Shabat-Blaschke products

Abstract

The aims of this thesis are to introduce a hyperbolic analogue of Grothendieck's dessins d'enfant, to give arithmetic properties of the coefficients of the Chebyshev-Blaschke products, and to prove some Landen-type identities for theta functions.

In 1979, Belyi proved that a connected compact Riemann surface $X$ is defined over the field $\overline{\mathbb{Q}}$ of algebraic numbers if and only if there exists a Belyi map, i.e. a nonconstant holomorphic map $f:X\rightarrow \widehat{\mathbb{C}}$ with at most $3$ critical values in the Riemann sphere $\widehat{\mathbb{C}}$. Inspired by Belyi's theorem, Grothendieck introduced the theory of dessin d'enfant in the hope of a better understanding of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. The dessins d'enfant and the related objects called the transitive monodromies are discrete combinatorial objects that determine uniquely the Belyi maps up to some equivalence. By introducing a Galois action on the dessins d'enfant, the structure of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ can then be revealed from the combinatorial properties of the dessins d'enfant. Special examples of dessins d'enfant are trees, and the Belyi maps that correspond to them are Shabat polynomials, i.e. polynomials with at most two finite critical values.

In chapter 2 of this thesis, we will establish a hyperbolic analogue of dessins d'enfant by replacing the Riemann sphere $\widehat{\mathbb{C}}$ with three marked points by the open unit disk $\mathbb{D}$ with two marked points. However, in this analogue the hyperbolic Belyi maps constructed from a given transitive monodromy are deformed, i.e. they depend on the hyperbolic distance between the two marked points under the Poincar\'{e} metric. Moreover, we will establish a hyperbolic analogue of Shabat's correspondence. In fact, the monodromies that correspond to a tree will also correspond to finite Blaschke products with at most two critical values in $\mathbb{D}$, and such finite Blaschke products will be referred to as Shabat-Blaschke products. We will also introduce and study the size of the hyperbolic dessin d'enfant of a Shabat-Blaschke product.

It is natural to ask if there is a hyperbolic analogue of Belyi's theorem when one replaces the Riemann sphere $\widehat{\mathbb{C}}$ by the open unit disk $\mathbb{D}$. To formulate such a result, we need to know what should be used to replace $\overline{\mathbb{Q}}$. Since the Chebyshev polynomials are examples of Belyi maps and their coefficients are integers, we are motivated to prove a hyperbolic analogue of this statement in Chapter 3. Chebyshev-Blaschke products, which are hyperbolic analogues of Chebyshev polynomials, were studied by Ng, Tsang and Wang recently. The Chebyshev-Blaschke products are examples of Shabat-Blaschke products. We prove that the Chebyshev-Blaschke products are defined over $$\mathbb{Z}\left[\sqrt{k},\sqrt{k\circ s_n}, \frac{\omega_1\circ s_n}{\omega_1}\right]\subseteq \overline{\mathbb{Q}(j)},$$ where $n$ is the degree of the Chebyshev-Blaschke product, $s_n$ is the scaling by $n$, $k$ and $\omega_1$ are defined in terms of Jacobi theta functions, and $j$ is the $j$-invariant. We also prove that the Chebyshev-Blaschke products are defined over $\mathbb{Z}[[q^{1/4}]]$, the ring of power series in $q^{1/4}$ over $\mathbb{Z}$, where $q=e^{2\pi i\tau}$. Finally, we also obtain a family of Landen-type identities for theta functions as byproducts, which degenerates to a family of trigonometric identities.