On the λ-invariant of Selmer groups arising from certain quadratic twists of Gross curves

2021/07/07 by Jianing Li, Li, Jianing
Mathematics · #11G05 #11R23 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11R23

paper · pdf · doi:10.48550/arxiv.2107.03027

minor revision

arxiv created 2021/09/14 · arxiv updated 2021/09/15

Abstract

Let q be a prime with q ≡ 7 \mod 8, and let K=ℚ(√(-q)). Then 2 splits in K, and we write \mathfrakp for either of the primes K above 2. Let K_∞ be the unique ℤ₂-extension of K unramified outside \mathfrakp with n-th layer K_n. For certain quadratic extensions F/K, we prove a simple exact formula for the λ-invariant of the Galois group of the maximal abelian 2-extension unramified outside \mathfrakp of the field F_∞ = FK_∞. Equivalently, our result determines the exact ℤ₂-corank of certain Selmer groups over F_∞ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to K of the Gross curve with complex multiplication defined over the Hilbert class field of K. We also discuss computations of the associated Selmer groups over K_n in the case when the λ-invariant is equal to 1.

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