569 research outputs found

    A note on p-adic Rankin--Selberg L-functions

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    We prove an interpolation formula for the values of certain pp-adic Rankin--Selberg LL-functions associated to non-ordinary modular forms.Comment: Updated version, with minor corrections. To appear in Canad. Math. Bulleti

    Density of classical points in eigenvarieties

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    In this short note, we study the geometry of the eigenvariety parametrizing p-adic automorphic forms for GL(1) over a number field, as constructed by Buzzard. We show that if K is not totally real and contains no CM subfield, points in this space arising from classical automorphic forms (i.e. algebraic Grossencharacters of K) are not Zariski-dense in the eigenvariety (as a rigid space); but the eigenvariety posesses a natural formal scheme model, and the set of classical points is Zariski-dense in the formal scheme. We also sketch the theory for GL(2) over an imaginary quadratic field, following Calegari and Mazur, emphasizing the strong formal similarity with the case of GL(1) over a general number field

    P-adic interpolation of metaplectic forms of cohomological type

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    Let G be a reductive algebraic group over a number field k. It is shown how Emerton's methods may be applied to the problem of p-adically interpolating the metaplectic forms on G, i.e. the automorphic forms on metaplectic covers of G, as long as the metaplectic covers involved split at the infinite places of k.Comment: 37 page

    P-adic Asai L-functions of Bianchi modular forms

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    The Asai (or twisted tensor) LL-function of a Bianchi modular form Ψ\Psi is the LL-function attached to the tensor induction to Q\mathbb{Q} of its associated Galois representation. In this paper, when Ψ\Psi is ordinary at pp we construct a pp-adic analogue of this LL-function: that is, a pp-adic measure on Zp×\mathbb{Z}_p^\times that interpolates the critical values of the Asai LL-function twisted by Dirichlet characters of pp-power conductor. The construction uses techniques analogous to those used by Lei, Zerbes and the first author in order to construct an Euler system attached to the Asai representation of a quadratic Hilbert modular form.Comment: Final version, to appear in Algebra & Number Theor

    Rankin--Eisenstein classes in Coleman families

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    We show that the Euler system associated to Rankin--Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in pp-adic Coleman families. We prove an explicit reciprocity law for these families, and use this to prove cases of the Bloch--Kato conjecture for Rankin--Selberg convolutions.Comment: Updated version, to appear in "Research in the Mathematical Sciences" (Robert Coleman memorial volume

    On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions

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    We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.Comment: To appear in Canad. J. Mat

    Rankin--Eisenstein classes and explicit reciprocity laws

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    We construct three-variable pp-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated LL-value does not vanish.Comment: Final version, to appear in Cambridge J Math; small correction to acknowlegement
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