Computing Vector-valued Modular Forms of Congruence Types and of Some Extension Types

Sammanfattning: This thesis explores applications of vector-valued modular forms of congruence and extension types to scalar-valued modular forms for congruence subgroups with a character, higher order modular forms, and iterated Eichler-Shimura integrals of depth one and two, including considerable generalizations thereof. In \textsc{Paper I} (co-authored with Martin Raum), we present an algorithm for computing bases for spaces of vector-valued modular forms of congruence type and of weight at least $2$ in terms of products of components of vector-valued Eisenstein series. Since the Fourier series expansions of these Eisenstein series are available, our algorithm can be used to compute Fourier series expansions of any vector-valued modular forms belonging to these spaces. It complements two available algorithms that (as opposed to ours) are limited to inductions of Dirichlet characters, and vector-valued modular forms of Weil type. Our algorithm is based on a representation theoretical interpretation of a theorem due to Raum and Xià. After a heuristic evaluation of the time-complexity, we compare our algorithm to the two available ones, highlighting the trade-offs between generality and performance. In \textsc{Paper II} (co-authored with Martin Raum and Albin Ahlbäck), we show that all Eichler integrals, and all ``generalized second order modular forms'' can be expressed as linear combinations of corresponding generalized second order Eisenstein series with coefficients in classical modular forms. We compute the Fourier series expansions of generalized second order Eisenstein series in level one, and provide their tail estimates via convexity bounds for additively twisted $L$-functions. As an application, we illustrate a bootstrapping procedure that yields numerical evaluations of, for instance, Eichler integrals from merely the associated cocycle. Finally, in \textsc{Paper III} (co-authored with Martin Raum), we provide an explicit vector-valued modular form whose top components are given by the depth two iterated Eichler-Shimura integral $I_{f,g}$, where $f$ and $g$ are cusp forms of weight $k\in\ZZ_{\geq 2}$. We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by $\cE_{f,g}$, that can be seen as a higher-depth generalization of the scalar-valued Eichler integral $\cE_f$ of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Pa\c{s}ol-Popa. We show that $\cE_{f,g}$ can be expressed in terms of sums of products of components of vector-valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form $\Delta$. This allows for effective computation of $\cE_{f,g}$.