Darmon points on elliptic curves over number fields of arbitrary signature

Abstract

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.

Exceptional zeros and $\mathcal {L}$-invariants of Bianchi modular forms

“…In particular, as for modular forms and modular symbols, we vary the overconvergent modular symbol attached to f over the edges of T p . In the case of a modular elliptic curve E over F , this distribution has been studied before; in particular, in [Tri06] Mak Trifkovic used it to construct Stark-Heegner points on E, whilst in the same setting the method of construction given here has been implemented independently by Xevi Guitart, Marc Masdeu and Haluk Sengun in [GMŞ15].…”

“…In this situation, the elliptic curve will have multiplicative reduction at p, so there is a Tate uniformiser q ∈ F × p such that E(F p ) ∼ = F × p /q Z . By analogy with the case of Hilbert modular forms, one would expect that: [GMŞ15], can be seen as providing strong evidence for this conjecture. More generally, for a form f of arbitrary weight, let ρ f be the Galois representation attached to f by Taylor (see [Tay94]).…”

“…In both [Dar01] and [Ort04], one obtains a link between the cohomology classes defined above and L-values of the form f at characters of conductor c by exploiting optimal embeddings of conductor c, that is, certain embeddings F × F ֒→ M 2 (F ). Heuristically, one should see such an embedding as giving rise to a cycle in the symmetric space corresponding to our level group, and hence it gives rise to a homology class that we can then pair our cohomology classes against (see, for example, [GMŞ15] for details of this approach).…”

Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

“…Taking π B -isotypical components we arrive at [12], Conjecture 2, for the case that the narrow class number of F is one and Conjecture 4.8 of [14] for the general case. The equivalence of their formulations and ours follows from Lemma 4.8.…”

Periods of Modular GL2-type Abelian Varieties and p-adic Integration

“…In the one-dimensional case, there is a huge amount of experimental evidence supporting the existence of elliptic curves attached to modular forms over non totally real fields (see [GHM78], [Cre84], [CW94], [GHY12], [GY12], [Jon14], [DGKMY15]). In this setting, [GMŞ15] contains two conjectural analytic constructions of the period lattice of the elliptic curve A f . The first construction concerns the complex lattice of A f , and is a generalization of Oda's conjecture to number fields having at least one real place, in the spirit of the work of Darmon-Logan [DL03] and Gartner [Gär12].…”