The modular curve $Y^1(N)$ parametrises pairs $(E,P)$, where $E$ is an elliptic curve and $P$ is a point of order $N$ on $E$, up to isomorphism. A unit on the affine curve $Y^1(N)$ is a holomorphic function that is nowhere zero and I will mention some applications of the group of units in the talk.
The main result is a way of generating generators (sic) of this group using a recurrence relation. The generators are essentially the defining equations of $Y^1(N)$ for $n < (N + 3)/2$. This result proves a conjecture of Maarten Derickx and Mark van Hoeij.[-]

Over the last few years I developed (partly jointly with coauthors) dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for certain hyperbolic surfaces/orbifolds L \ H with cusps (both of finite and infinite area; arithmetic and non-arithmetic).
Both types of transfer operators arise from discretizations of the geodesic flow on L \ H. The eigenfunctions with eigenvalue 1 of slow transfer operators characterize Maass cusp forms. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. The Fredholm determinant of the fast transfer operators equals the Selberg zeta function, which yields that the zeros of the Selberg zeta function (among which are the resonances) are determined by the eigenfunctions with eigenvalue 1 of the fast transfer operators. It is a natural question how the eigenspaces of these two types of transfer operators are related to each other.[-]

In this talk we review results on several types of harmonic weak Maass forms that are related to integral even weight newforms. We start with a brief introduction to the theory of harmonic weak Maass forms. These can be related to classical modular forms via a certain differential operator, the so-called $\chi $-operator. Starting with an integral weight newform, we will review different constructions of integral weight harmonic weak Maass forms via (generalized) Weierstrass zeta functions that map to the newform under the $\chi $-operator. A second construction via theta liftings gives a half-integral weight harmonic weak Maass form whose coefficients are given by periods of certain meromorphic modular forms with algebraic coefficients and periods of the integer even weight newform. This is joint work with Jens Funke, Michael Mertens, and Eugenia Rosu resp. Jan Bruinier and Markus Schwagenscheidt.[-]