On the search of genuine p-adic modular L-functions for GL(n)

Haruzo Hida

MSRI #1995-035

Recently many p-adic Galois representations are constructed out of
cohomological modular forms on the algebraic groups GL(n) over various
number fields.  These representations have values in GL(n,I) for an
integral domain I which is finite over a several variable power series
ring over the p-adic integer ring.  In this paper, I study a
conjectural theory of "genuiune" p-adic L-functions associated to such
big Galois representations into GL(n,I).  We give a precise conjecture
predicting the existence and the singularity of such p-adic
L-functions.  We also discuss its relation to known p-adic L-functions
and the factorization of motivic periods.