On the $\lambda$-invariant of Selmer groups arising from certain quadratic twists of Gross curves

Abstract

Let $q$ be a prime with $q\equiv 7\mathrm{mod}8$, 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 $\mathfrak{p}$ with $n$-th layer $K_n$. For certain quadratic extensions $F/K$, 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 $F_{\infty }=F{K}_{\infty }$. Equivalently, our result determines the exact ${\mathbb{Z}}_2$-corank of certain Selmer groups over $F_{\infty }$ 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 discuss computations of the associated Selmer groups over $K_n$ in the case when the $\lambda$-invariant is equal to $1$.

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Cited by in corpus (1)

• On the unramified Iwasawa module of a ${\mathbb{Z}}_p$-extension generated by division points of a CM elliptic curve