Codimension two cycles in Iwasawa theory and tensor product of Hida families
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Mathematische Annalen
Codimension two cycles in Iwasawa theory and tensor product of Hida families Antonio Lei1
· Bharathwaj Palvannan2
Received: 5 February 2019 / Revised: 16 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The purpose of this paper is to build on results in higher codimension Iwasawa theory. The setting of our results involves Galois representations arising as cyclotomic twist deformations associated to (1) the tensor product of two cuspidal Hida families F and G, and (2) the tensor product of three cuspidal Hida families F, G and H . On the analytic side, we consider (1) a pair of 3-variable Rankin–Selberg p-adic L-functions constructed by Hida and (2) a balanced 4-variable p-adic L-function (due to Hsieh and Yamana) and an unbalanced 4-variable p-adic L-function (whose existence is currently conjectural). In each of these setups, when the two p-adic L-functions generate a height two ideal in the corresponding deformation ring, we use codimension two cycles of that ring to relate them to a pair of pseudo-null modules. Mathematics Subject Classification Primary 11R23; Secondary 11F33 · 11R34 · 11S25
Communicated by W. Zhang. The first named author’s research is supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096.
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Bharathwaj Palvannan [email protected] Antonio Lei [email protected]
1
Département de mathématiques et de statistique, Université Laval, Pavillon Alexandre-Vachon, 1045 avenue de la Médecine, Québec G1V 0A6, Canada
2
National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4, Roosevelt Rd., Taipei 106, Taiwan
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A. Lei, B. Palvannan
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . Tensor product of two Hida families . . . . . . . . . . . Tensor product of three Hida families . . . . . . . . . . 1.1 Main result in higher codimension Iwasawa theory 1.2 Selmer groups and principal divisors . . . . . . . . 1.3 Pseudo-nullity conjectures in Iwasawa theory . . . 1.4 Concluding remarks . . . . . . . . . . . . . . . . . 2 Structural properties of Selmer groups . . . . . . . . . . 2.1 Non-primitive Selmer groups and principal divisors 2.2 Cyclotomic twist deformations . . . . . . . . . . . 2.3 Proof of Theorem 2 . . . . . . . . . . . . . . . . . 3 Hida families . . . . . . . . . . . . . . . . . . . . . . . 3.1 Tensor product of two Hida families . . . . . . . . 3.2 Tensor product of three Hida families . . . . . . . . 4 Description of Z(Q, Dd ,n ) and Z(Q, Dd ,n ) . . . . . . Description of Z(Q, Dd ,n ) . . . . . . . . . . . . . . . Description of Z(Q, Dd ,n ) . . . . . . . . . . . . . . . 5 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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