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Mod pq Galois representations and Serre's conjecture

Authors :
Chandrashekhar Khare
Ian Kiming
Source :
Journal of Number Theory. 98(2):329-347
Publication Year :
2003
Publisher :
Elsevier BV, 2003.

Abstract

Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one obtains compatible families of mod p representations. While the representations that occur in such a p-adic or mod p family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial. In this paper we investigate when a pair of mod p and mod q representations of the absolute Galois group of a number field K simultaneously arises from an {\it automorphic motive}: we do this in the 1-dimensional (Section 2) and 2-dimensional (Section 3: this time assuming $K={\mathbb Q}$) cases. In Section 3 we formulate a mod pq version of Serre's conjecture refining in part a question of Barry Mazur and Ken Ribet.<br />This is an older preprint that was made available elsewhere on Sep. 19, 2001

Details

ISSN :
0022314X
Volume :
98
Issue :
2
Database :
OpenAIRE
Journal :
Journal of Number Theory
Accession number :
edsair.doi.dedup.....2c26fbd3f04942f3b61487377bae46d8
Full Text :
https://doi.org/10.1016/s0022-314x(02)00043-4