Fundamental Domains for Some Hilbert Modular Groups

Abstract

In this thesis, our main aim is to discuss about the fundamental domains for the action of the specific Hilbert modular groups SL₂(𝒪_K) on ℍ × ℍ, where ℍ is the complex upper half-plane and K is a real quadratic field. These domains generalize the action of the classical modular group SL₂(ℤ) on ℍ.

For the Hilbert modular case, 𝒪_K is the ring of integers of a field ℚ(√d) with d > 0, and the corresponding Hilbert modular group SL₂(𝒪_K) acts on the product ℍ × ℍ.

In the first chapters, we recall group actions, define the modular group SL₂(ℤ), and see some properties of this group. We then construct the classical fundamental domain ℱ for SL₂(ℤ).

Next, we introduce some basics of hyperbolic geometry and present some important theorems. Then, we introduce the definition of a Dirichlet domain and apply it to the modular group to conclude that for suitable choices, a Dirichlet domain for SL₂(ℤ) coincides with the classical fundamental domain ℱ.

After that, we study some basics of algebraic number theory, to describe the ring of integers 𝒪_K in ℚ(√d) and to study the group SL₂(𝒪_K).

Finally, we study the cases K = ℚ(√5) and K = ℚ(√2), explaining how the regions ℛ∞, ℛ₀ may be described in these class number 1 fields. We also illustrate how Cohn's restriction for ℛ₀ reduces the possible inequalities and gives a partial description of the Hilbert fundamental domain ℛ. We then state the closest cusp reduction algorithm to find explicitly elements in SL₂(𝒪_K) that move points of ℍ × ℍ into the reduced region ℛ.