arXiv subjects

Jinbo Ren

Publications and source records attributed to Jinbo Ren.

5 recordsLinked to original sources

arXiv2023

Purely exponential parametrizations and their group-theoretic applications

This paper is mainly motivated by the analysis of the so-called Bounded Generation property (BG) of linear groups (in characteristic $0$), which is known to admit far-reaching group-theoretic implications. We achieve complete answers to certain longstanding open questions about Bounded Generation (sharpening considerably some earlier results). For instance, we prove that linear groups boundedly generated by semi-simple elements are necessarily virtually abelian. This is obtained as a corollary of sparseness of subsets which are likewise generated. In the paper in fact we go further, framing (BG) in the more general context of (Purely) Exponential Parametrizations (PEP) for subsets of affine spaces, a concept which unifies different issues. Using deep tools from Diophantine Geometry (including the Subspace Theorem), we systematically develop a theory showing in particular that for a (PEP) set over a number field, the asymptotic distribution of its points of Height at most $T$ is always $\sim c(\log T)^r$, with certain constants $c>0$ and $r\in \mathbb{Z}_{\geq 0}$. (This shape fits with a well-known viewpoint first put forward by Manin.)

math.NT↗

arXiv2022

Bounded Generation by semi-simple elements: quantitative results

We prove that for a number field $F$, the distribution of the points of a set $Σ\subset \mathbb{A}_F^n$ with a purely exponential parametrization, for example a set of matrices boundedly generated by semi-simple (diagonalizable) elements, is of at most logarithmic size when ordered by height. As a consequence, one obtains that a linear group $Γ\subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero admits a purely exponential parametrization if and only if it is finitely generated and the connected component of its Zariski closure is a torus. Our results are obtained via a key inequality about the heights of minimal $m$-tuples for purely exponential parametrizations. One main ingredient of our proof is Evertse's strengthening of the $S$-Unit Equation Theorem.

math.NT↗

arXiv2021

Non-virtually abelian anisotropic linear groups are not boundedly generated

We prove that if a linear group $Γ\subset \mathrm{GL}_n(K)$ 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 never boundedly generated. Our proof relies on Laurent's theorem from Diophantine geometry and properties of generic elements.

math.GR↗

arXiv2018

Conjugation of semisimple subgroups over real number fields of bounded degree

Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field extension of $k$ of degree bounded independently of the subgroups. Moreover, if $k$ is a real number field, we show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over the field of real numbers $\mathbb{R}$ are actually conjugate over a finite real extension of $k$ of degree bounded independently of the subgroups.

math.GR↗

arXiv2018

Applications of the hyperbolic Ax-Schanuel conjecture

In 2014, Pila and Tsimerman gave a proof of the Ax-Schanuel conjecture for the $j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax-Schanuel conjecture. In this article, we show that the hyperbolic Ax-Schanuel conjecture can be used to reduce the Zilber-Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila-Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila-Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber-Pink conjecture for curves in abelian varieties.

math.NT↗