In this paper, we prove an analogue of Kida's formula for the fine Selmer groups of elliptic curves. We study the growth behaviour of the fine Selmer groups in p-power degree extensions of the cyclotomic ${\mathbb{Z}}_p$-extension of a totally real number field and obtain a formula for the ${\mathbb{Z}}_p$ co-rank of the fine Selmer group of an elliptic curve.

中文翻译：

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精细Selmer椭圆曲线群的Kida公式的类似物

在本文中，我们证明了椭圆曲线的精细Selmer组的Kida公式的类似物。我们研究了在环原子的p-幂级数扩展中精细Selmer群的增长行为${\mathbb{ž}}_p$扩展全实数字段并获得公式 ${\mathbb{ž}}_p$ 椭圆曲线的精细Selmer组的协秩。