On the local-global divisibility of torsion points on elliptic curves\n and rm GL₂-type varieties

2016/09/01 by Florence Gillibert, Gillibert, Florence, Gabriele Ranieri +1 · 1 citation
Computer Science · Mathematics · #11G05 #11G10 #11R34 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1609.00410

openalex publication_date 2016/09/01 · openalex created_date 2022/08/27 · openalex updated_date 2026/07/28

Abstract

Let p be a prime number and let k be a number field. Let E be an\nelliptic curve defined over k. We prove that if p is odd, then the\nlocal-global divisibility by any power of p holds for the torsion points of\nE. We also show with an example that the hypothesis over p is necessary.\n We get a weak generalization of the result on elliptic curves to the larger\nfamily of rm GL₂-type varieties over k. In the special case of the\nabelian surfaces A/k with quaternionic multiplication over k we obtain that\nfor all prime p, except a finite number depending on A, the local-global\ndivisibility by any power of p holds for the torsion points of A\n

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