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A \(p\)-adic Simpson correspondence. (English) Zbl 1102.14022
The Narasimhan-Seshadri theorem gave a bijective correspondence between irreducible unitary representations of the (topological) fundamental group of a compact Riemann surface and stable vector bundles of degree \(0\) on the surface. Hitchin, Simpson and Corlette established a one to one correspondence between semisimple representations of the fundamental group of a complex Kähler manifold and Higgs bundles on the manifold.
In this interesting paper Faltings gives a \(p\)-adic analogue of these famous results. Using almost étale coverings and \(p\)-adic Hodge theory, he defines generalized representaions of the étale fundamental group of a curve over a \(p\)-adic field, these include usual representaions as a full subcategory. He establishes an equivalence between the category of generalized representations and the category of Higgs bundles on the curve. Under this equivalence, the Higgs bundles associated to usual representations are semistable of degree \(0\). It is not known if the converse is true, the converse is true for line bundles on curves over \(p\)-adic local fields. The equivalence is not canonical, it depends on certain choices, including a choice of an exponential function of the multiplicative group. Most of the constructions are in fact done and work in the more general set up of schemes with toroidal singularities (with some restrictions). Related interesting results are due to C. Deninger and A. Werner [Ann. Sci. Éc. Norm. Supér Sér. 38, No. 4, 553–597 (2005; Zbl 1087.14026); in: Number fields and function fields – two parallel worlds. Progr. Math. 239, 101–131 (2005; Zbl 1100.11019)].

14H60 Vector bundles on curves and their moduli
14G20 Local ground fields in algebraic geometry
14D20 Algebraic moduli problems, moduli of vector bundles
Full Text: DOI
[1] C. Deninger, A. Werner, Vector bundles and p-adic representations I, Mŭnster, 2003, preprint.
[2] C. Deninger, A. Werner, Bundles on p-adic curves and parallel transport, Mŭnster, 2004, preprint. · Zbl 1087.14026
[3] Faltings, G., Hodge-Tate structures and modular forms, Math. ann., 278, 133-149, (1987) · Zbl 0646.14026
[4] Faltings, G., Crystalline cohomology of semistable curves, and p-adic Galois-representations, J. algebraic geom., 1, 61-82, (1992) · Zbl 0813.14012
[5] Faltings, G., Curves and their fundamental groups (following Grothendieck, Tamagawa and mochizuki), Sem. bourbaki 840, astérisque, 252, 131-150, (1998) · Zbl 0933.14015
[6] Faltings, G., Almost étale extensions, Astérisque, 279, 185-270, (2002) · Zbl 1027.14011
[7] J.M. Fontaine, Le corps de périodes p-adiques, in: Périodes p-adiques, Astérisque, vol. 223, 1994. · Zbl 0940.14012
[8] J.M. Fontaine, Arithmétique des représentations galoisiennes p-adiques, Orsay 2000-24, preprint.
[9] J.M. Fontaine, Presque \(\mathbb{C}_p\)-représentations, Orsay 2002-12, preprint.
[10] Gabber, O.; Ramero, L., Almost ring theory, () · Zbl 1045.13002
[11] Simpson, C.T., Higgs bundles and local systems, Publ. math. IHES, 75, 5-95, (1992) · Zbl 0814.32003
[12] Vologodsky, V., Hodge structure on the fundamental group and its application to p-adic integration, Moscow math. J., 3, 205-247, (2003) · Zbl 1050.14013
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