1,884 research outputs found
Several variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations
Let F be a totally real field and G=GSp(4)_{/F}. In this paper, we show under
a weak assumption that, given a Hecke eigensystem lambda which is
(p,P)-ordinary for a fixed parabolic P in G, there exists a several variable
p-adic family underline{lambda} of Hecke eigensystems (all of them (p,P)-nearly
ordinary) which contains lambda. The assumption is that lambda is cohomological
for a regular coefficient system. If F=Q, the number of variables is three.
Moreover, in this case, we construct the three variable p-adic family
rho_{underline{lambda}} of Galois representations associated to
underline{lambda}. Finally, under geometric assumptions (which would be
satisfied if one proved that the Galois representations in the family come from
Grothendieck motives), we show that rho_{underline{lambda}} is nearly ordinary
for the dual parabolic of P. This text is an updated version of our first
preprint (issued in the "Prepublication de l'universite Paris-Nord") and will
appear in the "Annales Scientifiques de l' E N S"
Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of -adic -functions
We study the derivative of the standard -adic -function associated with
a -ordinary Siegel modular form (for a parabolic subgroup of
) when it presents a semi-stable trivial zero. This implies
part of Greenberg's conjecture on the order and leading coefficient of -adic
-functions at such trivial zero. We use the method of Greenberg--Stevens.
For the construction of the improved -adic -function we develop Hida
theory for non-cuspidal Siegel modular forms
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