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Abstract

Let $A$ denote a quadratic twist of the Gross curve over $H$ with complex multiplication by $K=\mathbb {Q}(\sqrt {-q})$, where $q$ is a prime with $q \equiv 7\, \mathrm {mod}\, 8$, and $H$ is the Hilbert class field of $K$. We assume that $A$ has good reduction at the primes of $H$ above $2$. Let $B = \mathrm {Res}_{H/K}(A)$ be the abelian variety which is the restriction of scalars of $A$ from $H$ to $K$. We study the structure of the Selmer group of $B$ over a certain infinite extension of $K$ generated by division points of $B$, whose Galois group over $K$ is a $2$-adic Lie group of dimension $2$. In particular, we apply our results to study the Selmer group of the elliptic curve $X_0(49)$ over its $2^\infty $-division field.