562 research outputs found
A note on p-adic Rankin--Selberg L-functions
We prove an interpolation formula for the values of certain -adic
Rankin--Selberg -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
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
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
The Asai (or twisted tensor) -function of a Bianchi modular form is
the -function attached to the tensor induction to of its
associated Galois representation. In this paper, when is ordinary at
we construct a -adic analogue of this -function: that is, a -adic
measure on that interpolates the critical values of the
Asai -function twisted by Dirichlet characters of -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
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 -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
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
We construct three-variable -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 -value does not vanish.Comment: Final version, to appear in Cambridge J Math; small correction to
acknowlegement
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