On congruence divisors of cusp forms as factors of the special values of their zeta functions

The Iwasawa Main Conjectures for GL2

“…That the factor in front of I g in the displayed equation belongs to F A follows from [Hi81], and by Lemma 12.2.2 it equals η f up to a unit if (irred) and (dist) hold forρ f . The arguments proving Theorem 12.3.1 are now easily adapted to prove this theorem.…”

Kummer's criterion for the special values of HeckeL-functions of imaginary quadratic fields and congruences among cusp forms

“…By combining these theorems with the previous results in [12] and [13], we obtain In the rest of this section, we shall give several preparatory results for the proof of these theorems:…”

“…Since arrow (i) of (0.1) has been proved in [12] and [13], we owe much to the idea of Ribet [21], for our method of proving Theorem 0.1. However, we should emphasize that our proof does not depend on the classification theory of finite flat group schemes (cf.…”

“…Since the ramification index of the base field of these group schemes goes beyond the limit of classification theory in our case, we have to invent a new method. To do this, the result of [13] is indispensable.…”

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The theme of this paper may be summarized by the triangle: Here, arrow (i) refers to our previous papers [9,12] and [13], in which we have shown that a fixed primitive cusp form f of Sk (F I(N)) has congruences with other cusp forms of Sk(F~(N)) modulo the special value at s=k of a certain zeta function off. In this paper, we will construct the second arrow for the cusp form f associated with a Hecke character of an imaginary quadratic field K, and consequently obtain the third.

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