$p$-adic families and Galois representations for $GS_p(4)$ and $GL(2)$

In this brief paper, we prove local-global compatibility for holomorphic Siegel modular forms with Iwahori level. In previous work, we proved a weaker version of this result (up to a quadratic twist) and one of the goals of this paper is to remove this quadratic twist by different methods, using $p$-adic families. We further study the local Galois representation at p for nonregular holomorphic Siegel modular forms. Then we apply the results to the setting of modular forms on $GL(2)$ over a quadratic imaginary field and prove results on the local Galois representation ℓ, as well as crystallinity results at $p$.

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Published 15 March 2013