: Arithmetic Fuchsian Groups

The class of Fuchsian groups that we are (by far) most interested in are the arithmetic groups. The easiest way to describe arithmetic Fuchsian groups is that class of groups containing all groups commensurable with PSL2(Z) and also all groups which uniformize compact Shimura curves. This is however, not a very wellmotivated definition: structurally, what do classical modular curves and Shimura...

Three Lectures on Shimura Curves

These notes are taken from a lecture series in three parts given at the University of Sydney in April 2006. In the rst part, we introduce Shimura curves for the non-expert, avoiding technicalities. In the other two sections, we discuss the moduli interpretation of Shimura curves and treat some of the computational aspects. Ever wondered what a Shimura curve is? They lie at the crossroads of man...

Summer School entitled “Harmonic Analysis, the Trace Formula, and Shimura

The main goal of the School was to introduce graduate students and young mathematicians to three broad and interrelated areas in the theory of automorphic forms. Much of the volume is comprised of the articles of Arthur, Kottwitz, and Milne. Although these articles are based on lectures given at the school, the authors have chosen to go well beyond what was discussed there, in order to provide ...

Linear Algebraic Groups: a Crash Course

Lectures on Shimura Curves 3: More Fuchsian Groups

Proof: Writing Y (Γ) = Γ\H, it is natural to ask (as we did in the “parabolic” case C/Λ) which elements σ ∈ PSL2(R) = Aut(H) descend to give automorphisms of Y (Γ). The condition we need is clearly that for all z ∈ H and γ ∈ Γ, there exists γ′ ∈ Γ such that σ(γz) = γ′(σz). Of course this holds for all z ⇐⇒ σγ = γ′σ; in other words the condition is that σΓσ−1 = Γ, i.e., that σ ∈ N(Γ). This defin...