Details of Grant 

EPSRC Reference:	EP/G050511/1
Title:	Congruences between automorphic forms
Principal Investigator:	Sasaki, Dr S
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Department:	Mathematics
Organisation:	Kings College London
Scheme:	Postdoc Research Fellowship
Starts:	21 October 2009	Ends:	20 October 2012	Value (£):	228,951
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 17 Mar 2009 Maths Postdoctoral Fellowships Interview Panel Announced 12 Feb 2009 Maths Postdoctoral Research Fellowships 2008/2009 Excluded
Summary on Grant Application Form
I propose to prove new cases of the strong Artin conjecture for totally odd two dimensional representations of the Galois group of a totally real field. More precisely, I propose to establish ``analytic continuation of overconvergent Hilbert modular eigenforms'' in the inert Hilbert case and generalise the main theorem of Buzzard-Taylor to the Hilbert case. Mimicking the approach of Taylor's ``Artin II'' paper, one should get modularity of certain mod p Galois representations subject to some local conditions, and many cases of Artin's conjecture should then become accessible. Secondly, I propose to show that there is another example of the correspondence predicted by Langlands between n-dimensional representations of the absolute Galois group of the rationals Q and (cuspidal) automorphic representations of GL(n) over Q. Following the geometric argument of Kisin-Lai for Hilbert modular forms, it should be possible to construct p-adic analytic families of automorphic Galois representations and associaten-dimensional Galois representations by ``specialisation'' to non-regular algebraic automorphic representations of GL(n) over Q (whose Hodge-Tate weights ``at infinity'' are not necessarily distinct). It may also be possible toprove that non-regular automorphic representations for GL(n) appear on the Chenevier's eigenvariety and associate Galois representations by specialising families of Galois representations Chenevier constructed. Another approach to look at congruences between automorphic forms of ``regular weight'' and ``non-regular weight'', analogous to Deligne-Serre, will also be considered.
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