2023/08/13 by Anwesh Ray, Ray, Anwesh
Arts and Humanities · Mathematics · #11R23 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2308.06673
openalex publication_date 2023/08/13 · openalex created_date 2023/08/16 · openalex updated_date 2026/07/28
Let E/ℚ be an elliptic curve and p be an odd prime number at which E has good ordinary reduction. Let Selp^∞(ℚ_∞, E) denote the p-primary Selmer group of E considered over the cyclotomic ℤp-extension of ℚ. The (algebraic) μ-invariant of Selp^∞(ℚ_∞, E) is denoted μp(E). Denote by ρE, p:Gal(ℚ/ℚ)→ GL2(ℤ/pℤ) the Galois representation on the p-torsion subgroup of E(ℚ). Greenberg conjectured that if ρE, p is reducible, then there is a rational isogeny E→ E' whose degree is a power of p, and such that μp(E')=0. In this article, we study this conjecture by showing that it is satisfied provided some purely Galois theoretic conditions hold that are expressed in terms of the representation ρE,p. In establishing our results, we leverage a theorem of Coates and Sujatha on the algebraic structure of the fine Selmer group. Furthermore, in the case when ρE, p is irreducible, we show that our hypotheses imply that μp(E)=0 provided the classical Iwasawa μ-invariant vanishes for the splitting field ℚ(E[p]):=ℚ^kerρE,p.