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Dan Yasaki

Publications and source records attributed to Dan Yasaki.

At least 19 recordsLinked to original sources
arXiv2025

Bianchi Modular Forms over Imaginary Quadratic Fields with arbitrary class group

Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be its ring of integers. For an integral ideal $\mathfrak{n}$ of $\mathcal{O}_K$, let $\Gamma_0({\mathfrak{n}})$ be the congruence subgroup of level ${\mathfrak{n}}$ consisting of matrices in $\operatorname{GL}_2{\mathcal{O}_K}$ that are upper triangular mod ${\mathfrak{n}}$. In this paper, we discuss techniques to compute the space of Bianchi modular forms of level $\Gamma_0({\mathfrak{n}})$ as a Hecke module in the case where $K$ has arbitrary class group. Our algorithms and computations extend and complement those carried out for fields of class number $1$, $2$, and $3$ by the first author, and by his students Bygott and Lingham in unpublished theses. We give details and several examples for $K=\mathbb{Q}(\sqrt{-17})$, whose class group is cyclic of order $4$, including a proof of modularity of an elliptic curve over this field. We also give an overview of the results obtained for a wide range of imaginary quadratic fields, which are tabulated in the L-functions and modular forms database (\href{https://www.lmfdb.org/}{LMFDB}).

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arXiv2023

On the cohomology of GL_2 and SL_2 over imaginary quadratic fields

We report on computations of the cohomology of GL_2(O_D) and SL_2(O_D), where D<0 is a fundamental discriminant. These computations go well beyond earlier results of Vogtmann and Scheutzow. We use the technique of homology of Voronoi complexes, and our computations recover the integral cohomology away from the primes 2, 3. We observed exponential growth in the torsion subgroup of H^2 as $D$ increases, and compared our data to bounds of Rohlfs.

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arXiv2021

Cohomology of congruence subgroups of SL_3(Z), Steinberg modules, and real quadratic fields

We investigate the homology of a congruence subgroup Gamma of SL_3(Z) with coefficients in the Steinberg modules St(Q^3) and St(E^3), where E is a real quadratic field and the coefficients are Q. By Borel-Serre duality, H_0(Gamma, St(Q^3)) is isomorphic to H^3(Gamma,Q). Taking the image of the connecting homomorphism H_1(Gamma, St(E^3)/St(Q^3)) \to H_0(Gamma, St(Q^3)), followed by the Borel-Serre isomorphism, we obtain a naturally defined Hecke-stable subspace H(Gamma,E) of H^3(Gamma,Q). We conjecture that H(Gamma,E) is independent of E and consists of the cuspidal cohomology H_cusp^3(Gamma,Q) plus a certain subspace of H^3(Gamma, Q)$ that is isomorphic to the sum of the cuspidal cohomologies of the maximal faces of the Borel-Serre boundary. We report on computer calculations of H(Gamma,E) for various Gamma, E which provide evidence for the conjecture. We give a partial heuristic for the conjecture.

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arXiv2021

Perfect Forms over Imaginary Quadratic Fields

In this work, we compute the perfect forms for all imaginary quadratic fields of absolute discriminant up to $5000$ and study the number and types of the polytopes that arise. We prove a bound on the combinatorial types of polytopes that can arise regardless of discriminant and give a volumetric argument for a lower bound on the number of perfect forms as well as a heuristic for a better lower bound for imaginary quadratic fields of sufficiently large absolute discriminant.

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arXiv2020

Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields

We develop methods for constructing explicit generators, modulo torsion, of the K_3-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic 3-space or on direct calculations in suitable pre-Bloch groups, and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite K_3-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for K_3 of any field, predict the precise power of 2 that should occur in the Lichtenbaum conjecture at -1 and prove that the latter prediction is valid for all abelian number fields.

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arXiv2020

Steinberg homology, modular forms, and real quadratic fields

We compare the homology of a congruence subgroup Gamma of GL_2(Z) with coefficients in the Steinberg modules over Q and over E, where E is a real quadratic field. If R is any commutative base ring, the last connecting homomorphism psi_{Gamma,E} in the long exact sequence of homology stemming from this comparison has image in H_0(Gamma, St(Q^2;R)) generated by classes z_βindexed by beta in E \ Q. We investigate this image. When R=C, H_0(Gamma, St(Q^2;C)) is isomorphic to a space of classical modular forms of weight 2, and the image lies inside the cuspidal part. In this case, z_beta is closely related to periods of modular forms over the geodesic in the upper half plane from beta to its conjugate beta'. Assuming GRH we prove that the image of $ψ_{Γ,E}$ equals the entire cuspidal part. When R=Z, we have an integral version of the situation. We define the cuspidal part of the Steinberg homology, H_0^cusp(Gamma, St(Q^2;Z)). Assuming GRH we prove that for any congruence subgroup, psi_{Gamma,E} always has finite index in H_0^cusp(Gamma, St(Q^2;Z)), and if Gamma=Gamma_1(N)^pm or Γ_1(N), then the image is all of H_0^cusp(Gamma, St(Q^2;Z)). If Gamma=Gamma_0(N)^pm or Gamma_0(N), we prove (still assuming GRH) an upper bound for the size of H_0^cusp(Gamma, St(Q^2;Z))/image(psi_{Gamma,E}). We conjecture that the results in this paragraph are true unconditionally. We also report on extensive computations of the image of psi_{Gamma,E} that we made for Gamma=Gamma_0(N)^pm and Gamma=Gamma_0(N). Based on these computations, we believe that the image of psi_{Gamma,E} is not all of H_0^cusp(Gamma, St(Q^2;Z)) for these groups, for general N.

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arXiv2019

Non-integrality of some Steinberg modules

We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartments. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartments if and only if the ring is Euclidean. We also construct new cohomology classes in the top dimensional cohomology group of the special linear group of some quadratic imaginary number rings.

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arXiv2019

An exploration of Nathanson's $g$-adic representations of integers

We use Nathanson's $g$-adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets $S$ to problems in additive number theory. If $S$ consists of all powers of a fixed integer $g$, we find explicit formulas for the smallest positive integer of a given length. This is related to finding the smallest positive integer expressible as a fixed number of sums and differences of powers of $g$. We also consider $S$ to be the set of all powers of all primes and bound the diameter of Cayley graph by relating it to Goldbach's conjecture.

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arXiv2018

On the cohomology of congruence subgroups of GL3 over the Eisenstein integers

Let F be the imaginary quadratic field of discriminant -3 and OF its ring of integers. Let Gamma be the arithmetic group GL_3 (OF), and for any ideal n subset OF let Gamma_0 (n) be the congruence subgroup of level n consisting of matrices with bottom row (0,0,*) bmod n. In this paper we compute the cohomology spaces H^{nu - 1} (Gamma_0 (n); C) as a Hecke module for various levels n, where nu is the virtual cohomological dimension of Gamma. This represents the first attempt at such computations for GL_3 over an imaginary quadratic field, and complements work of Grunewald--Helling--Mennicke and Cremona, who computed the cohomology of GL_2 over imaginary quadratic fields. In our results we observe a variety of phenomena, including cohomology classes that apparently correspond to nonselfdual cuspforms on GL_3/F.

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arXiv2017

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).

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arXiv2016

On the growth of torsion in the cohomology of arithmetic groups

Let G be a semisimple Lie group with associated symmetric space D, and let Gamma subset G be a cocompact arithmetic group. Let L be a lattice inside a Z Gamma-module arising from a rational finite-dimensional complex representation of G. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup H_i (Gamma_k; L )_tors as Gamma_k ranges over a tower of congruence subgroups of Gamma. In particular they conjectured that the ratio (log |H_i (Gamma_k ; L)_tors|)/[Gamma : Gamma_k] should tend to a nonzero limit if and only if i= (dim(D)-1)/2 and G is a group of deficiency 1. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including GL_n (Z) for n=3,4,5 and GL_2 (O) for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron--Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron--Venkatesh conjecture.

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arXiv2014

On the toplogical computation of K4 of the Gaussian and Eisenstein integers

In this paper we use topological tools to investigate the structure of the algebraic K-groups K_4 (Z[i]) and K_4 (Z[rho]), where i := sqrt{-1} and rho := (1+sqrt{-3})/2. We exploit the close connection between homology groups of GL_n(R) for n <= 5 and those of related classifying spaces, then compute the former using Voronoi's reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which GL_n(R) acts. Our main result is that K_4 (Z[i]) and K_4 (Z[rho]) have no p-torsion for p >= 5.

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arXiv2014

A table of elliptic curves over the cubic field of discriminant -23

Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the action of the Hecke operators. The goal of that paper was to test the modularity of elliptic curves over F. In the present paper, we complement and extend this prior work in two ways. First, we tabulate more elliptic curves than were found in our prior work by using various heuristics ("old and new" cohomology classes, dimensions of Eisenstein subspaces) to predict the existence of elliptic curves of various conductors, and then by 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.

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arXiv2013

On the cohomology of linear groups over imaginary quadratic fields

Let Gamma be the group GL_N (OO_D), where OO_D is the ring of integers in the imaginary quadratic field with discriminant D<0. In this paper we investigate the cohomology of Gamma for N=3,4 and for a selection of discriminants: D >= -24 when N=3, and D=-3,-4 when N=4. In particular we compute the integral cohomology of Gamma up to p-power torsion for small primes p. Our main tool is the polyhedral reduction theory for Gamma developed by Ash and Koecher. Our results extend work of Staffeldt, who treated the case n=3, D=-4. In a sequel to this paper, we will apply some of these results to the computations with the K-groups K_4 (OO_{D}), when D=-3,-4.

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arXiv2009

Hyperbolic tessellations associated to Bianchi groups

Let F/Q be 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, which descend give rise to hyperbolic tessellations of 3-dimensional hyperbolic space by ideal polytopes. We compute the structure of these polytopes for a range of imaginary quadratic fields.

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