### On ordinary forms and ordinary Galois representations, by Kirti Joshi and Chandrashekhar Khare

In this paper we study ordinary mod $p$ Galois representations and the
different ordinary newforms which give rise to this representation. As
a consequence this provides a complete converse to Deligne's theorem that
any ordinary newform gives rise to an ordinary Galois
representation. The results proven here are in some sense optimal. We
also study applications of Carayol's lemma: using this lemma we
show, for instance, that any local component of the ordinary Hecke
algebra of Hida attached to an ordinary newform is two
dimensional. This provides another proof of a well-known theorem of
Hida. We also raise some $p$-adic variants of our mod $p$ questions,
which seem interesting.

Kirti Joshi and Chandrashekhar Khare <kirti@motive.math.tifr.res.in,shekhar@motive.math.tifr.res.in>