Modular version of Nevanlinna theory

Abstract

Fundamental Theorem of Algebra implies that every degree $d$ polynomial with complex coefficient takes value $a \in \mathbb{C}$ exactly $d$ times, counting multiplicity. One also knows the growth of polynomial is governed by its degree $d$. In other words, Fundamental Theorem of Algebra establishes a relation between number of roots of a polynomial and its growth. Nevanlinna theory can be considered as an extension of Fundamental Theorem of Algebra to (transcendental) meromorphic functions $f$ in this manner. It introduces the counting function $N(f;r,a)$ (as an analog of the number of roots in polynomial), the proximity function $m(f;r,a)$ and Nevanlinna characteristic function $T(f;r)$ (as an analog of the growth of polynomial). Starting with Poisson-Jensen formula, the First Main Theorem of Nevanlinna theory reads: \[ m(f;r,a) + N(f;r,a) = T(f;r) + O(1). \] Roughly speaking, the Second Main Theorem says that mostly $N(f;r,a)$ predominates in the left-hand side of the equation above. These two theorems suggest the counting function $N(f;r,a)$ roughly grows like the characteristic function $T(f;r)$ for almost all $a \in \mathbb{C}$. This is the reason why Nevanlinna theory is also known as a quantitative version of Fundamental Theorem of Algebra for meromorphic functions. An introduction to classical Nevanlinna theory is given in Chapter 2.

It will be interesting to develop a version of Nevanlinna theory for a specific family of functions, which shares more properties and structure, say modular functions. Basic theory of modular functions will be introduced in Chapter 3. Let $f$ be a meromorphic modular function with respect to $SL_{2}(\mathbb{Z})$ satisfying $f(\infty) =1$. Rohrlich Theorem (see Chapter 4 for details) says that \begin{align*} \frac{3}{\pi} \iint_{\Gamma \backslash \mathcal{H}} \log |f(z)| \frac{dxdy}{y^{2}} = \log \prod_{z \in \Gamma \backslash \mathcal{H}} (y^{6} |\Delta(z)|)^{- \ord_{z}f} . \end{align*} Rohrlich Theorem is a modular analog of Jensen's formula. Since Jensen's formula, the starting point of classical Nevanlinna theory, has now an analog in modular version, it is natural to examine whether there exists a modular analog of Nevanlinna theory.

In this thesis, we examine the proof of Rohrlich's Theorem and generalize the result by eliminating the condition $f(\infty) =1$. To do this, we need the technique of regularizing Petersson inner product. In Chapter 6, we explain a common technique in regularizing inner product which is the Rankin-Selberg method (also known as unfolding trick). Theory of non-holomorphic Eisenstein series $E(z,s)$ is involved in Rankin-Selberg method (see Chapter 4 for theory of non-holomorphic Eisenstein series). In Chapter 7, we will try to develop modular version of Nevanlinna theory from the generalization of Rohrlich's Theorem. In particular, we obtain the first main theorem for this modular version of Nevanlinna theory. A comparison of modular Nevanlinna theory and classical Nevanlinna theory is also given in Chapter 7.