Dan Yasaki

Hyperbolic Tessellations Associated to Bianchi Groups

Let F/ℚ be a number field. The space of positive definite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into subcones. In the case of an imaginary quadratic field these subcones descend to hyperbolic space to give rise to tessellations of 3-dimensional hyperbolic space by ideal polytopes. We compute the structure of these polytopes for a range of imaginary quadratic fields.

Hecke operators and Hilbert modular forms

Let F be a real quadratic field with ring of integers O andwith class number 1. Let Γ be a congruence subgroup of GL2(O). Wedescribe a technique to compute the action of the Hecke operators on thecohomology H3(Γ;C). For F real quadratic this cohomology group containsthe cuspidal cohomology corresponding to cuspidal Hilbert modularforms of parallel weight 2. Hence this technique gives a way to computethe Hecke action on these Hilbert modular forms.

Modular Forms and Elliptic Curves over the Field of Fifth Roots of Unity

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.

Modular Forms And Elliptic Curves Over The Cubic Field Of Discriminant - 23

Let F be the cubic field of discriminant –23 and let be its ring of integers. By explicitly computing cohomology of congruence subgroups of , we computationally investigate modularity of elliptic curves over F.

A database of genus-2 curves over the rational numbers

We describe the construction of a database of genus 2 curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated L-function. This data has been incorporated into the L-Functions and Modular Forms Database (LMFDB).

A Table of Elliptic Curves over the Cubic Field of Discriminant–23

Let F be the cubic field of discriminant −23 and its ring of integers. Let Γ be the arithmetic group , and for every ideal , let be the congruence subgroup of level . In [Gunnells and Yasaki 13], the cohomology of various was computed, along with the action of the Hecke operators. The goal of that work was to test the modularity of elliptic curves over F. In the present article, we complement and extend those results in two ways. First, we tabulate more elliptic curves than were found earlier using various heuristics (“old and new” cohomology classes, dimensions of Eisenstein subspaces) to predict the existence of elliptic curves of various conductors, and then using more sophisticated search techniques (for instance, torsion subgroups, twisting, and the Cremona–Lingham algorithm) to find them. We then compute further invariants of these curves, such as their rank and representatives of all isogeny classes. Our enumeration includes conjecturally the first elliptic curves of ranks 1 and 2 over this field, which occur at levels of norm 719 and 9173 respectively.

Computing Modular Forms for GL2 over Certain Number Fields

The cohomology of an arithmetic group is built out of certain automorphic forms. This allows computational investigation of these automorphic forms using topological techniques. We discuss recent techniques developed for the explicit computation of the cohomology of congruence subgroups of GL2 over CM-quartic and complex cubic number fields as Hecke-modules.