On the $\lambda$-invariant of Selmer Groups Arising from Certain Quadratic Twists of Gross Curves

Abstract

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$. For certain quadratic and biquadratic extensions $\mathfrak{F}/K$, we prove a simple exact formula for the $\lambda$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $\mathfrak{F}_\infty = \mathfrak{F} K_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer groups over $\mathfrak{F}_\infty$ 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 exhibit computations of the associated Selmer groups over $K_n$ in the case when the $\lambda$-invariant is equal to $1$; here $K_n$ denotes the $n$-th layer of $K_\infty/K$.