2010/04/20 by Víctor Rotger, Rotger, Victor, M. Seveso +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1004.3513
openalex publication_date 2010/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f be a modular eigenform of even weight k>0 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 DFM(f) and an L-invariant LFM(f). The first goal of this paper is building a suitable p-adic integration theory that allows us to construct a monodromy module D(f) and an L-invariant L(f) in the spirit of Darmon. We conjecture both monodromy modules are isomorphic, and in particular the two L-invariants are equal. For the second goal of this note we assume the conjecture is true. 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 Mordell-Weil 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.