### Speakers

##### Monday 28 September

**Ulrich Derenthal** – *Strong approximation and descent*

**Leo Margolis** – *Torsion Units in Integral Group Rings* **(slides)**

**Simon Hampe** – *Realizing graphs of genus 4 as skeleta of tropical space curves* **(slides)**

**Sebastian Gutsche & Sebastian Posur** – *Cap – Categories, algorithms, and programming* **(slides)**

**Tim Netzer** – *Proving Kazhdan’s Property (T) with Semidefinite Programming* **(slides)**

##### Tuesday 29 September

**Matteo Varbaro** – *Dual graphs of projective schemes* **(slides)**

**Stefan Toman** – *The Complexity of the Radical Word Problem for Binomial Ideals* **(slides)**

**Thomas Markwig** – *How to compute tropical varieties over the -adic numbers?* **(slides)**

**Michael Stoll** – *Chabauty without the Mordell-Weil group* **(slides)**

**Ute Spreckels** – *On the order of abelian varieties over finite prime fields*

**Ulrich Thiel** – *Computational aspects of rational Cherednik algebras* **(slides)**

**Simon Keicher** – *Computing automorphisms of graded algebras and Mori dream spaces* **(slides)**

**Claus Fieker** – *Software Development Within the SPP1489: Number Theory ***(slides)**

##### Wednesday 30 September

**Michael Kemeny** – *Syzygies of curves via K3 surfaces*

**Marta Pieropan** – *Generalized Cox rings over nonclosed fields* **(slides)**

**Tobias Rossmann** – *Computing representation zeta functions of unipotent groups*

**Michel Börner** – *The functional equation for L-functions of hyperelliptic curves* **(slides)**

##### Thursday 01 October

**Wolfram Decker** – *Software Development Within the SPP1489* **(slides)**

**Tommaso Centeleghe** – *Computing integral Frobenius for Modular Abelian Surfaces* **(slides)**

**Bettina Eick** – *Nilpotent associative algebras and coclass theory* **(slides)**

**Simon King** – *Algorithmic isomorphism classification of modular cohomology rings of finite groups* **(slides)**

**Janko Boehm** – *Modular Techniques in Computational Algebraic Geometry* **(slides)**

**Christof Söger** – *Better triangulations in Normaliz 3.0 * **(slides)**

**Christian Eder** – *Improved Parallel Gaussian Elimination for Groebner Bases Computations in Finite Fields* **(slides)**

**Andreas Steenpaß** – *Gröbner Bases over Algebraic Number Fields* **(slides)**

**Sebastian Gutsche** – *Dockerimage für Schwerpunktssoftware* **(slides)**

##### Friday 02 October

**Steffen Müller** – *Quadratic Chabauty*

**Bartosz Naskrecki** – *Generalized Fermat equations – a progress report* **(slides)**

**Panagiotis Tsaknias** – *Generalizations of Maeda’s Conjecture*

**Alexander Rahm** – *Novel techniques for group homology calculation, and applications to Bianchi modular forms*

The lecture room for the talks is in building **31 (AVZ)**, room **E06**.

In the math department (building **69**), you can find a room for discussions and working (room **127**). The coffee breaks will be held in rooms **E13 **and **E15**.

Download schedule as pdf file.

### Abstracts:

**Ulrich Derenthal** – *Strong approximation and descent*

Colliot-Thélène and Sansuc introduced universal torsors and descent methods to study weak approximation and the Hasse principle for rational points on projective varieties over number fields. We generalize these methods to the setting of strong approximation and the Hasse principle for integral points on affine varieties and apply them to two classes of varieties; in one case, we apply work of Browning and Matthiesen based on additive combinatorics. This is joint work with Dasheng Wei.

**Leo Margolis** – *Torsion Units in Integral Group Rings*

Let be a finite group and the group of normalized units in the integral group ring of i.e. the units whose coefficients sum up to 1. Though intensively studied most questions concerning the torsion units of remain open, e.g. it is unknown if the orders of torsion units in and the orders of elements of coincide. The main question concerning the torsion part of , the so called Zassenhaus Conjecture, states that any torsion unit in is conjugate to an element of within the rational group algebra of . We will present some new results concerning the Zassenhaus Conjecture and related questions. The methods involved include the so called Luthar-Passi-Hertweck method which was recently published as a GAP package.

**Simon Hampe** – *Realizing graphs of genus 4 as skeleta of tropical space curves*

(This is joint work in progress with Ralph Morrison and Bernd Sturmfels)

In tropical geometry, there are two main notions of “curves”: One can either talk about abstract tropical curves, which are basically metric graphs. Or one can consider embedded tropical curves, which are weighted, balanced polyhedral complexes. Any embedded curve naturally gives rise to an abstract curve via a deformation retract to its “skeleton” and it is a natural question to ask, which graphs can be obtained in this manner.

In the case of plane curves, this question has been answered exhaustively by Sarah Brodsky, Michael Joswig, Ralph Morrison and Bernd Sturmfels. For genus up to 5, they give a full description of the combinatiorial types that are realizable. Their method involves computing all possible regular unimodular triangulations of certain polytopes.

In this talk we will focus on the case and ask for graphs that are realizable by tropical curves in . Classically, every smooth genus 4 curve has a canonical embedding that is the complete intersection of a cubic and a quadric. Finding such an intersection tropically boils down to finding a regular unimodular triangulation of a fourdimensional Cayley polytope. Enumerating all such triangulations is probably an impossible task, but one can still construct a large enough sample to either hope to find all missing types or to obtain enough supporting evidence to attempt a proof of non-realizability.

After a short introduction to the subject, I will quickly recap the results for plane curves. Then I will talk about our methods for obtaining large data samples of unimodular triangulations of the afore-mentioned Cayley polytope and the results of our analysis.

**Sebastian Gutsche & Sebastian Posur** – *Cap – Categories, algorithms, and programming*

In the talk we present CAP, which is a realization of categorical programming written in GAP.

CAP makes it possible to compute complicated mathematical structures, e.g., spectral sequences. This can be achieved using only

a small set of basic algorithms given by the existential quantifiers of ABELian categories, e.g., composition, kernel, direct sum.

In the talk we will explain the concepts of categorical programming and give a demonstration of the functionalities of CAP.

**Tim Netzer** – *Proving Kazhdan’s Property (T) with Semidefinite Programming*

Kazhdan’s Property (T) is a rather abstract property of a group, involving all its unitary representations on a Hilbert space. Surprisingly, it can be formulated in terms of sums of squares in the group algebra, and is thus accessible to semidefinite programming. Using this approach, we could give a new and easy proof of property (T) for the group and at the same time improve the so-called spectral gap significantly.

**Matteo Varbaro** – *Dual graphs of projective schemes*

Given a projective scheme , its dual graph is the graph whose vertices are given by the irreducible components of , and such that 2 vertices are connected by an edge iff the intersection of the 2 correspondent components is a codimension 1 subscheme of . A classical result of Hartshorne says that, if admits an arithmetically Cohen-Macaulay (aCM) embedding, then is connected. In a joint work with Bruno Benedetti and Barbara Bolognese, we improved the conclusion assuming that admits an arithmetically Gorenstein embedding: in this case, if the (Castelnuovo-Mumford) regularity of (in such an embedding) is , and the regularity of each irreducible component of is , then is -connected. We also proved that for any graph there exists a reduced projective curve admitting an aCM embedding in which all the irreducible components of are rational normal scrolls. During the talk we will discuss these features, some examples, experiments and open questions on dual graphs.

**Stefan Toman** – *The Complexity of the Radical Word Problem for Binomial Ideals*

The radical word problem is to decide given a polynomial and an ideal whether the polynomial is contained in the radical of the ideal. We compare the complexity of the radical word problem for different types of polynomial ideals, in particular toric and (pure) binomial ideals. For binomial ideals over fields with characteristic we show matching upper and lower bounds using their cellular decomposition. These bounds imply that the problem is coNP-complete for binomial ideals.

**Thomas Markwig** – *How to compute tropical varieties over the -adic numbers?*

Tropical geometry has attracted a lot of attention over the past years due to its many interesting applications. Tropical varieties associated to an algebraic variety over a field with (possibly trivial) valuation can be described with the aid of initial ideals. This allows, in principle, to use standard bases techniques to compute tropical varieties. So far this approach was algorithmically only fully developed and implemented for the rational numbers with trivial valuation or the rational functions with the -adic valuation as base fields. We have extended this to the case of the rationals with -adic valuation using standard bases over the integers. In the talk I will try to explain the basic methods and the problems one has to deal with.

**Michael Stoll** – *Chabauty without the Mordell-Weil group*

Chabauty’s method is the standard method one uses to determine the set of rational points on a curve. In its usual form, it requires the knowledge of explicit generators of a subgroup of finite index of the group of rational points on the Jacobian variety of the curve, which in practice can be rather hard to find, in particular when the genus is not very small.

In this talk, we present a variant of the method that avoids the necessity of finding rational points on the Jacobian; it works with a Selmer group instead. Even though the method is not guaranteed to work, heuristically it is expected to do so in many cases. For example, it can be applied to solve the generalized Fermat equation for primes up to 53.

**Ute Spreckels** – *On the order of abelian varieties over finite prime fields*

Let be an abelian variety of dimension over a number field . We consider the case that has CM as well as the non-CM case. Let be a prime ideal of with residue field such that has good reduction at . Denote the number of -rational points on modulo by . For prime let be the density of primes such that divides . The Galois group of the -division field of can be embedded in the general symplectic group of degree over and is the proportion of matrices with eigenvalue in . Firstly, by counting matrices with eigenvalue in we compute for abelian varieties with endomorphism ring and , or odd. We extend a conjecture of Koblitz concerning the number of such that is prime from elliptic curves to such abelian varieties. Secondly, let be a simple principally abelian variety which has CM by the CM-field . For contained in , is a torus in . The splitting behavior of in determines a maximal torus containing which we compute for arbitrary . By assuming that is such a maximal torus we are able to count matrices with eigenvalue in . Finally, we generalize Koblitz’ conjecture to CM abelian varieties of arbitrary dimension.

**Ulrich Thiel** – *Computational aspects of rational Cherednik algebras*

For any finite group acting as a reflection group on a finite-dimensional complex vector space Etingof and Ginzburg introduced in 2002 the so-called rational Cherednik algebras. These are non-commutative deformations of the skew coordinate ring of the symplectic singularity . They seem to encode a lot of information about this singularity, but also about Hecke algebras and finite groups of Lie type (more generally, spetses) attached to .

Rational Cherednik algebras are not only interesting from the theoretical side but also from the computational side, on which I will concentrate in my talk. I will address algorithms to:

- do explicit computations in these algebras,
- compute a presentation of the center at (the so-called Calogero–Mo\-ser

space), - compute the -fixed points of the Calogero–Moser

space (the so-called Calogero–Moser families), - compute Poisson brackets and the zero-dimensional symplectic leaves of the

Calogero–Moser space.

One important result we could obtain in this way is that for the exceptional Weyl group the Calogero–Moser families equal the Lusztig families for all parameters, thus proving the Gordon–Martino conjecture for . This is a further evidence for a deep connection between Calogero–Moser spaces and finite groups of Lie type.

This is joint work with Cédric Bonnafé (Université Montpellier 2).

**Simon Keicher** – *Computing automorphisms of graded algebras and Mori dream spaces*

We present algorithms to compute the automorphism group of integral, finitely generated algebras that are graded by a finitely generated abelian group. We apply our methods to compute automorphism groups of Mori dream spaces. As an example, we compute automorphism groups of Fano varieties in the computer algebra system Singular.

**Wolfram Decker** – *Software Development Within the SPP1489 Part I*

A major goal of the priority programme SPP1489 is to create a free and open source arrangement

of computer algebra systems and libraries specializing in the areas covered by the programme.

In this talk, I will give an overview on what has been achieved so far and outline ideas for future

development. More details on the recent number theory part of this will be given in the lecture by Claus Fieker.

**Claus Fieker** – *Software Development Within the SPP1489 Part II*

Over the last year(s) we have developed (basic) number theory software from

scratch in Kaiserslautern with the explicit aim to have fast, competitive,

class groups in large degree fields and to supply fundamental number

theoretical software infrastructure to other projects. To further this

aim, we revisited many classical algorithms for example for ideal arithmetic

and, in several cases, came up with new, much faster, algorithms.

On the practical side, the software has a part in c, antic, extending Bill

Hart’s flint2 library and a layer in julia, a modern JIT-compiled language.

This is both a progress report, a glimpse into the future and a call for

comments, it represents joint work with Bill Hart and Tommy Hofmann.

**Michael Kemeny** – *Syzygies of curves via K3 surfaces*

The talk would cover recent developments on two conjectures about the shape of the free resolution of the ideal of projectively embedded curves. In particular, I would outline the recent proof of the Prym-Green conjecture for syzygies of odd genus Prym-canonical curves as well as the generic Green-Lazarsfeld conjecture (joint with Gabi Farkas). These conjectures can be seen as generalisations of the famous generic Green’s conjecture as proved by Voisin. Our proofs involves the geometry of curves on K3 surfaces in a deep way. If time permits I will also discuss recent progress on the higher level analogue of the Prym-Green conjecture.

**Marta Pieropan** – *Generalized Cox rings over nonclosed fields*

The relations between Cox rings and universal torsors for varieties over algebraically closed fields of characteristic 0 are well known. See, for example, the book “Cox rings” by Arzhantsev, Derenthal, Hausen and Laface, 2015. Universal torsors are certain torsors under quasi-tori that have been introduced by Colliot-Thélène and Sansuc in the 1970s for varieties over arbitrary fields. In joint work with U. Derenthal, we introduce a notion of generalized Cox rings associated to torsors under quasi-tori that extends the usual notion of Picard graded Cox rings. We define the generalized Cox rings from an axiomatic point of view and we discuss their classification and their existence over nonclosed fields. After a general presentation, this talk will focus on the computation of pullbacks of finitely generated generalized Cox rings illustrated by an example of arithmetic interest.

**Tobias Rossmann** – *Computing representation zeta functions of unipotent groups*

We discuss methods for computing local and topological zeta functions arising from the enumeration of irreducible complex representations of arithmetic groups associated with unipotent algebraic groups.

**Michel Börner** – *The functional equation for L-functions of
hyperelliptic curves*

We introduce an algorithm (written in Sage) for the L-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local L-factor and the conductor exponent at the primes of bad reduction. The method we use works for any superelliptic curve over a number field. We present several families of examples, e.g. hyperelliptic curves of genus 2,…,6 with semistable reduction everywhere.

**Steve Linton** – *Combining Computational Tools for Discrete Mathematics*

In this talk I will give a survey of recent and ongoing developments in

techniques for combining multiple software systems to solve complex problems. These

can be distinct systems — combining tools from group theory and algebraic geometry to

study representation theory, for instance — or multiple copies of the same system — e.g.

using hundreds of copies of GAP in parallel to solve huge enumeration problems.

I will discuss different technical approaches to these tasks, and also some of the potential

mathematical and semantic pitfalls that arise. I will in particular give a brief introduction to

some of the activities of the recently started OpenDreamKit proposal, which will be

exploring some of these issues further over the next four years.

**Tommaso Centeleghe** – *Computing integral Frobenius for Modular Abelian Surfaces*

Let be a weight 2 cuspidal eigenform on , and let be the corresponding factor of the Jacobian of the modular curve. Assume that is a surface. The goal of this project is to construct an algorithm for computing the Galois structure of the -adic Tate modules at the unramfied primes . The key fact is that we focus on the integral Tate module as opposed to its rational version . In order to study the action of on the former, the charateristic polynomial will not suffice in general. We will discuss

a strategy for solving the problem.

**Bettina Eick** – *Nilpotent associative algebras and coclass theory*

The classification of nilpotent associative algebras is a wide open research project. We show how the coclass can be used to obtain new insights into such a classification. For this purpose we define the coclass graph . Its vertices correspond one-to-one to the isomorphism types of nilpotent associative algebras of coclass over a field and there is an edge if the

nilpotency classes of and differ by 1 and is isomorphic to . In the course of this project we have investigated the graphs in detail using a combination of theoretical and algorithmic tools. This talk gives an overview on the obtained results.

**Simon King** – *Algorithmic isomorphism classification of modular cohomology rings of finite groups*

It was proved by Jon Carlson that there are only finitely many isomorphism types of modular cohomology rings for finite 2-groups of fixed coclass. A proof of the analogous statement for finite p-groups (p>2) has recently been announced by Antonio Díaz Ramos. Since the proofs are not constructive, it is unclear what the isomorphism types actually look like. Charles Leedham-Green and Bettina Eick define coclass families of groups and conjecture that the cohomology rings within each group are isomorphic up to finitely many exceptions. But how many exceptions?

To explore these questions further, it is useful to study the isomorphism types by computer experiments. In joint work with Bettina Eick, we developed an algorithm to decide whether two graded-commutative algebras with certain finiteness properties are graded isomorphic, and classified the modular cohomology rings of all p-groups of order less than 100.

**Janko Boehm** – *Modular Techniques in Computational Algebraic Geometry*

Computations over the rational numbers often suffer from intermediate coefficient growth. One approach to this problem is to determine the result modulo a number of primes and then lift to the rationals. This method is guaranteed to work if we use a sufficiently large set of good primes. In many applications, however, there is no efficient way of excluding bad primes. We develop a new technique for rational reconstruction which will nevertheless return the correct result, provided the number of good primes in the set is large enough. We discuss applications of this technique in computational algebraic geometry.

This is joint work with Claus Fieker, Wolfram Decker, Gerhard Pfister, and Santiago Laplagne.

**Christof Söger** – *Better triangulations in Normaliz 3.0 *

Normaliz is a well-established computer program for rational cones and polytopes. Its main computation goals are Hilbert bases and Ehrhart series. We give an overview of the program and discuss the new features of Normaliz 3.0. In particular we will describe how we find triangulations which have a better computational behaviour. This includes the reordering of the generators, approximating a rational polytope and refining a triangulation using integer linear programming.

**Christian Eder** – *Improved Parallel Gaussian Elimination for Groebner Bases Computations in Finite Fields*

We present a GPLv2 open source C library for linear algebra specialized for eliminating matrices generated during GB computations in algorithms like F4 or F5. We improve on the initial ideas of Faugere

and Lachartre (FL). Our approach takes even more advantage of the very special structure the corresponding matrices have: quasi unit-triangular sparse matrices with patterns in the data. Optimizing this reduction step is crucial for the overall GB computation.

We first present improved data structures for storing these FL matrices in binary format, namely by compressing the repeated data in the rows and the column indexes, before gzip-ing the result. We can save up to an order of magnitude in space, allowing us to produce and share a large database of such matrices for benchmarking and testing purpose.

We show efficient blocked data structures for computing the reduced forms with specialized AXPY and TRSM operations, that take advantage of the patterns in the matrices and their sparsity. For instance, a special multiline storage allows cache friendly storage of sparse rows.

We also reduce the number of operations, in a parallel friendly fashion, by changing the order of the operations in the elimination and by not computing the full row echelon form.

Finally, we present experimental results for sequential and parallel computations on NUMA architectures. With our new implementation we get a 5-10\% speed-up for the sequential algorithm depending on

the rank. We also get better scaling up until 32 (non hyper-threaded) cores instead of 16: we have speed-ups around or for bigger benchmarks. We also save more than twice the amount of memory used during the computation.

**Andreas Steenpaß** – *Gröbner Bases over Algebraic Number Fields*

Although Buchberger’s algorithm, in theory, allows us to compute Gröbner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field

, a simple extension of , where is an algebraic number, and let be the minimal polynomial of . In this

talk we present a new efficient method to compute Gröbner bases in polynomial rings over the algebraic number field . Starting from the ideas of Noro [1], we proceed by joining to the ideal to be considered, adding as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2,3,4], that is, by inferring information in characteristic zero from information in characteristic . For suitable primes $p$, the minimal polynomial is reducible over . This allows us to apply modular methods once again, on a second level, with respect to the factors of . The algorithm thus resembles a divide and conquer strategy and is in particular easily parallelizable. At current state, the algorithm is probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithm, which has been implemented in Singular, outperforms other known methods by far.

[1] M. Noro. An efficient implementation for computing Gröbner bases

over algebraic number fields. In

Mathematical software – ICMS 2006. Second

international congress on mathematical software,

Castro Urdiales, Spain, September 1–3, 2006.

Proceedings, pages 99–109. Springer, 2006.

[2] E. A. Arnold. Modular algorithms for computing Gröbner bases. J.

Symb. Comput., 35(4):403–419, 2003.

[3] J. B\”ohm, W. Decker, C. Fieker, and G. Pfister. The

use of bad primes in rational reconstruction.

To appear in Math. Comp., 2015. http://arxiv.org/abs

\linebreak

/1207.1651.

[4] N. Idrees, G. Pfister, and S. Steidel. Parallelization of

modular algorithms. J. Symb. Comput.,

46(6):672–684, 2011.

**Sebastian Gutsche** – *Dockerimage für Schwerpunktssoftware*

Um ein einfaches Installieren der im Schwerpunkt entwickelten Software zu ermöglichen, habe ich ein Dockerimage erstellt, welches einen Großteil dieser Software enthält. Das Image bietet eine Möglichkeit, ohne Aufwand lauffähige Versionen der CASs auf diversen Architekturen zu benutzen und aktuell zu halten. Im Vortrag gebe ich ein kurzes Tutorial zu Docker und dem Image unter Linux/Windows/OS X.

**Steffen Müller** – *Quadratic Chabauty*

I will talk about joint work with Jennifer Balakrishnan and Amnon Besser on a -adic method to compute integral points on hyperelliptic curves. If time permits, I will also discuss work in progress on extending this method in several directions.

**Bartosz Naskrecki** – *Generalized Fermat equations – a progress report*

During this talk I will explain the progress in joint work with Nuno Freitas and Michael Stoll on the solution of generalized Fermat equations of the type for a prime . We will explain the local computations that arise during the analysis of the twisted modular curves involved and some descent calculations that were performed. In the final part I will explain the remaining questions and perspectives towards a full resolution of before mentioned equations.

**Panagiotis Tsaknias** – *Generalizations of Maeda’s Conjecture*

I will report on joint work with L. Dieulefait, currently in progress, on generalizations of the Maeda conjecture. I will provide a precise generalized version of its weak form regarding the number of newform Galois orbits for arbitrary level and trivial nebentypus. I will also describe further ways to generalize the original conjecture (e.g. non-trivial nebentypus, Hilbert modular forms, strong form for arbitrary levels).

**Alexander Rahm** – *Novel techniques for group homology calculation, and applications to Bianchi modular forms*

In this talk, we present work on determining the dimensions of the spaces of cuspidal Bianchi modular forms, locating several of the very rare instances that are not lifts of classical modular forms. Classical modular forms are the subject of the now proven Taniyama-Shimura conjecture; and the proof has allowed to complete Fermat’s Last Theorem. The Taniyama-Shimura conjecture is nowadays called the modularity theorem, and attaches an elliptic curve to each classical modular form, corresponding via the L-series. There are deep number-theoretical reasons for expecting that this kind of correspondence can extend to the Bianchi modular forms, attaching Abelian varieties to them. Bianchi modular forms over an imaginary quadratic field K are automorphic forms of cohomological type associated to the Bianchi group of K, the latter being the rank two special linear group over the ring of integers in K. Even though modern studies of Bianchi modular forms go back to the mid !

1960’s,

most of the fundamental problems surrounding their theory are still wide open. In this project, we have made use of the speaker’s constructive answer to a question of Jean-Pierre Serre on the Borel-Serre compactification of the Bianchi orbifolds, which had been open for 40 years. This has permitted heavy machine calculations joint with M. Haluk Sengun, locating several of the very rare instances of Bianchi modular forms in a large array of dicriminants and weights, such that these forms are not lifts of classical modular forms.

We will also discuss further applications of the study of the Bianchi orbifolds, like the cohomology of discrete groups, for which a new technique, called torsion subcomplex reduction, has been developed by the speaker based on observations on the Bianchi groups, and has allowed to find formulas for the torsion in the cohomology not only for the Bianchi groups, but also for several other classes of discrete groups so far.

The same technique, combined with some additional studies, further yields formulas for the Chen-Ruan orbifold cohomology of the Bianchi orbifolds.

And an adaptation of the technique to Bredon homology has yielded the equivariant K-homology of the Bianchi groups; as the latter satisfy the Baum-Connes conjecture, this K-homology is isomorphic to the operator K-theory of the Bianchi groups, which would be extremely hard to compute directly.