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Kida's formula for imaginary quadratic fields with p = 2

Kida’s formula in classical Iwasawa theory relates the $\lambda$-invariants of certain Iwasawa modules in $p$-extensions. This formula is analogous to the Riemann-Hurwitz formula for curves. After a brief explanation of this analogy, I will focus on the Coates-Wiles $\mathbb{Z}_p$ extensions arising from study of CM elliptic curves. In this case, the formula was proved by Winberg and by Michel in 1990s for $p>2$, and I will explain that the formula holds also for $p = 2$. The case $p = 2$ has applications on arithmetic of certain Gross elliptic curves, applying the formula for $\mathbb{Q}(\sqrt{-q})$ for a prime $q≡7 \pmod{8}$.