On congruence modules related to Hilbert Eisenstein series
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Mathematische Zeitschrift
On congruence modules related to Hilbert Eisenstein series Sheng-Chi Shih1 Received: 3 December 2018 / Accepted: 14 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute their constant terms by computing local integrals. In the second part, we prove a control theorem for one-variable ordinary -adic Hilbert modular forms following Hida’s work on the space of multivariable ordinary -adic Hilbert cusp forms. In part three, we compute congruence modules related to Hilbert Eisenstein series through an analog of Ohta’s methods.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 Hilbert modular forms . . . . . . . . . . . . . . . . 3 Automorphic forms . . . . . . . . . . . . . . . . . 4 -adic modular forms . . . . . . . . . . . . . . . . 5 Hilbert modular varieties and p-adic modular forms 6 Main results . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction Let R be an integral domain with quotient field Q(R). We consider a short exact sequence of flat R-modules i
p
→B− → C → 0. 0→ A− Suppose that we are given splitting maps after tensoring with Q(R) over R, i.e., we have t
s
0 ← A ⊗ R Q(R) ← − B ⊗ R Q(R) ← − C ⊗ R Q(R) ← 0. The congruence module attached to these data is defined by Cs := C/ p(B ∩ s(C)).
B 1
Sheng-Chi Shih [email protected] UMR 8524-Laboratoire Paul Painlevé, CNRS, University of Lille, 59000 Lille, France
123
S.-C. Shih
Congruence modules have been studied by many people in different settings. For instance, Ohta [22] computed the congruence module associated with the sequence res
0 → S ord (; ) E → M ord (; ) E −→ → 0, where “res” is the residue map, and M ord (; ) and S ord (; ) are respectively the spaces of ordinary -adic modular forms and ordinary -adic cusp forms. Here = o[[T ]] for some extension o of Z p . In this paper, we generalize Ohta’s work to the setting of Hilbert modular forms. In order to achieve this goal, we review important facts about p-adic and -adic Hilbert modular forms, and prove crucial results about Eisenstein series and cusps through their adelic construction. For the above examples and our main results (Theorem 1.1), we require that splittings are Hecke-equivariant. Moreover, there exist canonical splittings which are considered in the computation of these congruence m
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