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

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• 作者：Florence Gillibert ; Gabriele Ranieri ; ^{gabriele.ranieri@pucv.cl}
• 关键词：11R34 ; 11G05 ; 11G10
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：174
• 期：Complete
• 页码：202-220
• 全文大小：414 K
• 卷排序：174

文摘

Let p be a prime number and let k   be a number field. Let EE be an elliptic curve defined over k. We prove that if p is odd, then the local–global divisibility by any power of p   holds for the torsion points of EE. We also show with an example that the hypothesis over p is necessary.We get a weak generalization of the result on elliptic curves to the larger family of GL2GL2-type varieties over k  . In the special case of the abelian surfaces A/kA/k with quaternionic multiplication over k we obtain that for all prime numbers p  , except a finite number depending only on the isomorphism class of the ring Endk(A)Endk(A), the local–global divisibility by any power of p   holds for the torsion points of AA.