Higher coherent cohomology and \(p\)-adic modular forms of singular weights
``In this article, we investigate the theory of \(p\)-adic families of automorphic forms for the group \(\text{GSp}_4/\mathbb Q\) whose component at infinity has singular Harish-Chandra parameter and is a nondegenerate limit of discrete series. The automorphic forms we consider can be realized in the coherent cohomology of an appropriate automorphic vector bundle over a Siegel threefold (see [\textit{M. Harris}, Proc. Lond. Math. Soc. (3) 59, No. 1, 1--22 (1989; Zbl 0711.14011)]). The Siegel threefolds are finite unions of arithmetic quotients of the \(3\)-dimensional Siegel upper half-space. They have a modular interpretation as moduli spaces of abelian surfaces with polarization and level structure and they have canonical models over number fields. Using this coherent realization, one can prove that the Hecke parameters of these automorphic forms are defined over number fields, and one can construct, using congruences, compatible systems of \(4\)-dimensional Galois representations (see, e.g., [the author and \textit{B. Stroh}, Ann. Math. Qué. 40, No. 1, 167--202 (2016; Zbl 1357.14033); \textit{R. Taylor}, Duke Math. J. 63, No. 2, 281--332 (1991; Zbl 0810.11033)]).'' The author constructs \(p\)-adic families of (cuspidal) cohomology classes for the sheaves \(\{\Omega^{(k,2)}\}_{k\geq 0}\) (attached to the representation \(\text{Sym}^k\text{St} \otimes \det^2 \text{St}\) of the group \(\text{GL}_2\)) in degree \(0\) and \(1\) (Theorems 1.1 and 1.2). ``It is conjectured that for every simple abelian surface \(A\) over \(\mathbb Q\) there should exist a cuspidal automorphic form \(\pi\) on \(\text{GSp}_4\mathbb Q\) such that the spin \(L\)-function of \(\pi\) and the \(L\)-function of \(H^1(A)\) coincide. When \(\text{End}(A) \not=\mathbb Z\), this is known (see, e.g., [\textit{M. Kisin}, Invent. Math. 178, No. 3, 587--634 (2009; Zbl 1304.11043); \textit{H. Yoshida}, Invent. Math. 60, 193--248 (1980; Zbl 0453.10022); \textit{A. Brumer} and \textit{K. Kramer}, Trans. Am. Math. Soc. 366, No. 5, 2463--2516 (2014; Zbl 1285.11087)] for a precise conjecture in the case in which \(\text{End}(A) = \mathbb Z\)). These automorphic forms are of the type we have considered so far, as their component at infinity should be a limit of discrete series and they should realize in the cuspidal coherent cohomology of the sheaf \(\Omega^{(0,2)}\).'' \textit{H. Hida} [Asian J. Math. 2, No. 4, 667--710 (1998; Zbl 0963.11027)] initiated the study of ordinary Betti cohomology on locally symmetric spaces associated to \(\text{GL}_n\) over arbitrary number fields. When \(n\geq 3\) (or \(n\geq 2\) and \(F\) is not totally real), the non-Eisenstein cohomology is concentrated in more than one degree. To some extent, what we present here is the beginning of a coherent analogue of his theory. The analogy is that in both situations the interesting cohomology is naturally supported in several concsecutive degrees (see the Introduction of [\textit{F. Calegari} and \textit{D. Geraghty}, Invent. Math. 211, No. 1, 297--433 (2018; Zbl 1476.11078)]).
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