We explain a construction of explicit extensions — of rational Hodge structures and of
-adic
Galois representations — in a simple context: the cohomology of
relative
to
.
These extensions are naturally related to Dirichlet characters, and
we connect the nonsplitting of these extensions to the values at
and
of associated
Dirichlet
-functions
. We
highlight the close parallels between the proofs of nonsplitting in both the Hodge-theoretic
and
-adic
cases, emphasizing the use of de Rham theory. We also indicate connections with
Euler systems along with variations on these constructions in the setting of modular
curves. This paper is intended as an introduction to some of the key ideas in
forthcoming constructions of Galois cohomology classes and Euler systems in a range
of settings.
Keywords
$L$-values, Galois extensions, mixed Hodge structures,
Euler systems
