Computational Aspects of Classical and Hilbert Modular Forms

Doctoral thesis (2017)

The main topic of this thesis is the study of classical and Hilbert modular forms and computational aspects of their q-expansions. The coefficients of q-expansions of eigenforms are particularly ... [more ▼]

The main topic of this thesis is the study of classical and Hilbert modular forms and computational aspects of their q-expansions. The coefficients of q-expansions of eigenforms are particularly interesting because of their arithmetic significance. Most notably, modular forms are an essential ingredient in Andrew Wiles’s proof of Fermat’s last theorem. This thesis consists of two parts: the first part concerns the distribution of the coefficients of a given classical eigenform; the second part studies computational aspects of the adelic q-expansion of Hilbert modular forms of weight 1. Part I of this thesis is an adapted version of the article On the Distribution of Frobenius of Weight 2 Eigenforms with Quadratic Coefficient Field published in Experimental Mathematics [38]. It presents a heuristic model that settles the following question related to the Sato-Tate and Lang-Trotter conjectures: given a normalised eigenform of weight 2 with quadratic coefficient field, what is the asymptotic behaviour of the number of primes p such that the p-th coefficient of this eigenform is a rational integer? Our work contributes to this problem in two ways. First, we provide an explicit heuristic model that describes the asymptotic behaviour in terms of the associated Galois representation. Secondly, we show that this model holds under reasonable assumptions and present numerical evidence that supports these assumptions. Part II concerns the study of (adelic) q-expansions of Hilbert modular forms. Our main achievements are the design, proof and implementation of several algorithms that compute the adelic q-expansions of Hilbert modular forms of weight 1 over C and over finite fields. One reason we are studying such q-expansions is that their coefficients (conjecturally) describe the arithmetic of Galois extensions of a totally real number field with Galois group in GL 2 (F p ) that are unramified at p. Using the adelic q-expansions of Hilbert modular forms of higher weight, these algorithms enable the explicit computation of Hilbert modular forms of any weight over C and the computation of Hilbert modular forms of parallel weight both over C and in positive characteristics. The main improvement to existing methods is that this algorithm can be applied in (partial) weight 1, which fills the gap left by standard computational methods. Moreover, the algorithm computes in all characteristics simultaneously. More precisely, we prove that, under certain conditions in higher weight, the output of the algorithm for given level N and quadratic character E includes a finite set of primes L such that all Hilbert modular forms of given parallel weight, level N and quadratic character E over F p are liftable for all primes p outside the set L. In particular, testing primes in the set L enabled the computation of examples of non-liftable Hilbert modular forms of weight 1. [less ▲]