zbMATH — the first resource for mathematics

\(\mathcal L\)-invariants and Darmon cycles attached to modular forms. (English) Zbl 1292.11069
Summary: Let \(f\) be a modular eigenform of even weight \(k\geq 2\) and new at a prime \(p\) dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to \(f\) a monodromy module \(\mathbf{D}^{FM}_f\) and an \(\mathcal{L}\)-invariant \(\mathcal{L}^{FM}_f\). The first goal of this paper is building a suitable \(p\)-adic integration theory that allows us to construct a new monodromy module \(\mathbf{D}_f\) and \({\mathcal{L}}\)-invariant \(\mathcal{L}_f\), in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two \(\mathcal{L}\)-invariants are equal.
Let \(K\) be a real quadratic field and assume the sign of the functional equation of the \(L\)-series of \(f\) over \(K\) is \(-1\). The Bloch-Beilinson conjectures suggest that there should be a supply of elements in the Selmer group of the motive attached to \(f\) over the tower of narrow ring class fields of \(K\). Generalizing work of Darmon for \(k=2\), we give a construction of local cohomology classes which we expect to arise from global classes and satisfy an explicit reciprocity law, accounting for the above prediction.

11G18 Arithmetic aspects of modular and Shimura varieties
Full Text: DOI arXiv
[1] Ash, A., Stevens, G.: Cohomology of arithmetic groups and congruences between sys- tems of Hecke eigenvalues. J. Reine Angew. Math. 365, 192-220 (1986) · Zbl 0596.10026 · crelle:GDZPPN002203324 · eudml:152810
[2] Bertolini M., Darmon, H., Green, P.: Periods and points attached to quadratic algebras. In: Heegner Points and Rankin L-series, Math. Sci. Res. Inst. Publ. 49, Cambridge Univ. Press, 323-367 (2004) · Zbl 1173.11328 · www.msri.org
[3] Bertolini, M., Darmon, H., Iovita, A.: Families of automorphic forms on defi- nite quaternion algebras and Teitelbaum’s conjecture. Astérisque 331, 29-64 (2010) · Zbl 1251.11033 · smf4.emath.fr
[4] Bertolini, M., Darmon, H., Iovita, A., Spiess, M.: Teitelbaum’s exceptional zero conjec- ture in the anticyclotomic setting. Amer. J. Math. 124, 411-449 (2002) · Zbl 1079.11036 · doi:10.1353/ajm.2002.0009 · muse.jhu.edu
[5] Bloch, S., Kato, K.: L-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift I, Progr. Math. 108, Birkhäuser, 333-400 (1993) · Zbl 0768.14001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.