Overconvergent cohomology and quaternionic Darmon points

Guitart, Xavier; Masdeu, Marc

doi:10.1112/jlms/jdu036

Cited by 14 publications

(24 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case of curves defined over totally real fields F , so far the most compelling evidence for the conjectural rationality of Darmon points comes from explicit numerical verifications (cf. [12][13][14][26][27][28]). In this section, we describe how the new constructions of Darmon points introduced in the present article, in which the curves are defined over fields F of mixed signature, can be performed in certain cases.…”

Section: Effective Methods and Numerical Evidencementioning

confidence: 99%

“…(b) Non-archimedean computations. When F = Q and K is real quadratic the computations where performed in [12,14] (see also [28]) for B M 2 (Q), and in [27] for B a division algebra.…”

Section: Effective Methods and Numerical Evidencementioning

confidence: 99%

Section: Non-archimedean Computationsmentioning

confidence: 99%

“…The first explicit methods for computing p-adic Darmon points associated to quaternion division algebras were introduced in [27]. Although in the setting of [27] the base field is F = Q, K is real quadratic, and B/Q is an indefinite division algebra, as we will see in this section, the methods, in fact, can be easily adapted to work also in more general settings. That is to say, the algorithms of [27] can be suitably modified so that they also allow for the computation of some of the new instances of Darmon points that were introduced in Section 4.…”

Section: Non-archimedean Computationsmentioning

confidence: 99%

“…The first archimedean construction was also given by Darmon himself [11,Section 8], in the case where K is almost totally real (ATR) and a certain Heegner-type condition is satisfied, and later generalized by Gärtner [21]. These conjectures are supported by abundant numerical evidence [12][13][14][26][27][28]. Actually, a theme that runs in parallel with providing constructions of Darmon points is that of finding the algorithms that allow for their effective computation in order to test the conjectures.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Darmon points on elliptic curves over number fields of arbitrary signature

Guitart

Masdeu

Sengun

2015

Proc. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Effective Methods and Numerical Evidencementioning

confidence: 99%

“…(b) Non-archimedean computations. When F = Q and K is real quadratic the computations where performed in [12,14] (see also [28]) for B M 2 (Q), and in [27] for B a division algebra.…”

Section: Effective Methods and Numerical Evidencementioning

confidence: 99%

Section: Non-archimedean Computationsmentioning

confidence: 99%

Section: Non-archimedean Computationsmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Darmon points on elliptic curves over number fields of arbitrary signature

Guitart

Masdeu

Sengun

2015

Proc. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Diagonal restrictions of p-adic Eisenstein families

2020

View full text Add to dashboard Cite

A. We compute the diagonal restriction of the rst derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series a ached to a general (odd) character of the narrow class group of a real quadratic eld, and express the Fourier coe cients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of [DV1] at appropriate real quadratic points of Drinfeld's p-adic upper half-plane. is can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier [GZ, §7] which arose in their "analytic proof" of the factorisation of di erences of singular moduli, and whose inspiration can be traced to Siegel's proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real eld. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic elds based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross-Stark units and Stark-Heegner points in terms of the rst derivatives of certain twisted Rankin triple product p-adic Lfunctions.

show abstract

Parabolic eigenvarieties via overconvergent cohomology

Salazar¹,

Williams²

2021

Math. Z.

View full text Add to dashboard Cite

Let $$\mathcal {G}$$ G be a connected reductive group over $$\mathbf {Q}$$ Q such that $$G = \mathcal {G}/\mathbf {Q}_p$$ G = G / Q p is quasi-split, and let $$Q \subset G$$ Q ⊂ G be a parabolic subgroup. We introduce parahoric overconvergent cohomology groups with respect to Q, and prove a classicality theorem showing that the small slope parts of these groups coincide with those of classical cohomology. This allows the use of overconvergent cohomology at parahoric, rather than Iwahoric, level, and provides flexible lifting theorems that appear to be particularly well-adapted to arithmetic applications. When Q is a Borel, we recover the usual theory of overconvergent cohomology, and our classicality theorem gives a stronger slope bound than in the existing literature. We use our theory to construct Q-parabolic eigenvarieties, which parametrise p-adic families of systems of Hecke eigenvalues that are finite slope at Q, but that allow infinite slope away from Q.

show abstract

Overconvergent cohomology and quaternionic Darmon points

Cited by 14 publications

References 17 publications

Darmon points on elliptic curves over number fields of arbitrary signature

Darmon points on elliptic curves over number fields of arbitrary signature

Diagonal restrictions of p-adic Eisenstein families

Parabolic eigenvarieties via overconvergent cohomology

Contact Info

Product

Resources

About