ON THE SEARCH OF GENUINE p-ADIC

MODULAR L-FUNCTIONS FOR GL(n)

Haruzo Hida

Abstract. 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 “genuine” 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.

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