Stark-Heegner points and the cohomology of quaternionic Shimura varieties

Greenberg, Matthew

doi:10.1215/00127094-2009-017

Cited by 47 publications

(104 citation statements)

References 17 publications

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“…Using Theorem 2, we deduce Conjecture 2 of [Greenberg 2009] in the case where the base field is ‫;ޑ‬ see Section 8 for details. The proof of Theorem 2 falls into two steps.…”

Section: Samit Dasgupta and Matthew Greenbergmentioning

confidence: 89%

“…In [Greenberg 2009], a p-adic analytic construction of local Stark-Heegner points on E was presented, generalizing a construction of Darmon [2001] applicable when p is inert in K and the primes dividing N split in K . The local definition of Stark-Heegner points given in [Greenberg 2009] is contingent upon Conjecture 2 [ibid.] over the base field ‫;ޑ‬ this now follows from Theorem 2.…”

Section: Samit Dasgupta and Matthew Greenbergmentioning

confidence: 99%

“…Following Darmon [2001], a conjectural method for doing this was proposed in [Greenberg 2009], as follows.…”

Section: Introductionmentioning

confidence: 99%

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ℒ-invariants and Shimura curves

Dasgupta

Greenberg

2012

ANT

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In earlier work, the second named author described how to extract Darmon-style ᏸ-invariants from modular forms on Shimura curves that are special at p. In this paper, we show that these ᏸ-invariants are preserved by the Jacquet-Langlands correspondence. As a consequence, we prove the second named author's period conjecture in the case where the base field is ‫.ޑ‬ As a further application of our methods, we use integrals of Hida families to describe Stark-Heegner points in terms of a certain Abel-Jacobi map.

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“…Using Theorem 2, we deduce Conjecture 2 of [Greenberg 2009] in the case where the base field is ‫;ޑ‬ see Section 8 for details. The proof of Theorem 2 falls into two steps.…”

Section: Samit Dasgupta and Matthew Greenbergmentioning

confidence: 89%

Section: Samit Dasgupta and Matthew Greenbergmentioning

confidence: 99%

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ℒ-invariants and Shimura curves

Dasgupta

Greenberg

2012

ANT

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“…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.…”

Section: 4mentioning

confidence: 99%

Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

Bergunde

Gehrmann

2018

Trans. Amer. Math. Soc.

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Abstract. Let E be a quadratic extension of a totally real number field. We construct Stickelberger elements for Hilbert modular forms of parallel weight 2 in anticyclotomic extensions of E. Extending methods developed by Dasgupta and Spieß from the multiplicative group to an arbitrary one-dimensional torus we bound the order of vanishing of these Stickelberger elements from below and, in the analytic rank zero situation, we give a description of their leading terms via automorphic L-invariants. If the field E is totally imaginary, we use the p-adic uniformization of Shimura curves to show the equality between automorphic and arithmetic L-invariants. This generalizes a result of Bertolini and Darmon from the case that the ground field is the field of rationals to arbitrary totally real number fields.

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“…However, several conjectural constructions have emerged in the last years under the generic name of Stark-Heegner points, or also Darmon points as the first such construction was introduced in [Dar01]. Variants of this initial construction applying to several different settings have been proposed since then, for instance in [Das05], [Gre09], [LRV09], [Gär11a], and [GRZ12]. The leitmotif of these methods is the analytic construction of algebraic points on ring class fields of quadratic extensions K/F which, unlike the classical case, are not CM.…”

Section: Introductionmentioning

confidence: 99%

Computation of ATR Darmon Points on Nongeometrically Modular Elliptic Curves

Guitart

Masdeu

2013

Experimental Mathematics

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Abstract. ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which in general the Heegner point method is not available. So far the only numerical evidence, provided by Darmon-Logan and Gärtner, concerned curves arising as quotients of Shimura curves. In those special cases the ATR points can be obtained from the already existing Heegner points, thanks to results of Zhang and Darmon-Rotger-Zhao.In this paper we compute for the first time an algebraic ATR point on a curve which is not uniformizable by any Shimura curve, thus providing the first piece of numerical evidence that Darmon's construction works beyond geometric modularity. To this purpose we improve the method proposed by Darmon and Logan by removing the requirement that the real quadratic base field be norm-euclidean, and accelerating the numerical integration of Hilbert modular forms.

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Stark-Heegner points and the cohomology of quaternionic Shimura varieties

Abstract: Abstract

Cited by 47 publications

References 17 publications

ℒ-invariants and Shimura curves

ℒ-invariants and Shimura curves

Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

Computation of ATR Darmon Points on Nongeometrically Modular Elliptic Curves

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