arXiv subjects

Lorenzo La Porta

Publications and source records attributed to Lorenzo La Porta.

4 recordsLinked to original sources

arXiv2026

On the local-global principle for twists of abelian varieties and Galois representations

This paper investigates the validity of a local-global principle for finite twists of a large class of objects endowed with a continuous action of the absolute Galois group of a given number field, such as abelian varieties, modular forms and Galois representations. Our aim is to determine when, for $m$ a positive integer, twists that are given locally by characters of order $m$ are realised by a global character of the same order. We define and study a ``Tate--Shafarevich cohomology set'' that governs the obstruction to the local-global principle for $m$-atic twists and we prove that this set is finite. Finally, we apply our results and prove several instances of the local-global principle in various examples.

math.NT↗

arXiv2026

Decidability of singularities in the Ekedahl--Oort stratification

For an abelian type Shimura variety and an odd prime $p$ of good reduction, we characterize the regularity in codimension one of Zariski closures of Ekedahl--Oort strata in terms of the Frobenius action on the root datum. We give an algorithm that detects codimension one singularities for arbitrary Ekedahl--Oort strata. When the Shimura datum is of split type, we relate the singularities of Ekedahl--Oort strata to a stack of $G$-zips over the complex numbers. We study the existence of generalized Hasse invariants on this stack.

math.NT↗

arXiv2025

Singularities in the Ekedahl--Oort stratification

We give conceptual and combinatorial criteria for the normality and Cohen--Macaulayness of unions of Ekedahl--Oort strata in the special fiber of abelian type Shimura varieties. For unions of two strata, one of the two having codimension one in the closure of the other, we determine exactly when their union is smooth. We provide explicit numerical criteria for the smoothness of any one-dimensional EO-stratum closure of Shimura varieties. For groups of type $\mathsf{B}_n$, we describe the smooth and normal loci of all EO-strata closures. We construct reduced strata Hasse invariants of explicit weights on EO-strata of codimension at most $n-1$, showing that their Zariski closures are local complete intersections. We also provide a closed form formula for their cycle classes.

math.NT↗

arXiv2023

Generalised theta operators on unitary Shimura varieties

The main result of this paper is the construction of a new class of weight shifting operators, similar to the theta operators of arXiv:1902.10911, arXiv:1712.06969 and others, which are defined on the lower Ekedahl-Oort strata of the geometric special fibre of unitary Shimura varieties of signature $(n-1, 1)$ at a good prime $p$, split in the in the reflex field $E$, which we assume to be quadratic imaginary. These operators act on certain graded sheaves which are obtained from the arithmetic structure of the EO strata, in particular the $p$-rank on each stratum. We expect these operators to have applications to the study of Hecke-eigensystems of modular forms modulo $p$ and generalisations of the weight part of Serre's conjecture.

math.NT↗