AbstractWe prove (under certain assumptions) the irreducibility of the limit $$\sigma _2$$
σ
2
of a sequence of irreducible essentially self-dual Galois representations $$\sigma _k: G_{{\mathbf {Q}}} \rightarrow {{\,\mathrm{GL}\,}}_4(\overline{{\mathbf {Q}}}_p)$$
σ
k
:
G
Q
→
GL
4
(
Q
¯
p
)
(as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to $$1 \oplus \rho \oplus \chi $$
1
⊕
ρ
⊕
χ
with $$\rho $$
ρ
irreducible, two-dimensional of determinant $$\chi $$
χ
, where $$\chi $$
χ
is the mod p cyclotomic character. More precisely, we assume that $$\sigma _k$$
σ
k
are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as $$k\rightarrow 2$$
k
→
2
appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to $$\rho $$
ρ
) which we assume are non-zero.
On Galois Representations via Siegel Modular Forms of Genus Two
