On the coefficients of the characteristic series of the  U -operator

Abstract

A conceptual proof is given of the fact that the coefficients of the characteristic series of the U-operator acting on families of overconvegent modular forms lie in the Iwasawa algebra.

Local to global compatibility on the eigencurve

“…As observed on page 5 of [11] the Eisestein family over B has q-expansion in Λ [[q]], where Λ is the Iwasawa algebra. Let Λ K denote the Iwasawa algebra over the ring of integers of K. Viewing κ as a parameter on B, l −(m+n) κ(l n ) becomes an element of Λ K .…”

“…We deduce that at any cusp e g (q) ∈ Λ K [[q]]. Now we can apply the results of Thm 2.1 [11], which states that e g must extend uniquely over B, replacing Λ with Λ K and Z with X(U (N ) p × Γ 1 (q)) ≥1 . The proof is identical to that given in [11].…”

“…Essentially this theorem is a generalisation of Coleman's construction of Hecke operators in [12]. As such we follow Coleman's approach, first giving a proof at the centre of B * and then extending to all of B using theorem 2.1 of [11]. We will be making extensive use of q-expansions so must first work over a finite extension of Q p and then use a descent argument.…”

“…Now we can apply the results of Thm 2.1 [11], which states that e g must extend uniquely over B, replacing Λ with Λ K and Z with X(U (N ) p × Γ 1 (q)) ≥1 . The proof is identical to that given in [11]. Note that by construction e g has trivial action of the Diamond operators at p so is of level U 0 (q).…”

Overconvergent modular forms and their explicit arithmetic

“…. , κ n ](t) is the power series in t obtained by taking the corresponding finite differences on the coefficients of P (κ, t) of t, which are elements of the Iwasawa algebra by Coleman [Col97a]. The theory of finite differences then shows that upon increasing the number of interpolation points, the nth Newton series p-adically approaches the series P (κ, t).…”