Kubert-Lang modular units, Bernoulli-Hurwitz numbers, Wieferich places and Galois representations

In this talk I will discuss the size of the image of a
certain Galois representation (originally studied by Rohrlich) that
arises as a deformation of the usual p-adic Galois representation
associated to the Tate module of an elliptic curve E. In particular,
we will show that if E has CM by K then, for almost all inert primes
in K, the image of the larger representation is “as large as
possible”, that is, it is the full inverse image in SL(2,Z_p[[X]]) of
a Cartan subgroup of SL(2,Z_p). If p splits in K, then the same result
holds as long as a certain Bernoulli-Hurwitz number is a p-adic unit
which, in turn, is equivalent to a prime ideal not being a Wieferich
place. The proof rests on the theory of elliptic units developed by
Ramachandra, Robert, Kubert and Lang, and on the two-variable main
conjecture of Iwasawa theory for quadratic imaginary fields.