On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms

Order, Jeanine Van

doi:10.4153/cjm-2012-002-x

Cited by 8 publications

(16 citation statements)

References 47 publications

(121 reference statements)

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“…Finally, this construction allows us to reduce the associated Iwasawa main conjecture to a nonvanishing criterion for these p-adic Lfunctions via the theorem of Howard [11,Theorem 3.2.3(c)] (cf. also [23,Theorem 1.3]), as we explain below. In particular, Howard's criterion (Conjecture 5.1) has the following applications to Iwasawa main conjectures.…”

Section: Introductionmentioning

confidence: 92%

“…We then obtain from Theorem 1.3 the following result, following the discussion in [11, Theorem 3.2.3 (c)] (cf. also [23,Theorem 1.3]).…”

Section: Let L +mentioning

confidence: 98%

“…Let Sel p ∞ (f , K p ∞ ) denote the p ∞ -Selmer group associated to f in the tower K p ∞ /K. We refer the reader to [10], [13] or [23] for definitions. Let…”

Section: Introductionmentioning

confidence: 99%

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On the quaternionic p-adic L-functions associated to Hilbert modular eigenforms

Van Order

2011

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We construct p-adic L-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions via the Jacquet-Langlands correspondence, generalizing works of Bertolini-Darmon, Vatsal and others. The construction given here is adelic, which allows us to deduce a precise interpolation formula from a Waldspurger type theorem, as well as a formula for the dihedral µ-invariant. We also make a note of Howard's nonvanishing criterion for these p-adic L-functions, which can be used to reduce the associated Iwasawa main conjecture to a certain nontriviality criterion for families of p-adic L-functions. Contents 1. Introduction 1 2. Some preliminaries 5 3. Modular forms on totally definite quaternion algebras 8 4. Construction of measures 17 5. Howard's criterion 28 References 30denote the specialization of λ to ρ. Let P denote the maximal ideal of O. We define the µ-invariant µ = µ(λ) associated to an element λ ∈ Λ to be the largest exponent c such that λ ∈ P c Λ. Let π = π f denote the cuspidal automorphic representation of GL 2 over F associated to f , withthe L-series of the adjoint representation of π. LetWe can state the criterion of Conjecture 5.1 in the following more explicit way. Let us now assume for simplicity that N is prime to the relative discriminant of K over F . Fix a positive integer k. Let us define a set of admissible primes L k of F , all of which are inert in K, with the condition that for any ideal n in the set S k of squarefree products of primes in L k , there exists a nontrivial eigenform f (n) of level nN such that we have the following congruence on Hecke eigenvalues:

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Section: Introductionmentioning

confidence: 92%

“…We then obtain from Theorem 1.3 the following result, following the discussion in [11, Theorem 3.2.3 (c)] (cf. also [23,Theorem 1.3]).…”

Section: Let L +mentioning

confidence: 98%

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On the quaternionic p-adic L-functions associated to Hilbert modular eigenforms

Van Order

2011

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“…In particular, the divisibilities (17) of Proposition 4.3 would follow from the dihedral/anticyclotomic main conjectures for f in the dihedral/anticyclotomic Z d pextension of K, where d = [F : Q]. For results in this direction, see for instance the generalizations to totally real fields of work of Bertolini-Darmon [1] (as well as Pollack-Weston [31]) by Longo [27] and the author [42]. For the equality condition (18) of Proposition 4.3, see the result of Howard [17,Theorem 3.2.3] with the main result of Pollack-Weston [31].…”

Section: Divisibility Criteriamentioning

confidence: 98%

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

Order

2012

Journal of Algebra

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We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate padic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori-Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate-Shafarevich group of an elliptic curve in the Z 2 p -extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange.

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“…We refer the reader to the e.g. works [3], [29], [24], or [38] for some of the many known Iwasawa theoretic results in this direction.…”

Section: Introductionmentioning

confidence: 99%

On the dihedral Euler characteristics of Selmer groups of Abelian varieties

Order¹

2015

Arithmetic and Geometry

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This note shows how to use the framework of Euler characteristic formulae to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures, and in particular how to verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this setting. When the Selmer group is cotorsion with respect to the associated Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with respect to the associated Iwasawa algebra, we give a conjectural description of the Euler characteristic of the cotorsion submodule, and explain how to deduce inequalities from the associated main conjecture divisibilities of Perrin-Riou and Howard. U

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On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms

Cited by 8 publications

References 47 publications

On the quaternionic p-adic L-functions associated to Hilbert modular eigenforms

On the quaternionic p-adic L-functions associated to Hilbert modular eigenforms

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

On the dihedral Euler characteristics of Selmer groups of Abelian varieties

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