Schedule for: 21w5205 - Lattices and Cohomology of Arithmetic Groups: Geometric and Computational Viewpoints (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In a joint work with Nikolay Bogachev, Alexander Kolpakov, and Leone Slavich we discovered an interesting connection between totally geodesic subspaces of a hyperbolic manifold or orbifold and finite subgroups of the commensurator of its fundamental group. We call the totally geodesic subspaces associated to the finite subgroups by fc- subspaces. It appears that these subspaces have some remarkable properties. We show that in an arithmetic orbifold all totally geodesic subspaces of sufficiently small codimension are fc and there are infinitely many of them, while in non-arithmetic cases there are only finitely many fc- subspaces and their number is bounded in terms of volume. In particular, all totally geodesic subspaces of an arithmetic hyperbolic 3-orbifold are fc-subspaces. In the talk, I will begin with an introduction to the topic and then discuss some results and their proofs.

In this talk, I will survey known results about the (co)homology of SLn(Z) and its finite index subgroups. I will focus on homological stability and representation stability phenomena. In addition to stability results where the homological degree remains fixed, I will talk about new (partially conjectural) stability patterns near the virtual cohomological dimension. These patterns involve comparing cohomology groups in different cohomological degrees. This talk will include joint work with Brück, Church, Nagpal, Patzt, Reinhold, Sroka, and Wilson.

The principal congruence subgroup of SL_n(Z) of prime level p is the kernel of the mod p map SL_n(Z) to SL_n(Z/pZ). Its cohomology vanishes in degrees above n(n-1)/2. Lee and Szczarba gave a comparison map of its cohomology in top degree n(n-1)/2 to the top homology of an "oriented" version of the Tits building of F_p. We prove this map is surjective for all primes p and injective if and only if p=2,3,5. In particular, the case p=5 is a new and complete computation of the top cohomology. This is joint work with Jeremy Miller and Andrew Putman.

In this talk I will describe some current efforts to understand the high-degree rational cohomology of SL_n(Z), or more generally the cohomology of SL_n(O) when O is a number ring. I will survey some results, conjectures, and ongoing work toward this goal. We will see that a key approach is to construct appropriately "small" flat resolutions of an SL_n(O)-representation called the Steinberg module, and overview how we may hope to accomplish this by studying the topology of certain associated simplicial complexes. This talk includes work joint with Brück, Kupers, Miller, Patzt, Sroka, and Yasaki.

The relationship between special values of L-functions modular forms and Selmer group of modular p-adic Galois representations is a major theme in number theory. Given the developing mod p Langlands program, it is natural to ask whether there some kind of mod p analogue of the above theme. Notice that mod p modular forms do not have associated L-functions! In this talk, I will report on ongoing work with Lewis Combes in which we formulate, and computationally test, a connection between Selmer groups of mod p Galois representations and mod p Bianchi modular forms. This is inspired by a speculation of Calegari and Venkatesh.

In the first part of the talk, I will recall the definition and properties of the "Grayson-Stuhler filtration" of a Euclidean lattice, The definition is modeled on - and share some properties of - the Harder-Narasimhan filtration of vector bundles over curves. It has potential applications towards computation of the cohomology of arithmetic groups. In the second part, I will report on recent works about the "tensor product conjecture" (preservation of semistability under tensor product), with focus on isodual lattices (over Z, or more generally over the ring of integers of a number field).

The fundamental domain of an arithmetic Fuchsian group $\Gamma$ reveals a lot of interesting information about the group. An algorithm to compute this fundamental domain in practice was given by Voight, and it was later expanded by Page to the case of arithmetic Kleinian groups. Page's version features a probabilistic enumeration of group elements, which performs significantly better in practice. In this talk, we describe work to improve the geometric algorithms, and specialize Page's enumeration down to Fuchsian groups, to produce a final algorithm that is much more efficient. Optimal choices of constants in the enumeration are given by heuristics, which are supported by large amounts of data. This algorithm has been implemented in PARI/GP, and we demonstrate its practicality by comparing running times versus the live Magma implementation of Voight's algorithm.

Given an imaginary quadratic field, there is a finite number of equivalence classes of perfect forms over that field. We investigate these forms in the rank 2 case using a Voronoi's reduction theory. We show that the perfect forms cannot get too complicated, which allows us to give a lower bound on the number such perfect forms in terms of the discriminant of the field and the value of the Dedekind zeta function at 2. This is joint work with Kristen Scheckelhoff and Kalani Thalagoda.

A joint work of Bui Anh Tuan, Matthias Wendt and the speaker aims at pushing forward the frontiers of computations on mod p Farrell-Tate cohomology for arithmetic groups. We deal with p-rank 1 cases different from PSL(2). The conjugacy classification of cyclic subgroups Z/pZ, of order p, is reduced to the classification of modules of Z/pZ group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell-Tate cohomology computations, we discuss in detail the homological torsion in PGL(3) over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case. A copy of the paper is available on the BIRS workshop homepage.

Eisenstein cocyles are elements in the group cohomology of GL(n) that parametrize special values of L-functions. I will report on joint work with J. Flórez and C. Karabulut on our construction of Eisenstein cocyles over imaginary quadratic fields $K$, proving the integrality of Hecke L-functions attached to degree $n$ extensions of $K$. This gives a new proof of a result previously obtained by P. Colmez and L. Schneps, and most recently by N. Bergeron, P. Charollois, and L. Garcia. Time permitting, I will discuss work in progress on the interpolation of these special values via a p-adic L-function.

In this talk, we will explain how to use hypergeometric functions to compute period lattices of generalized Legendre curves based on the work of Archinard and Wolfart and automorphic forms on arithmetic triangle groups based on the work of Yang. From which we will see how some recent developments on hypergeometric functions over finite fields can be used to compute the action of Hecke operators on automorphic forms on arithmetic triangle groups.

In 1908 Voronoi introduced an algorithm that solves the lattice packing problem in any dimension in finite time. Voronoi showed that any lattice with optimal packing density must correspond to a so-called perfect (quadratic) form and his algorithm enumerates the finitely many perfect forms up to similarity in a fixed dimension. However, the number of non-similar perfect forms grows quickly in the dimension and as a result Voronoi’s algorithm has only been completely executed up to dimension 8. We discuss an upper bound on the number of perfect forms and the challenges that arise for completing Voronoi's algorithm in dimension 9.

Motivated by the work of Meyer, Millichap and Trapp [MMT] and by Thurston, I shall present an elementary polyhedral approach to study and deduce results about the arithmeticity and commensurability of an infinite family of hyperbolic link complements M_n for n>2. The manifold M_n is the complement of the 3-sphere by the (2n)-link chain. The hyperbolic structure of M_n stems from an ideal right-angled polyhedron that can be cut into four copies of an ideal right-angled n-gonal antiprism. Each of these polyhedra gives rise to a hyperbolic Coxeter orbifold that is commensurable to a hyperbolic orbifold with a single cusp. Vinberg's arithmeticity criterion and certain cusp density and volume computations allow us to reproduce some of the main results in [MMT] about M_n in a comparatively elementary and direct way. This approach works in several other cases of link complements as well.

I will discuss work with M. Frączyk and S. Hurtado which implies the following statements: given a semisimple Lie group G there is a constant C such that for any (torsion-free) lattice Γ\Gamma in G, the size of the torsion subgroups of all its homology groups is at most C^v where v is its covolume in G. We prove this by constructing a simplicial complex with O(v) vertices and bounded degree which is a classifying space for Γ\Gamma, solving a conjecture of T. Gelander

Let $F$ be an imaginary cylic sextic field with discriminant $\Delta$ and the ring of integers $O_F$. The size function $h^0$ for $F$ is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. By Van der Geer and Schoof's conjecture, on the set of all (isometric) ideal lattices of covolume $\sqrt{|\Delta|}$ the function $h^0$ attains its maximum at the trivial ideal lattice $O_F$. In this talk we will discuss the main idea to prove that the conjecture holds for $F$.

Lattice-based cryptography relies on generating random bases which are difficult to fully reduce. Given a lattice basis (such as the private basis for a cryptosystem), all other bases are related by multiplication by matrices in GL(n, Z). We compare the strengths of various methods to sample random elements of SL(n, Z), finding some are stronger than others with respect to the problem of recognizing rotations of the Z^n lattice. In particular, the standard algorithm of multiplying unipotent generators together (as implemented in Magma's RandomSLnZ command) generates instances of this last problem which can be efficiently broken, even in dimensions nearing 1,500. We also can efficiently break the random basis generation method in one of the NIST Post-Quantum Cryptography competition submissions (DRS). Other random basis generation algorithms (some older, some newer) are described which appear to be much stronger.

We present an Hermitian version of the floating-point LLL reduction algorithm of Nguyen and Stehlé (2009). This new variant works on imaginary quadratic fields which are norm-Euclidean and also for some adequate cyclotomic fields. An optimized C++ implementation has been performed and results show a significant improvement for Hermitian lattices reduction of dimension N when compared to floating-point LLL reduction on the corresponding Euclidean lattice of dimension 2N . We demonstrate our implementation in the special case of Gaussian integers.

After revisiting the basics of algorithmic reduction theory for lattices under a more algebraic geometric prism, we present generic strategies to enhance the reduction over algebraic lattices over number fields (a.k.a. hermitian vector bundles over arithmetic curves) and see how we can leverage symplectic symmetries to design faster processes. Such techniques can be used to parallelize and speed up the core computations in algorithmic number theory and for the tractable cohomologies of arithmetic groups.

Over several work that I did, I use a combination of tools from group theory, polyhedral geometry in order to compute geometric or topological information. I have now shifted most of my programs to a C++ framework in order to achieve the best performance. All of the software is open source and I will present what has been done, the issues and what can be done in the future. I will present here what parts are relevant to lattice and cohomology theories. ---The foundational part of a lot of this is "permutalib" which is a permutation group library that allows to compute set-stabilizer and other operations needed for polyhedral computation which is 10 times faster than GAP. ---A direct application of it is the computation of the automorphism group of polytope. Another fundamental construction is the canonical form of a polytope which greatly helps with enumeration tasks. ---This also translates into an algorithm for the computation of the canonical form of a quadratic form. An illustration of this was the enumeration of C-type in dimension 6 where we found 55 million types in reasonable time. ---We also provide efficient algorithms for dual description using symmetries where we achieve a two-fold improvement over GAP. ---We also provide an implementation of the Vinberg algorithm using all the above that allows us to solve some 19 dimensional examples easily. The point of this presentation is not really to concentrate on specific problems but to show approaches that allow us to treat large problems.

The famous arithmeticity and superrigidity results of Margulis allow to classify lattices in higher rank Lie groups up to commensurability. It is known that two non-commensurable lattices can still be profinitely commensurable, i.e., their profinite completions have isomorphic open subgroups. In this talk I will explain how lattices in higher rank can be classified up to profinite commensurability (modulo the congruence subgroup problem). We will see that profinitely commensurable lattices exist in most simple Lie groups of higher rank. More surprisingly, such examples cannot exist in the complex Lie groups of type E_8, F_4 and G_2. This is based on joint work with Holger Kammeyer.