2025/11/03 by Alessandrì, Jessica, Paladino, Laura
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #Analytic Number Theory Research
paper · doi:10.48550/arxiv.2511.02078
Let p be a prime number and n a positive integer. Let E be an elliptic curve defined over a number field k. It is known that the local-global divisibility by p holds in E/k, but for powers of pn counterexamples may appear. The validity or the failing of the Hasse principle depends on the elliptic curve E and the field k and, consequently, on the group Gal(k(E[pn])/k). For which kind of these groups does the principle hold? For which of them can we find a counterexample? The answer to these questions was known for n=1,2, but for n≥ 3 they were still open. We show some conditions on the generators of Gal(k(E[pn])/k) implying an affirmative answer to the local-global divisibility by pn in E over k, for every n≥ 2. We also prove that these conditions are necessary by producing counterexamples in the case when they do not hold. These last results generalize to every power pn, a result obtained by Ranieri for n=2.