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Nearly ordinary rank four Galois representations and $$p$$-adic Siegel modular forms. With an appendix by Don Blasius. (English) Zbl 1159.11018
Let $$\pi$$ be a cuspidal representation (defined over $${\mathbb Q}$$) occurring in the middle degree cohomology of the Siegel threefold (of a given level) with coefficients in a local system of highest weight $$(a, b)$$ with $$a\geq b\geq 0$$. Let $$p$$ be a prime, prime to the level, and such that $$p- 1> a+ b+ 3$$.
Let $$T$$ be the localization of the universal nearly ordinary Hecke algebra at the maximal ideal corresponding to $$(\pi,p)$$. It is an algebra over an Iwasawa algebra $$\Lambda$$ in three variables. Let $$R$$ be the universal deformation ring of nearly ordinary symplectic Galois representations congruent to the residual Galois representation associated to $$(\pi, p)$$. It is also endowed with a natural structure of $$\Lambda$$-modules. The author conjectured in his monograph [Deformations of Galois representations and Hecke algebras, Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad (1996; Zbl 1009.11033)] that $$R$$ and $$T$$ are finite and flat as $$\Lambda$$-modules and that the natural map $$R\to T$$, obtained from the universal property of $$R$$, is an isomorphism. In this article, the author uses the isomorphism theorem proved in [A. Genestier and the author, Astérisque 302, 177–290 (2005; Zbl 1142.11036)] to prove these conjectures (Theorem 4) under a technical assumption assuring that the ordinary deformations of the residual representation associated to $$(\pi, p)$$ are crystalline. This is the main result of the first part of the article.
In the second part of the paper, the author shows that if the Galois representation on the $$p$$-adic Tate module of an abelian surface over $${\mathbb Q}$$ is residually modular (and satisfies certain technical assumptions) then this Galois representation arises from an overconvergent Siegel cusp form of weight $$(2, 2)$$. The question whether this form is in fact classical remains open.

##### MSC:
 11F80 Galois representations 11F33 Congruences for modular and $$p$$-adic modular forms 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11G18 Arithmetic aspects of modular and Shimura varieties 11F70 Representation-theoretic methods; automorphic representations over local and global fields
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