Fourier coefficients of the overconvergent generalized eigenform associated to a CM form

Abstract

Let f be a modular form with complex multiplication. If f has critical slope, then Coleman's classicality theorem implies that there is a padic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as f . We give a formula for the Fourier coefficiets of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of p-adic overconvergent forms containing f .

A geometric view on Iwasawa theory

“…Note that if a p-irregular classical weight k 2 points were to exist, they would be of non-critical slope and Coleman's Classicality Theorem [18] would imply that the corresponding generalized eigenspace consists only of classical forms (see [12]). Thus the second case above is (conjecturally) the only one yielding cuspidal-overconvergent generalized eigenforms, and their q-expansions have recently been computed by Hsu [40] (note that the technical condition preceding Theorem 1.1 in loc. cit.…”

“…In his quest [1,2] to attach p-adic L-functions to classical eigenforms of critical slope, J. Bellaïche classified the possible such examples in weight k 2, and concluded that conjecturally the only genuine overconvergent generalized eigenforms are critical CM forms, whose Fourier coefficients were recently computed by Hsu [40]. The first to take up the task in weight 1 were H. Darmon, A. Lauder and V. Rotger [21] who expressed the Fourier coefficients of a certain p-adic overconvergent weight 1 generalized eigenform in terms of p-adic logarithms of algebraic numbers in ring class fields of real quadratic fields.…”

A geometric view on Iwasawa theory

“…Note that if an irregular classical weight k ⩾ 2 points were to exist, they would be of noncritical slope and Coleman's Classicality Theorem [16] would imply that the corresponding generalized eigenspace consists only of classical forms. Thus the second case above is (conjecturally) the only one yielding cuspidal-overconvergent generalized eigenforms, and their q-expansions have recently been computed by Hsu [36] (note that the technical condition preceding Theorem 1.1 in loc. cit.…”

“…The very exclusive club of genuine overconvergent generalized eigenforms S † w(f ) ⟦f ⟧ 0 is a natural supplement of the classical subspace in S † w(f ) ⟦f ⟧. In his quest [2,1] to attach p-adic L-functions to classical eigenforms of critical slope, J. Bellaïche classified the possible such examples in weight k ⩾ 2, and concluded that conjecturally the only genuine overconvergent generalized eigenforms are critical CM forms, whose Fourier coefficients were recently computed by Hsu [36]. The first to take up the task in weight 1 were H. Darmon, A. Lauder and V. Rotger [18] who expressed the Fourier coefficients of a certain p-adic overconvergent weight 1 generalized eigenform in terms of p-adic logarithms of algebraic numbers in ring class fields of real quadratic fields.…”