### Computation of a universal deformation ring, by Ariane Mezard

We compute the universal deformation ring of an odd
Galois two dimensional representation of Gal$(M/Q)$
with an upper triangular image, where $M$ is the
maximal abelian pro-$p$-extension of $F_{\infty}$ unramified outside a
finite set of places S, $F_{\infty}$ being a free pro-$p$-extension
of a subextension $F$ of the field $K$ fixed by the kernel of the representation. We
establish a link between the latter universal deformation ring and the
universal deformation ring of the representation of
Gal$(K_S/Q)$, where $K_S$ is the maximal
pro-$p$-extension of $K$ unramified outside $S$. We then give some
examples.\\
This paper was accepted for publication in the Mathematical Proceedings of the Cambridge philosophical society (May 99).

Ariane Mezard <mezard@ujf-grenoble.fr>