On the local behaviour of ordinary $\Lambda$-adic representations

On classical weight one forms in Hida families

“…The following finiteness result for the number of classical weight one specializations in a non-CM family was proved in the course of the proof of the main result of [GV04] (see in particular the proof of the implications (ii) =⇒ (iii) =⇒ (iv) of Prop. 14 in that paper).…”

“…Assume that F has infinitely many classical weight one specializations. By [GV04,Prop. 14, (ii) =⇒ (iii)], F then contains infinitely many CM forms with CM by the same imaginary quadratic field, call it K. A quick check shows that the argument also works for p = 2. In [GV04,Prop. 14, (iii) =⇒ (iv)], a somewhat lengthy argument (for odd primes p) was given to show that this forces F to be a CM family.…”

“…By [GV04,Prop. 14, (ii) =⇒ (iii)], F then contains infinitely many CM forms with CM by the same imaginary quadratic field, call it K. A quick check shows that the argument also works for p = 2. In [GV04,Prop. 14, (iii) =⇒ (iv)], a somewhat lengthy argument (for odd primes p) was given to show that this forces F to be a CM family. While this argument had the advantage of being explicit, in that it explicitly constructed a Λ-adic Hecke character Φ which interpolates the finite order characters ϕ corresponding to the weight one forms above, it is possible to give a much shorter proof of this part, as follows.…”

“…However, it is known that a non-CM family F admits only finitely many classical weight one specializations [GV04]. The first goal of this paper is to make this finiteness result effective by giving explicit bounds on the number of such specializations.…”

Deformations of locally abelian Galois representations and unramified extensions

“…Our next interest is to solve the question: "Is the generator b divisible by p or not ?" This question may be related to the work of Ghate and Vatsal [6].…”