Schedule for: 22w5161 - Arithmetic Aspects of Algebraic Groups

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

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

An algebraic group $G$ over a field $F$ is called special if for every field extension $K/F$ all $G$-torsors (principle homogeneous $G$-spaces) over $K$ are trivial. Examples of special groups are special and general linear groups, symplectic groups. A. Grothendieck classified special groups over an algebraically closed field. In 2016, M. Huruguen classified special reductive groups over arbitrary fields. We improve the classification given by Huruguen.

The algebraic form of Hilbert's 13th Problem asks for the resolvent degree rd(n) of the general polynomial f(x) = x^n + a_1 x^{n-1} + ... + a_n of degree n, where a_1, ..., a_n are independent variables. Here rd(n) is the minimal integer d such that every root of f(x) can be obtained in a finite number of steps, starting with C(a_1, ..., a_n) and adjoining an algebraic function in <= d variables at each step. It is known that rd(n) = 1 for every n <= 5. It is not known whether or not rd(n) is bounded as n tends to infinity; it is not even known whether or not rd(n) > 1 for any n. Recently Farb and Wolfson defined the resolvent degree rd_k(G), where G is a finite group and k is a field of characteristic 0. In this setting rd(n) = rd_C(S_n), where S_n is the symmetric group on n letters and C is the field of complex numbers. In this talk I will define rd_k(G) for any field k and any algebraic group G over k. Surprisingly, Hilbert's 13th Problem simplifies when G is connected. In particular, I will explain why rd_k(G) <= 5 for an arbitrary connected algebraic group G defined over an arbitrary field k.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

We describe the first Galois cohomology set $H^1(R,G)$ of a connected reductive group $G$ over the field of real numbers $\mathbb{R}$ by the method of Borel and Serre and by the method of Kac.

This work was motivated by the problem of describing cyclic Galois extensions and differential crossed product algebras in mixed characteristic, with the the goal of liftingfrom arbitrary characteristic $p$ rings to suitable characteristic $0$ rings. The first step was the construction of Artin-Schreier like polynomials and extensions in mixed characteristic, where the group acts by $sigma(x) = \rho x+1$ ($\rho^p = 1$). This leads to Azumaya algebras $A$ defined by $xy-\rho yx = 1$. Of course, we want to generalize to higher degrees than $p$. This leads to an Albert like criterion for extending cyclic Galois extensions of rings and to the definition and study of almost-cyclic Azumaya algebras, generalizing $A$ above.

Elements in the Brauer group of an elliptic curve $E$ may be described as tensor products of symbol algebras over the function field of $E$ by the Merkurjev-Suslin Theorem. The symbol length is the smallest number $n$ so that every element in the Brauer group can be expressed as a tensor product of at most $n$ symbols. In this talk, I will describe bounds on the symbol length of $E$. In particular, I will show that the symbol length in the prime torsion for a prime $q$ of a CM elliptic curve over a number field is bounded above by $q+1$. This is joint work with Mateo Attanasio, Caroline Choi, and Andrei Mandelshtam.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Techniques involving reduction are very common in number theory and arithmetic geometry. In particular, elliptic curves and general abelian varieties having good reduction have been the subject of very intensive investigations over the years. The purpose of this talk is to report on recent work that focuses on good reduction in the context of reductive linear algebraic groups over higher-dimensional fields. (This is joint work with V. Chernousov and A. Rapinchuk.)

We produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Our key tools are the classical Demazure formula for the characteristic map and the elementary properties of divided-difference operators. Using this algorithm we confirm all previously known bounds by Karpenko and Devyatov as well as we produce new bounds (e.g. for groups of types $F_4$, adjoint $E_6$, for some semisimple groups).

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We provide a uniform bound for the index of cohomology classes in $H^i(F, \mu_\ell^{\otimes i-1})$ when $F$ is a semiglobal field (i.e., a one variable function field over a complete discretely valued field $K$). The bound is given in terms of the analogous data for the residue field of $K$ and its finitely generated extensions of transcendence degree at most one. We obtain an explicit bound in example cases when the information on the residue field is known. This is joint work with David Harbater and Daniel Krashen

Given a smooth, geometrically connected and projective curve C defined over a finite field k, let K=k(C) be the function field of rational functions on C. The Tamagawa number \tau(G) of a semisimple K-group G is defined as the covolume of the discrete group G(K) (embedded diagonally) in the adelic group G(A) with respect to the Tamagawa measure. The Weil conjecture, recently proved by Gaitsgory and Lurie, states that if G is simply-connected then \tau(G)=1. Our aim is to prove (this work is in progress), without relying on the Weil conjecture, the following fact: Let G be a quasi-split inner form of a split semisimple and simply-connected K-group G_1. Then \tau(G)=\tau(G_1). This Theorem can serve as a part of an alternative proof to the Weil conjecture. Joint work with Gunter Harder, Ralf Kohl and Claudia Schoemann

Let $G$ be a Kac-Moody group over $\mathbb Q$. This is a group associated to an infinite dimensional Lie algebra. The discrete form $G(\mathbb Z)$ of $G$ has many appearances in physical applications. Mathematical description of $G(\mathbb Z)$ is given by Tits. Tits' functorial definition of these groups is not suitable for our purposes. We made some progress on finding a relationship between different discrete forms of $G$. Our results generalize Chevalley's fundamental theorem on the integrality for finite dimensional semisimple Lie groups. We will report on our joint work with Lisa Carbone (Rutgers), Dongwen Liu (Zhejiang) and Scott S Murray (Toronto).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

An abstract group $\Gamma$ has the property of {\bf bounded generation} if it is equal to a product of finitely many fixed cyclic groups. Being a purely combinatorial property of groups, bounded generation has a number of interesting consequences and applications in different areas including Kazhdan's constants computation, semi-simple rigidity, the Margulis-Zimmer conjecture and the Serre's congruence subgroup problem. In this talk, I will prove that if a linear group $\Gamma \subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero is boundedly generated by semi-simple (diagonalizable) elements then it is virtually solvable. As a consequence, one obtains that infinite $S$-arithmetic subgroups of absolutely almost simple anisotropic algebraic groups over number fields are {\it never} boundedly generated. Our proof relies on the subspace theorem (a far-reaching generalization of Roth's Fields medal work) from Diophantine approximation and properties of generic elements. This is joint work with Corvaja, A. Rapinchuk and Zannier.

It is known that conjugacy classes of elements in orthogonal groups over number fields have finite width. In this talk we will present evidence that the same is true for conjugacy classes of elements in higher rank arithmetic groups of orthogonal type. We will show that the question whether a conjugacy class of such a group has a finite width can be viewed as a congruence subgroup problem on a non-standard saturated model of the group. This is joint work with Nir Avni.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Joint work with A. Lourdeaux and A. Pianzola. In the talk we sketch a proof that Albert algebras arising from the first Tits construction are determined uniquely up to an isomorphism by the Rost cohomological invariant $g_3$.

We discuss recent results on bounded elementary generation and bounded commutator width for Chevalley groups over Dedekind rings of arithmetic type in positive characteristic. In particular, Chevalley groups of rank greater than 1 over polynomial rings and Chevalley groups of arbitrary rank over Laurent polynomial rings (in both cases the coefficients are taken from a finite field) are boundedly elementarily generated. We establish rather plausible explicit bounds. We discuss applications to Kac-Moody groups and various model theoretic consequences. Finally, we plan to state conjectures which look quite tempting. joint work with B. Kunyavskii, N. Vavilov

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

During the last decades long-standing conjectures in number theory have been reformulated and, subsequently, some of them successfully solved using the homogeneous dynamics approach. The approach is based on the description of the closures of orbits for the natural action of subgroups H of an algebraic group G on the homogeneous space G/Γ where Γ is an arithmetic subgroup of G. The closures of such orbits are well-understood when H is unipotent and considerably less when H is a torus. We will describe some recent results for the action of maximal (split or non-split) tori on G/Γ and related applications.

I will discuss a recent work with Tali Kaufman. We show that if arithmetic groups with certain properties exist, then one can construct good 2-query locally testable codes. The device that enables this is a novel notion of sheaves on simplicial complexes. The latter are typically taken to be quotients of an affine building by an arithmetic group, e.g., Ramanujan complexes.

We discuss a weak Hasse principle for a smooth intersection of two quadrics in P^5 and connections to period index questions for the associated hyperelliptic curves.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.