On 2-rank of the ideal class groups of algebraic number fields.
We formulate and prove an analogue of the noncommutative Iwasawa main conjecture for -adic Lie extensions of a separated scheme of finite type over a finite field of characteristic prime to .
Let p be an odd prime number. We prove the existence of certain infinite families of imaginary quadratic fields in which p splits and for which the Iwasawa λ-invariant of the cyclotomic ℤₚ-extension is equal to 1.
We give precise estimates for the number of classical weight one specializations of a non-CM family of ordinary cuspidal eigenforms. We also provide examples to show how uniqueness fails with respect to membership of weight one forms in families.