Coset bounds for algebraic geometric codes

Bounding the Number of Maximal Torsion Cosets on Subvarieties of Algebraic Tori

We obtain bounds on the number of maximal torsion cosets for algebraic subvarieties of n-tori, defined over the rationals, using model theoretic methods.

Solving algebraic equations in roots of unity

This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an algebraic subvariety of the complex algebraic $n$-torus ${\mathbb G}_{\rm m}^n$. In contrast to earlier works that give the bounds of polynomial growth in the maximum total degree of defining polynomials, the proofs of our results are constructive. This…

The finiteness of the genus of a finite-dimensional division algebra, and some generalizations

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple algebraic groups of type $\textsf{G}_2$. These applications involve the double cosets of adele groups of algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of algebraic groups and hence have been actively analyzed, similar question over more…

The number of maximal torsion cosets in subvarieties of tori

We present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus $\mathbb{G}_{\textrm{m}}^n$. Our first main result gives a bound in terms of the degree of the defining polynomials. A second result gives a bound in terms of the toric degree of the subvariety. As a consequence, we prove the conjectures of Ruppert and of Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively…

$p$-adic 有限维单重群的几何引理对偶与第二对偶定理

Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$-adic reductive groups

本文研究了$p$-adic有限维单重群中几何引理对偶与第二对偶定理的关系，并推广了Bezrukavnikov-Kazhdan关于第二对偶性显式描述的结果。

Geometric modularity for algebraic and non-algebraic weights

In this short paper we generalise a result of Diamond--Sasaki connecting geometric modularity of algebraic weights to geometric modularity of non-algebraic weights and vice versa. In particular, we show that geometric modularity of non-algebraic weights implies geometric modularity of multiple algebraic weights, which is easy to see. More significantly, we show that geometric modularity of multiple algebraic weights sometimes implies geometric modularity of a non-algebraic weight. This connects work of Diamond--Sasaki to earlier work on generalisations of the weight part of Serre's conjecture.