arXiv subjects

Alex Torzewski

Publications and source records attributed to Alex Torzewski.

4 recordsLinked to original sources

arXiv2023

On the existence of free sublattices of bounded index and arithmetic applications

Let $\mathcal{O}$ be a Dedekind domain whose field of fractions $K$ is a global field. Let $A$ be a finite-dimensional separable $K$-algebra and let $\Lambda$ be an $\mathcal{O}$-order in $A$. Let $n$ be a positive integer and suppose that $X$ is a $\Lambda$-lattice such that $K \otimes_{\mathcal{O}} X$ is free of rank $n$ over $A$. Then $X$ contains a (non-unique) free $\Lambda$-sublattice of rank $n$. The main result of the present article is to show there exists such a sublattice $Y$ such that the generalised module index $[X : Y]_{\mathcal{O}}$ has explicit upper bounds with respect to division that are independent of $X$ and can be chosen to satisfy certain conditions. We give examples of applications to the approximation of normal integral bases and strong Minkowski units, and to the Galois module structure of rational points over abelian varieties.

math.NT↗

arXiv2022

Bounds on the number of rational points of curves in families

In this note, we give an alternative proof of uniform boundedness of the number of integral points of smooth projective curves over a fixed number field with good reduction outside of a fixed set of primes. We use that due to Bertin-Romagny, the Kodaira-Parshin families constructed by Lawrence-Venkatesh can themselves be assembled into a family. We then repeat Lawrence-Venkatesh's study of the p-adic period map, together with the comparison of nearby fibres.

math.NT↗

arXiv2018

Functoriality of motivic lifts of the canonical construction

Let (G,X) be a Shimura datum and K a neat open compact subgroup of $G(\mathbb{A}_f)$. Under mild hypothesis on (G,X), the canonical construction associates a variation of Hodge structure on $\textrm{Sh}_K(G,X)(\mathbb{C})$ to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of the canonical construction which takes values in relative Chow motives over $\textrm{Sh}_K(G,X)$ and is functorial in (G,X). Using the formalism of mixed Shimura varieties, we show that such a motivic lift exists on the full subcategory of representations of Hodge type {(-1,0),(0,-1)}. If (G,X) is equipped with a choice of PEL-datum, Ancona has defined a motivic lift for all representations of G. We show that this is independent of the choice of PEL-datum and give criteria for it to be compatible with base change.

math.AG↗

arXiv2017

Regulator constants of integral representations of finite groups

Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_{p}[G]$-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser--Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p[G]$-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

math.RT↗