Schedule for: 20w5133 - Arithmetic Aspects of Algebraic Groups (Online)

Let $F$ be a field which is prime to $p$ closed. We show that any twisted polynomial in $D[y,\sigma]$ of degree at most $p-1$ ($K/F$ a cyclic Galois extension of degree $p$) splits into linear factors. As an application we show that a standard Kummer space is a degree $p$ symbol algebra $D=(a,b)_{p,F}$ generates the multiplicative group $D^{\times}$. We also show that $GL_p(F)$ is also generated by any standard Kummer space.

In this talk we shall explain a method to translate arithmetic information to algebraic data in the context of the study of the unramified Brauer group of tori.

Any central simple algebra $A$ over a field $K$ is a form of a matrix algebra. Further $A/K$ comes equipped with a reduced norm map which is obtained by twisting the determinant function. Every element in the commutator subgroup $[A^*, A^*]$ has reduced norm 1 and hence lies in $SL_1(A)$, the group of reduced norm one elements of A. Whether the reverse inclusion holds was formulated as a question in 1943 by Tannaka and Artin in terms of the triviality of the reduced Whitehead group $SK_1(A) := SL_1(A)/[A^*, A^*]$. $$ $$ Platonov negatively settled the Tannaka-Artin question by giving a counter example over a cohomological dimension (cd) 4 base field. In the same paper however, the triviality of $SK_1(A)$ was shown for all algebras over cd at most 2 fields. In this talk, we investigate the situation for $l$-torsion algebras over a class of cd 3 fields of some arithmetic flavour, namely function fields of $p$-adic curves where l is any prime not equal to p. We partially answer a question of Suslin by proving the triviality of the reduced Whitehead group for these algebras. The proof relies on the techniques of patching as developed by Harbater-Hartmann-Krashen and exploits the arithmetic of these fields.

I will give a brief sketch of why the number of special unitary groups of quaternionic skew-hermitian forms with good reduction at a set of discrete valuations is finite for certain fields. This answers a conjecture of Chernousov, Rapinchuk and Rapinchuk for groups of this type.

A large family of classical arithmetic problems (in algebraic groups) including $$ $$ (a) Finding rational solutions of the so-called trigonometric Diophantine equation $F(\cos 2\pi x_i, \sin 2\pi x_i)=0$, where $F$ is an irreducible multivariate polynomial with rational coefficients; $$ $$ (b) Determining all $\lambda \in \mathbb{C}$ such that $(2,\sqrt{2(2-\lambda)})$ and $(3, \sqrt{6(3-\lambda)})$ are both torsion points of the elliptic curve $y^2=x(x-1)(x-\lambda)$; $$ $$ can be regarded as special cases of the Zilber-Pink conjecture in Diophantine geometry. In this short talk, I will explain how we use tools from mathematical logic to attack this conjecture. In particular, I will present a series partial results toward the Zilber-Pink conjecture, including those proved by Christopher Daw and myself.

I will overview work with Uri Bader, David Fisher, and Nick Miller on superrigidity of certain representations of lattices in $SO(n,1)$ and $SU(n,1)$. Our primary application of this superrigidity theorem is to prove arithmeticitiy of finite volume real or complex hyperbolic manifold containing infinitely many maximal totally geodesic submanifolds, answering a question due independently to Alan Reid and Curtis McMullen.

Joint work with S. Alsaody and A. Pianzola. In the first part of the talk we remind the definition of an $R$-equivalence (introduced by Manin), state the Tits-Weiss conjecture on generation of structure groups of Albert algebras by $U$-operators and the Kneser-Tits conjecture for isotropic groups. In the second part of the talk we focus on computation of $R$-equivalence classes for groups of type $E_6$. As applications of our result we prove the Tits-Weiss conjecture and the Kneser-Tits conjecture for some isotropic groups of type $E_7$ and $E_8$.

Please turn on your cameras for the "group photo" -- a screenshot in Zoom's Gallery view.

One of the most fundamental results underlying the theory of abelian varieties is "rigidity" -- that is, that any k-scheme morphism of abelian varieties which preserves identities is actually a k-group homomorphism. This result depends crucially upon the properness of such varieties. For affine groups in general, there is no analogous rigidity statement. We will nevertheless show that such a rigidity result holds for unirational groups (which are always affine) satisfying certain conditions, and discuss several implications.

The Clifford algebra is an object intimately connected with the theory of quadratic forms and orthogonal groups. This classical notion of Clifford algebras associated to quadratic forms can be generalized to higher degree. In this talk, I will discuss a generic version of the Clifford algebra associated to a binary cubic form. This algebra defines a nontrivial Brauer class in the Brauer group of a relative elliptic curve – the Jacobian of the universal genus one curve obtained from the Clifford algebra. This is joint work in progress with Rajesh Kulkarni.

In its classical setting, the Congruence Subgroup Problem (CSP) asks whether every finite index subgroup of $GL_{n}(\mathbb{Z})$ contains a principal congruence subgroup of the form \[ \ker(GL_{n}(\mathbb{Z})\to GL_{n}(\mathbb{Z}/m\mathbb{Z})) \] for some $m\in\mathbb{Z}$. It was known already in the 19th century that for $n=2$ the answer is negative, and actually $GL_{2}(\mathbb{Z})$ has many finite index subgroups which do not come from congruence considerations. On the other hand, quite surprisingly, it was proved in the sixties by Mennicke and by Bass-Lazard-Serre that for $n\geq 3$ the answer to the CSP is affirmative. This breakthrough led to a rich theory of the CSP for general arithmetic groups. $$ $$ Viewing $GL_{n}(\mathbb{Z})\cong Aut(\mathbb{Z}^{n})$ as the automorphism group of $\Gamma=\mathbb{Z}^{n}$, one can generalize the CSP to automorphism groups as follows: Let $\Gamma$ be a finitely generated group; does every finite index subgroup of $Aut(\Gamma)$ contain a principal congruence subgroup of the form \[ \ker(Aut(\Gamma)\rightarrow Aut(\Gamma/M)) \] for some finite index characteristic subgroup $M\leq\Gamma$? Considering this generalization, there are very few results when $\Gamma$ is non-abelian. For example, only in 2001 Asada proved, using concepts from Algebraic Geometry, that $Aut(F_{2})$ has an affirmative answer to the CSP, when $F_{2}$ is the free group on two generators. For $Aut(F_{n})$ when $n\geq 3$ the problem is still unsettled. $$ $$ In the talk I will give a survey of some recent results regarding the case where $\Gamma$ is non-abelian. We will see that when $\Gamma$ is a nilpotent group the CSP for $Aut(\Gamma)$ is completely determined by the CSP for arithmetic groups. We will also see that when $\Gamma$ is a finitely generated free metabelian group the picture changes and we have a dichotomy between $n=2,3$ and $n\geq 4$.