On 7-division fields of CM elliptic curves

2021/06/08 by Jessica Alessandrì, Alessandrì, Jessica, Laura Paladino +1
Arts and Humanities · Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.2106.04579

openalex publication_date 2021/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be a CM elliptic curve defined over a number field K, with Weiestrass form y³=x³+bx or y²=x³+c. For every positive integer m, we denote by E[m] the m-torsion subgroup of E and by K_m:=K(E[m]) the m-th division field, i.e. the extension of K generated by the coordinates of the points in E[m]. We classify all fields K₇. In particular we give explicit generators for K₇/K and produce all Galois groups \textrmGal(K₇/K). We also show some applications to the Local-Global Divisibility Problem and to modular curves.

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