JournalscmhVol. 87, No. 2pp. 303–353

Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms

  • Matteo Longo

    Università di Padova, Italy
Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms cover
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Abstract

Let F/QF/\mathbb{Q} be a totally real extension and ff an Hilbert modular cusp form of level n\mathfrak{n}, with trivial central character and parallel weight 2, which is an eigenform for the action of the Hecke algebra. Fix a prime n\wp | \mathfrak{n} of FF of residual characteristic pp. Let K/FK/F be a quadratic totally imaginary extension and KK_{\wp^\infty} be the \wp-anticyclotomic Zp\mathbb{Z}_p-extension of KK. The main result of this paper, generalizing the analogous result [5] of Bertolini and Darmon, states that, under suitable arithmetic assumptions and some technical restrictions, the characteristic power series of the Pontryagin dual of the Selmer group attached to (f,K)(f,K_{\wp^\infty}) divides the pp-adic LL-function attached to (f,K)(f,K_{\wp^\infty}), thus proving one direction of the Anticyclotomic Main Conjecture for Hilbert modular forms. Arithmetic applications are given.

Cite this article

Matteo Longo, Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms. Comment. Math. Helv. 87 (2012), no. 2, pp. 303–353

DOI 10.4171/CMH/255