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edited Nov 13, 2015 at 8:52

This is a very rich and active subject. There are lots of different approaches to the problem, giving more or less strong results -- you can try to interpolate any or all of { Hecke eigenvalues, Fourier coefficients, L-values, Galois representations }, for forms satisfying various different flavours of finite-slope condition, while the weight varies in families having different numbers of parameters. Here are a selection of the important works on this:

For ordinary families appearing in Betti cohomology, see Tilouine--Urban, "Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations", Ann Sci ENS, 1999.
For non-ordinary families appearing in functions on a compact form of $Sp(4)$, try the PhD thesis of Daniel Snaith (aka "Caribou"), (Imperial College, early 2000's).
For a coherent-cohomology approach, more in the style of Coleman's work for $GL(2)$, try Andrei Jorza's thesis (Princeton, 2010), or for rather stronger results Andreatta--Iovita--Pilloni (Annals, this year).

These are just the references I know that treat Siegel modular forms specifically; there are other references that treat general reductive groups from which one can extract something for $Sp(4)$.

This is a very rich and active subject. There are lots of different approaches to the problem, giving more or less strong results. Here are a selection of the important works on this:

For ordinary families appearing in Betti cohomology, see Tilouine--Urban, "Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations", Ann Sci ENS, 1999.
For non-ordinary families appearing in functions on a compact form of $Sp(4)$, try the PhD thesis of Daniel Snaith (aka "Caribou"), (Imperial College, early 2000's).
For a coherent-cohomology approach, more in the style of Coleman's work for $GL(2)$, try Andrei Jorza's thesis (Princeton, 2010), or for rather stronger results Andreatta--Iovita--Pilloni (Annals, this year).

These are just the references I know that treat Siegel modular forms specifically; there are other references that treat general reductive groups from which one can extract something for $Sp(4)$.

For ordinary families appearing in Betti cohomology, see Tilouine--Urban, "Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations", Ann Sci ENS, 1999.
For non-ordinary families appearing in functions on a compact form of $Sp(4)$, try the PhD thesis of Daniel Snaith (aka "Caribou"), (Imperial College, early 2000's).
For a coherent-cohomology approach, more in the style of Coleman's work for $GL(2)$, try Andrei Jorza's thesis (Princeton, 2010), or for rather stronger results Andreatta--Iovita--Pilloni (Annals, this year).

These are just the references I know that treat Siegel modular forms specifically; there are other references that treat general reductive groups from which one can extract something for $Sp(4)$.

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created Nov 13, 2015 at 8:43

David Loeffler

This is a very rich and active subject. There are lots of different approaches to the problem, giving more or less strong results. Here are a selection of the important works on this:

For ordinary families appearing in Betti cohomology, see Tilouine--Urban, "Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations", Ann Sci ENS, 1999.
For non-ordinary families appearing in functions on a compact form of $Sp(4)$, try the PhD thesis of Daniel Snaith (aka "Caribou"), (Imperial College, early 2000's).
For a coherent-cohomology approach, more in the style of Coleman's work for $GL(2)$, try Andrei Jorza's thesis (Princeton, 2010), or for rather stronger results Andreatta--Iovita--Pilloni (Annals, this year).

These are just the references I know that treat Siegel modular forms specifically; there are other references that treat general reductive groups from which one can extract something for $Sp(4)$.