Workshop (Week 3)
Week 3: June, 27 June 30, 2022
Workshop on Cohomology, Geometry and Explicit Number Theory
This workshop is mainly aimed at PhD students and young researchers, with some talks given by experts of the fields.
Talks will be given from Monday June 27 early afternoon to June 30 late afternoon. The morning of Friday 1st of July will be free.
The final list of speakers include:
 Angelica Babei (McMaster University, Canada)
 Title: A family of $\phi$congruence subgroups of the modular group
 Abstract: In this talk, we introduce families of subgroups of finite index in the modular group, generalizing the congruence subgroups. One source of such families is studying homomorphisms of the modular group into linear algebraic groups over finite fields. In particular, we examine a family of noncongruence subgroups arising from a map into a quasiunipotent group. Using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve $y^2=x^31728$ defined by the commutator subgroup of the modular group, we provide a detailed discussion of the corresponding modular forms.
 Calista Bernard (University of Minnesota, USA)
 Title: Applications of twisted homology operations for E_nalgebras
 Abstract: An E_nalgebra is a space equipped with a multiplication that is commutative up to homotopy. Such spaces arise naturally in geometric topology, number theory, and mathematical physics; some examples include classifying spaces of braid groups, spaces of long knots, and classifying spaces of general linear groups. The structure of an E_nalgebra gives rise to operations on the homology of a space, and these operations prove quite useful for studying homology. In the 70s F. Cohen and J.P. May gave a complete description of operations on the mod p homology of E_nalgebras, and more recently I have worked on generalizing their results to homology with twisted coefficients. In this talk I will give a brief introduction to E_nalgebras and the theory of operations, and I will then discuss work in progress on applications of this theory to studying the homology of special linear groups SL_n(Z) and to studying the twisted homology of mixed braid groups.
 Tobias Braun (RWTH Aachen, Germany)
 Benjamin Brück (ETHZ, Switzerland)
 Title: Highdimensional rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{Sp}_{2n}(\mathbb{Z})$
 Abstract: By a result of ChurchPutman, the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes in "codimension one", i.e. $H^{{n \choose 2} 1}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 3$, where ${n \choose 2}$ is the virtual cohomological dimension of $\operatorname{SL}_n(\mathbb{Z})$. I will talk about work in progress on two generalisations of this result: The first project is joint work with MillerPatztSrokaWilson (see https://arxiv.org/abs/2204.11967). We show that the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes in codimension two, i.e. $H^{{n \choose 2} 2}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 3$. The second project is joint with PatztSroka. Its aim is to study whether the rational cohomology of the symplectic group $\operatorname{Sp}_{2n}(\mathbb{Z})$ vanishes in codimension one, i.e. whether $H^{n^2 1}(\operatorname{Sp}_{2n}(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 2$.
 Kieran Child (University of Bristol, UK)
 Title: Computation of weight 1 modular forms
 Abstract: A major achievement of modern number theory is the proof of a bijection between odd, irreducible, 2dimensional Artin representations and holomorphic weight 1 Hecke eigenforms. Despite this result, concrete examples have proven difficult to produce owing to weight 1 being noncohomological, and the contribution to the discrete spectrum from modular forms being inseparable from the contribution from Maass forms. In this talk, I will cover recent work towards an improved method for computing weight 1 forms, and report on the implementation of this method by which we computed all such forms up to level 10,000.
 Lewis Combes (University of Sheffield, UK)
 Title: Computing Selmer groups attached to mod p Galois representations
 Abstract: Selmer groups attached to a padic Galois representation have been studied thoroughly, but their mod p cousins have so far received less attention. In this talk we explain the construction of the padic Selmer group, how it translates to the mod p setting, and give some progress on understanding the ranks of some Selmer groups over various fields.
 Petru Constantinescu (MPI Bonn, Germany)
 Title: On the distribution of modular symbols and cohomology classes
 Abstract: Motivated by a series of conjectures of Mazur, Rubin and Stein, the study of the arithmetic statistics of modular symbols has received a lot of attention in recent years. In this talk, I will highlight several results about the distribution of modular symbols, including their Gaussian distribution and the residual equidistribution modulo p. I will also discuss about generalisations to cohomology classes in higher dimensions. Part of this talk is joint work with Asbjørn Nordentoft.
 Oussama Hamza (Western University, Canada)
 Title: Hilbert series and mild groups
 Abstract:
Let $p$ be an odd prime number and $G$ a finitely generated pro$p$ group. Define $I(G)$ the augmentation ideal of the group algebra of $G$ over $F_p$ and define the Hilbert series of $G$ by: $G(t):=sum_{n\in \NN} \dim_{\F_p} I^n(G)/I^{n+1}(G)$.The series $G(t)$ gives several information on $G$. First, during the $60$'s, Golod and Shafarevich used Hilbert series to relate the number of generators and relations defining $G$, to the cardinality of $G$. Also, if $G(t)$ satisfies some equalities, we can read the cohomological dimension of $G$. Between 1980 and 2000's, Anick and Labute, introduced a sufficient and easy condition on the relations of pro$p$group $G$, such that $G(t)$ satisfies some equality ensuring that $G$ is of cohomological dimension less than two. Groups satisfying this sufficient condition are called mild.If we consider $K$ a number field, then using mild groups, we can construct extensions$L/K$, in which infinitely many primes split completely, and with prescribed Hilbert series Gal(L/K)(t). Consequently Gal(L/K) is finitely generated, infinitely presented and of cohomological dimension $2$.
 Richard Hill (University College London, UK) * joint session with COGENT regular seminar *
 Title:Fractional weight modular forms
 Abstract: (This is joint work with Eberhard Freitag). It has been known since the 1930s that for all positive rational numbers p/q, there exist holomorphic modular forms on SL(2,R) with weight p/q. This contrasts with the situation for Sp(2n,R) with n >1, where one has only integral and halfintegral weight forms. Until recently, it was an open question whether there is any other Lie group (other than SL_2(R)) with holomorphic modular forms whose weight is neither integral nor halfintegral. In this talk I will describe how we recently found examples of holomorphic modular forms of weight 1/3 on the group SU(2,1).
 Zachary Himes (Purdue University, USA)
 Title: On not the rational dualizing module for $\text{Aut}(F_n)$
 Abstract: BestvinaFeighn proved that $\text{Aut}(F_n)$ is a rational duality group, i.e. there is a $\mathbb{Q}[\text{Aut}(F_n)]$module, called the rational dualizing module, and a form of Poincar\'e duality relating the rational cohomology of $\text{Aut}(F_n)$ to its homology with coefficients in this module. BestvinaFeighn's proof does not give an explicit combinatorial description of the rational dualizing module of $\text{Aut}(F_n)$. But, inspired by BorelSerre's description of the rational dualizing module of arithmetic groups, HatcherVogtmann constructed an analogous module for Aut(F_n) and asked if it is the rational dualizing module. In work with Miller, Nariman, and Putman, we show that HatcherVogtmann's module is not the rational dualizing module.
 Michael Lipnowski (McGill University, Canada) * joint session with COGENT regular seminar *
 Title: Rigid meromorphic cocycles for orthogonal groups
 Abstract: In the early 2000s, Darmon initiated a fruitful study of analogies between Hilbert modular surfaces and quotients Y := SL_2(ZZ[1/p]) \ H x H_p, where H is the complex upper half plane and H_p is Drinfeld's padic upper half plane. As Y mixes complex and padic topologies, making direct sense of Y as an analytic space seems difficult. Nonetheless, Y supports a large collection of exotic special points  corresponding to the units of real quadratic fields which are inert at p  and DarmonVonk have described an incarnation of meromorphic functions on Y, so called rigid meromorphic cocycles. This talk describes joint work with Henri Darmon and Lennart Gehrmann, in which we study generalizations Y' of the space Y to orthogonal groups G for quadratic spaces over QQ of arbitrary real signature. The spaces Y' support large collections of exotic special points  corresponding to subtori of G of maximal real rank  and we define explicit rigid meromorphic cocycles on Y'; these RMCs are analogous to meromorphic functions on orthogonal Shimura varieties with prescribed special divisors first studied by Borcherds, and they generalize the RMCs constructed by DarmonVonk. We will also discuss some computations suggesting that values of our RMCs at special points might realize new instances of explicit class field theory.
 Tobias Moede (TU Braunschweig, Germany)
 Title: Coclass theory for nilpotent associative algebras
 Abstract: The coclass of a finite pgroup of order p^n and class c is defined as nc. Using coclass as the primary invariant in the investigation of finite pgroups turned
out to be a very fruitful approach. Together with Bettina Eick, we have developed a coclass theory for nilpotent associative algebras over fields. A central tool are the coclass graphs associated with the algebras of a fixed coclass. The graphs for coclass zero are well understood. We give a full description for coclass one and explore graphs for higher coclasses. We prove several structural results for coclass graphs, which yield results in the flavor of the coclass theorems for finite pgroups. The most striking observation in our experimental data is that for finite fields all of these graphs seem to exhibit a periodic pattern. A similar periodicity in the graphs for finite pgroups has been proved independently by du Sautoy using the theory of zeta functions and by Eick & LeedhamGreen using cohomological methods. We give an outlook on how a proof might proceed and how the periodicity may be exploited to describe the infinitely many nilpotent associative Falgebras of a fixed coclass by a finite set of data.
 Andrew Putman (University of Notre Dame, USA)
 Title: The Steinberg representation is irreducible
 Abstract: The Steinberg representation is a topologicallydefined representation of groups like GL_n(k) that plays a fundamental role in the cohomology of arithmetic groups. The main theorem I will discuss says that for infinite fields k, the Steinberg representation is irreducible. For finite fields, this is a classical theorem of Steinberg. No background in representation theory will be assumed. This is joint work with Andrew Snowden.
 Alexander Rahm (Université de la Polynésie Française, France)
 Title: Verification of the Quillen conjecture in the rank 2 imaginary quadratic case
 Abstract: We confirm a conjecture of Quillen in the case of the mod 2 cohomology of arithmetic groups SL(2, A[1/2]), where A is an imaginary quadratic ring of integers. To make explicit the free module structure on the cohomology ring conjectured by Quillen, we compute the mod 2 cohomology of SL(2, B[1/2]) with B the imaginary quadratic ring of discriminant 8 (obtained as the ring of integers from the imaginary quadratic field generated by the squareroot of 2) via the amalgamated decomposition of the latter group. This is joint work with Tuan Anh Bui.
 Leonie Scheeren (RWTH Aachen, Germany)
 Title: The Voronoi Algorithm for calculating unit groups
 Abstract: Following an algorithm by Coulangeon, Nebe, Braun and Schönnenbeck to compute unit groups of orders in finite simple Qalgebras, we will again talk about the Voronoi algorithm for finding perfect forms. In this case we will work with a generalized version by Opgenorth in the context of dual cones and consider a combination with BassSerre theory regarding groups acting on trees via automorphisms, which yields information about the structure of said unit groups, which will be acting on a Voronoigraph. In particular we are going to focus on finding unit groups of maximal orders in indefinite Quaternion algebras over Q and compute an example.
 Kalani Thalagoda (UNC Greensboro, USA)
 Title: Bianchi modular forms
 Abstract: Bianchi Modular Forms are generalizations of classical modular forms for imaginary quadratic fields. Similar to the classical case, we can use the theory of modular symbols for computation. However, when the class group of the field is nontrivial, we can only compute certain components of the modular form. However, we can use techniques developed by Cremona and his students to extract the Bianchi Modular forms as Hecke eigensystems. In this talk, I will go over some of these computational techniques for the field $\mathbb{Q}(\sqrt{17})$ which has class number $4$.
 Haowen Zhang (Sorbonne Université, France)
 Title: BrauerManin and cohomological obstructions to rational points
 Abstract: In the problem of deciding integer or rational solutions of polynomial equations (i.e. finding integer/rational points of a variety), we often first look at the “local” solutions over all the Q_p. When does a collection of local solutions give rise to an honest global solution over Q? There are necessary conditions given by certain cohomological groups like the Brauer group, and these conditions sometimes turn to be sufficient, defining the BrauerManin obstruction. We can also use such tools to study the problem of weakapproximation. Going beyond number fields, we can consider such problems of certain function fields of the shape C(X,Y), C((X,Y)), Q_p(X) etc and I’ll try to present some of my results on such problems.
