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

2026/02/19 by Nirvana Coppola, Lorenzo La Porta, Matteo Longo
#math.NT

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Abstract

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.

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