Overconvergent cohomology and quaternionic Darmon points

We develop the (co)homological tools that make effective the construction of the quaternionic Darmon points introduced by Matthew Greenberg. In addition, we use the overconvergent cohomology techniques of Pollack--Pollack to allow for the efficient calculation of such points. Finally, we provide the first numerical evidence supporting the conjectures on their rationality.Comment: Fixed some minor typos, added authors' affiliatio

Overconvergent quaternionic forms and anticyclotomic p-adic L-functions

We reinterpret the explicit construction of Gross points given by Chida-Hsieh as a non-Archimedian analogue of the standard geodesic cycle (i∞)-(0) on the Poincaré upper half plane. This analogy allows us to consider certain distributions, which can be regarded as anticyclotomic p-adic L-functions for modular forms of non-critical slope following the overconvergent strategy à la Stevens. We also give a geometric interpretation of their Gross points for the case of weight two forms. Our construction generalizes those of Bertolini-Darmon, Bertolini-Darmon-Iovita-Spiess,-and Chida-Hsieh and shows a certain integrality of the interpolation formula even for non-ordinary forms

Darmon points on elliptic curves over number fields of arbitrary signature

We present new constructions of complex and $p$-adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture

Darmon points on elliptic curves over number fields of arbitrary signature

We present new constructions of complex and p-adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture

Stark-Heegner cycles attached to Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch--Kato conjecture predicts that this should force the existence of non-trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Barrera Salazar and the second author to construct certain P-adic Abel--Jacobi maps, which we use to propose a construction of such classes via "Stark--Heegner cycles". This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms

THE Λ-ADIC SHIMURA-SHINTANI-WALDSPURGER CORRESPONDENCE

International audienceWe generalize the Λ-adic Shintani lifting for GL 2 (Q) to indefinite quaternion algebras over Q