On some p-adic Galois representations and form class groups

2020/09/29 by Ho Yun Jung, Ja Kyung Koo, Jung, Ho Yun +5 · 1 citation
Mathematics · #11E08 #11F80 #11R29 #11R37 #14K22 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2009.13837

openalex publication_date 2020/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be an imaginary quadratic field of discriminant d_K with ring of integers O_K. When K is different from ℚ(√(-1)) and ℚ(√(-3)), we consider a certain specific model for the elliptic curve E_K with j(E_K)=j(O_K) which is defined over ℚ(j(E_K)). In this paper, for each positive integer N we compare the extension field of ℚ generated by the coordinates of N-torsion points on E_K with the ray class field K_(N) of K modulo NO_K. By using this result we investigate the image of a p-adic Galois representation attached to E_K for a prime p, in terms of class field theory. Second, we construct the definite form class group of discriminant d_K and level N which is isomorphic to Gal(K_(N)/ℚ).

Cited by

• A simplified algorithmic realization of Galois actions on special values of modular functions

Related

• Models of CM elliptic curves with a prescribed ℓ-adic Galois image
• Les représentations ell-adiques associées aux courbes elliptiques sur \Q_p
• Arithmetic properties of orders in imaginary quadratic fields
• Local torsion on elliptic curves and the deformation theory of Galois representations
• Class fields generated by coordinates of elliptic curves