Torsion subgroups of rational elliptic curves over the compositum of all\n D₄ extensions of the rational numbers

2017/10/14 by Harris B. Daniels, Daniels, Harris B.
Computer Science · Mathematics · Social Sciences · #11G05 #11R21 #12F10 #14H52 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Geometric and Algebraic Topology #Historical and Political Studies #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1710.05228

openalex publication_date 2017/10/14 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

Let E/\ℚ be an elliptic curve and let \ℚ(D₄^\∞) be\nthe compositum of all extensions of \ℚ whose Galois closure has\nGalois group isomorphic to a quotient of a subdirect product of a finite number\nof transitive subgroups of D₄. In this article we first show that\n\ℚ(D₄^\∞) is in fact the compositum of all D₄ extensions of\n\ℚ and then we prove that the torsion subgroup of\nE(\ℚ(D₄^\∞)) is finite and determine the 24 possibilities for\nits structure. We also give a complete classification of the elliptic curves\nthat have each possible torsion structure in terms of their j-invariants.\n

Citations

• The modular curves X0(65) and X0(91) and rational isogeny

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