A Note on Asymptotic Class Number Upper Bounds in p-adic Lie Extensions

Let p be an odd prime and F_∞ a p-adic Lie extension of a number field F with Galois group G. Suppose that G is a compact pro-p p-adic Lie group with no torsion and that it contains a closed normal subgroup H such that G/H ≅Z_p. Under various assumptions, we establish asymptotic upper bounds for the growth of p-exponents of the class groups in the said p-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that H ≅Z_p.