The well-known Riemann-Hurwitz formula for Riemann surfaces (or the corresponding formulas of the same name for curves/function fields) is used in genus computations. In 1979, Yûji Kida proved a strikingly analogous formula in [Kid80] for p-extensions of CM-fields (p an odd prime) which is similarly used to compute Iwasawa λ -invariants. However, the relationship between Kida’s formula and the statement for surfaces is not entirely clear since the proofs are of a very different flavor. Also, there were a few hypotheses for Kida’s result which were not fully satisfying; for example, Kida’s formula requires CM-fields rather than more general number fields and excludes the prime p = 2. Around a year after Kida’s result was published, Kenkichi Iwasawa used Galois cohomology in [Iwa81] to establish a more general formula (about representations) that did not exclude the prime p = 2 nor need the CM-field assumption. Moreover, Kida’s formula follows as a corollary from Iwasawa’s formula. We’ll prove a slight generalization of Iwasawa’s formula and use this to give a new proof of a result of Kida in [Kid79] and Ferrero in [Fer80] which computes λ-invariants in imaginary quadratic extensions for the prime p = 2. We go on to produce special generalizations of Iwasawa’s formula in the case of cyclic p-extensions; these formulas can be realized as statements about Q(p)-representations, and, in the cases of degree p or p², about p-adic integral representations. One upshot of these formulas is a vanishing criterion for λ-invariants which generalizes a result of Takashi Fukuda et al. in [FKOT97]. Other applications include new congruences and inequalities for λ-invariants that cannot be gleaned from Iwasawa’s formula. Lastly, we give a scheme theoretic approach to produce a general formula for finite, separable morphisms of Dedekind schemes which simultaneously encompasses the classical Riemann-Hurwitz formula and Iwasawa’s formula.
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