The sign of Galois representations attached to automorphic forms for unitary groups

Bellaïche, Joël; Chenevier, Gaëtan

doi:10.1112/s0010437x11005264

Cited by 69 publications

(137 citation statements)

References 6 publications

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“…The arguments are identical to the ones used to prove Proposition 7.2.8 in [2]. The uniqueness follows from Lemma 2.5 and the fact that the E i are reduced and…”

Section: Proof For This Letmentioning

confidence: 69%

“…cit., (ii) follows from the construction, and (iii) is Lemma 5.9 in [5]. The uniqueness is proved as in Proposition 7.2.8 of [2]. Remark 2.3.…”

Section: Introductionmentioning

confidence: 94%

“…We now use the arguments of [2] to show how to extend certain maps defined on an accumulation and Zariski-dense subset to morphisms between eigenvarieties. Let E be an eigenvariety.…”

Section: Introductionmentioning

confidence: 99%

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A p-adic Labesse–Langlands transfer

Ludwig

2017

manuscripta math.

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Abstract. We prove a p-adic Labesse-Langlands transfer from the group of units in a definite quaternion algebra to its subgroup of norm one elements. More precisely, given an eigenvariety for the first group, we show that there exists an eigenvariety for the second group and a morphism between them that extends the classical Langlands transfer. In order to find a suitable target eigenvariety for the transfer we formalise a notion of Langlands compatibility of tame levels. Proving the existence of Langlands compatible tame levels is the key to pass from the classical transfer on the level of L-packets to a map between classical points of eigenvarieties, which is then amenable for interpolation to give the p-adic transfer.

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“…The arguments are identical to the ones used to prove Proposition 7.2.8 in [2]. The uniqueness follows from Lemma 2.5 and the fact that the E i are reduced and…”

Section: Proof For This Letmentioning

confidence: 69%

“…cit., (ii) follows from the construction, and (iii) is Lemma 5.9 in [5]. The uniqueness is proved as in Proposition 7.2.8 of [2]. Remark 2.3.…”

Section: Introductionmentioning

confidence: 94%

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A p-adic Labesse–Langlands transfer

Ludwig

2017

manuscripta math.

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“…Note that, by [7,Lemma 5.8], E is equi-dimensional of dimension n + 1 (see [ This pseudo-character is obtained by the techniques illustrated in [3] using other properties of the eigenvariety.…”

Section: Ordinary Locus Of the Eigenvariety And Its Propertiesmentioning

confidence: 99%

“…Moreover, in contrast with the second case, we compute the exact dimension of t D 0 in the fourth case without assuming any transcendence conjecture. (3) Observe that, to compute the dimensions of the tangent spaces in the cases considered in Theorem 1 above, we need to assume the Leopoldt conjecture for K at the very least. But, in [4], Bellaïche and Dimitrov compute the dimensions of these tangent spaces in all the cases when F = Q without any conditions.…”

Section: Introductionmentioning

confidence: 99%

On the eigenvariety of Hilbert modular forms at classical parallel weight one points with dihedral projective image

Deo

2017

Trans. Amer. Math. Soc.

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Abstract. We show that the p-adic eigenvariety constructed by AndreattaIovita-Pilloni, parameterizing cuspidal Hilbert modular eigenforms defined over a totally real field F , is smooth at certain classical parallel weight one points which are regular at every place of F above p and also determine whether the map to the weight space at those points isétale or not. We prove these results assuming the Leopoldt conjecture for certain quadratic extensions of F in some cases, assuming the p-adic Schanuel conjecture in some cases and unconditionally in some cases, using the deformation theory of Galois representations. As a consequence, we also determine whether the cuspidal part of the 1-dimensional parallel weight eigenvariety, constructed by Kisin-Lai, is smooth or not at those points.

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Families of L-Functions and Their Symmetry

Sarnak

Shin

Templier

2016

Families of Automorphic Forms and the Trace Formula

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Abstract. In [100] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [32], [118], [73] [49], [72] and especially [108] by the second and third-named authors which make it possible to give a conjectural answer for the symmetry type of a family and in particular the universality class predicted in [68] for the distribution of the zeros near s = 1 2 . In this note we carry this out after introducing some basic invariants associated to a family. Definition of families and ConjecturesThe zoo of automorphic cusp forms π on G = GL n over Q correspond bijectively to their standard completed L-functions Λ(s, π) and they constitute a countable set containing species of different types. For example there are self-dual forms, ones corresponding to finite Galois representations, to Hasse-Weil zeta functions of varieties defined over Q, to Maass forms, etc. From a number of points of view (including the nontrivial problem of isolating special forms) one is led to study such Λ(s, π)'s in families in which the π's have similar characteristics. Some applications demand the understanding of the behavior of the L-functions as π varies over a family. Other applications involve questions about an individual L-function. In practice a family is investigated as it arises.For example the density theorems of Bombieri [13] and Vinogradov [113] are concerned with showing that in a suitable sense most Dirichlet L-functions have few violations of the Riemann hypothesis, and as such it is a powerful substitute for the latter. Other examples are the GL 2 subconvexity results which are proved by deforming the given form in a family (see [60] and [85] for accounts). In the analogous function field setting the notion of a family of zeta functions is well defined, coming from the notion of a family of varieties defined over a base. Here too the power of deforming in a family in order to understand individual members is amply demonstrated in the work of Deligne [30]. In the number field setting there is no formal definition of a family F of L-functions.Our aim is to give a working definition for the formation of a family which will correspond to parametrized subsets of A(G), the set of isobaric automorphic representations on G(A). As far as we can tell these include almost all families that have been studied. For the most part our families can be investigated using the trace formula, monodromy groups in arithmetic geometry and the geometry of numbers, and these lead to a determination of the distribution of the zeros near s = 1 2 of members of the family. For the high zeros of a given Λ(s, π), it was shown in [98] that the local scaled spacing statistics follows the universal GUE

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The sign of Galois representations attached to automorphic forms for unitary groups

Cited by 69 publications

References 6 publications

A p-adic Labesse–Langlands transfer

A p-adic Labesse–Langlands transfer

On the eigenvariety of Hilbert modular forms at classical parallel weight one points with dihedral projective image

Families of L-Functions and Their Symmetry

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