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

TL;DR

This paper derives an explicit formula for the $mbda$-invariant of Selmer groups associated with quadratic twists of Gross curves over certain $b$-extensions, advancing understanding of their arithmetic properties.

Contribution

It provides a simple exact formula for the $mbda$-invariant of Selmer groups for quadratic twists of CM abelian varieties over specific $b$-extensions.

Findings

Explicit $mbda$-invariant formula for certain quadratic twists.

Determination of the $b$-corank of Selmer groups over $F_b$-extensions.

Computations of Selmer groups when $mbda=1$.

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 $\mathfrak{p}$ 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 $F_\infty = FK_\infty$. 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 $1$.