Hecke operators for algebraic modular forms

• Heckeoperatoren für algebraische Modulformen

Schönnenbeck, Sebastian; Nebe, Gabriele (Thesis advisor); Hartmann, Julia (Thesis advisor); Hiß, Gerhard (Thesis advisor)

Aachen (2016)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2016

Abstract

Let $k$ be a totally real algebraic number field with ring of finite adeles $\hat{k}$ and $\mathbb{G}$ a connected, semisimple, linear algebraic group over $k$ such that $\mathbb{G}(k \otimes \mathbb{R})$ is compact. In 1999 Benedict H. Gross noticed that in this situation one can set up a theory of automorphic forms for $\mathbb{G}$ in an entirely algebraic way, which led to the notion of algebraic modular forms. The space of algebraic modular forms, which depends on an open compact subgroup $K$ of $\mathbb{G}(\hat{k})$ (the so-called level) and an irreducible representation $V$ of $\mathbb{G}(k)$ (the so-called weight), is always finite-dimensional and comes equipped with an action of the Hecke algebra of $\mathbb{G}(\hat{k})$ with respect to $K$. One of the primary goals in the field is to compute this action explicitly in given examples.In this thesis we start by reviewing the theory of integral forms of algebraic groups as well as basics of algebraic modular forms and Hecke algebras. Furthermore we describe the classical algorithmic approach to computing the action of Hecke operators by decomposing certain double cosets with respect to $K$ into right cosets. Afterwards we introduce intertwining operators and Eichler elements and study their behaviour. These operators are often computable with significantly less effort than in the standard approach and allow the computation of two Hecke operators at once by leveraging the adjacency relation of the Euclidean building of $\mathbb{G}$. We call the subalgebra of $H_K$ that is generated by the different Eichler elements the Eichler subalgebra and prove that for simply-connected $\mathbb{G}$ it is necessarily a polynomial ring. Moreover we investigate in which cases the Eichler subalgebra already coincides with the full Hecke algebra, i.e. in which cases the Eichler elements already generate $H_K$.In addition to these theoretical considerations we implemented the presented algorithms for the algebraic group $G_2$ and for compact forms of symplectic groups. We present the necessary theoretical background on these groups and give an overview of the capabilities of our implementation.Our explicit computations for $G_2$ and $\mathrm{Sp}_6$ can be used to check for the existence of so-called lifts of algebraic modular forms. We describe the necessary background on the Satake homomorphism, compute its image in the specific example at hand, determine what possible lifts should look like, and find some modular forms which appear to be lifts.Finally we present some alternative uses of our algorithms to the study of $S$-arithmetic subgroups. Namely we compute a free resolution of the integers via the action on the Euclidean building and determine a finite set of generators together with a system of defining relations.