Ideal class groups of number fields and Bloch-Kato's Tate-Shafarevich groups for symmetric powers of elliptic curves

2022/04/16 by Naoto Dainobu, Dainobu, Naoto
Mathematics · #11G05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Primary 11R29 #Secondary 11R34

paper · pdf · doi:10.48550/arxiv.2204.07759

openalex publication_date 2022/04/16 · openalex created_date 2022/04/28 · openalex updated_date 2026/07/28

Abstract

For an elliptic curve E over ℚ, putting K=ℚ(E[p]) which is the p-th division field of E for an odd prime p, we study the ideal class group Cl_K of K as a Gal(K/ℚ)-module. More precisely, for any j with 1\leqslant j \leqslant p-2, we give a condition that Cl_K⊗ \mathbbF_p has the symmetric power Sym^j E[p] of E[p] as its quotient Gal(K/ℚ)-module, in terms of Bloch-Kato's Tate-Shafarevich group of Sym^j V_p E. Here V_p E denotes the rational p-adic Tate module of E. This is a partial generalization of a result of Prasad and Shekhar for the case j=1.

Related

• Ideal class groups of division fields of elliptic curves and everywhere unramified rational points
• On p-adic L-functions of elliptic curves and the ideal class groups of the division fields
• Visibility of ideal classes
• Ideal class groups of number fields associated to modular Galois representations
• Horizontal variation of Tate--Shafarevich groups