Essential image of the p-adic Simpson correspondence: Faltings' conjecture that all semistable slope-zero Higgs bundles arise from genuine p-adic representations

Over a smooth proper curve $X$ over a $p$-adic field, Faltings' p-adic Simpson correspondence (2005, completed by Abbes-Gros-Tsuji 2016) attaches a Higgs bundle to every continuous $p$-adic representation of the geometric etale fundamental group $\pi_1(X_{\bar K},x)$, by embedding genuine representations into a larger category of generalized representations to which an equivalence with Higgs bundles is available. The central open question of the theory, raised in Faltings' original work and repeatedly reformulated since, is to characterize the essential image of the genuine representations: exactly which Higgs bundles arise. Faltings conjectured, for curves, that all semistable Higgs bundles of slope zero arise -- the p-adic analogue of the Narasimhan-Seshadri and Corlette-Simpson theorems. The rank-one case is understood (Faltings via the Jacobian logarithm; Heuer intrinsically: the image is the topologically torsion locus, which over coefficient fields isomorphic to $\mathbb{C}_p$ is the degree-zero locus), but for rank at least 2 the question is open despite substantial progress: Deninger-Werner parallel transport, Xu's equivalence with potentially strongly semistable reduction, and the geometrization program via $v$-bundles and moduli stacks (Heuer; Anschütz-Heuer-Le Bras; Heuer-Xu). A claimed negative answer (Andreatta, 2024) was withdrawn in 2025.

Background

The complex model. Let $Y$ be a smooth projective variety over the field of complex numbers. A Higgs bundle on $Y$ is a pair $(M,\theta)$ consisting of an algebraic vector bundle $M$ and an $\mathcal{O}_Y$-linear map $\theta\colon M\to M\otimes\Omega^1_Y$ (the Higgs field) satisfying $\theta\wedge\theta=0$. On a curve this condition is vacuous. The slope of $M$ is $\mu(M)=\deg(M)/\operatorname{rk}(M)$, where $\deg M=\deg\det M$, and a Higgs bundle is semistable if $\mu(N)\le\mu(M)$ for every nonzero proper subbundle $N\subseteq M$ stable under $\theta$; semistable of slope zero is equivalent to semistable of degree zero. The complex Simpson correspondence (with the Narasimhan-Seshadri theorem as the curve prototype) relates complex representations of the fundamental group of $Y$ to Higgs bundles, and its difficult part -- identifying which Higgs bundles come from representations -- is answered by semistability together with degree zero. The p-adic problem asks for the analogous statement when the coefficient field is a p-adic field. The p-adic setting. Let $K$ be a finite extension of $\mathbb{Q}_p$, let $\bar K$ be an algebraic closure, let $\mathbb{C}=\widehat{\bar K}$ be the completed algebraic closure (for every finite $K$ this field is isomorphic to $\mathbb{C}_p$, the completed algebraic closure of $\mathbb{Q}_p$), and let $X$ be a smooth, proper, geometrically connected curve over $K$ with geometric base point $x$. The geometric etale fundamental group $\pi_1(X_{\bar K},x)$ is the profinite group classifying finite etale covers of $X_{\bar K}$, and a genuine representation is a continuous group homomorphism $\pi_1(X_{\bar K},x)\to\mathrm{GL}_n(\mathbb{C})$. Since $\mathbb{C}$ is totally disconnected, no transcendental construction relates such representations to coherent cohomology, so any analogue of the complex correspondence must be arithmetic. Faltings' construction. Faltings (2005) introduced the p-adic Simpson correspondence: instead of working with genuine representations directly, one embeds them into a category of generalized representations -- sheaves of modules with a semilinear action of $\pi_1(X_{\bar K},x)$, living in Faltings' ringed topos built from a proper model $\mathfrak X$ of $X$ over $\mathcal{O}_K$ -- and constructs a functor from generalized representations to Higgs bundles on $X_{\mathbb{C}}$. The functor depends on auxiliary choices: a p-adic exponential (a splitting of the logarithm on the multiplicative group) and lifts of the variety to Faltings' sheaf of algebras $A_2$, a dual-number-type square-zero thickening of the structure sheaf by $\Omega^1$. When the $\pi_1$-action is sufficiently $p$-divisible (the smallness condition, roughly $\alpha>1/(p-1)$ in Faltings' parameter), the restricted functor is an equivalence between small generalized representations and small Higgs bundles; for curves, Faltings sketched a descent to all objects using the multiplication-by-$p^n$ maps on the Jacobian. Abbes-Gros-Tsuji (2016) completed and generalized the construction via the theory of Higgs-Tate algebras (sheaves of p-adic period rings on the formal model that serve as rings of coefficients for generalized representations), establishing the small correspondence in all dimensions. Further aspects of the theory are a canonical, choice-free formulation of the correspondence (progress: Heuer's geometrization, Anschütz-Heuer-Le Bras, Pan-Yu's lift-independence results) and the role of connections (the Riemann-Hilbert side of Liu-Zhu) and Frobenius structures (Tsuji's Higgs isocrystals) in the theory. What is known about the image. Every genuine representation induces a generalized representation, hence a Higgs bundle. Faltings claimed that these Higgs bundles are semistable of degree zero; this was proved in later work (Xu, Theorem 1.1.9(i); Heuer-Xu). In rank one the image is completely understood: via the logarithm sequence on the Jacobian $J$, the characters $\pi_1(X_{\bar K},x)\to\mathbb{C}^{\times}$ correspond to Higgs line bundles whose underlying line bundle lies in the topologically torsion subgroup $J(\mathbb{C})'$ (the preimage of the torsion of the special fiber under reduction); over coefficient fields isomorphic to $\mathbb{C}_p$, in particular over the field $\mathbb{C}$ just defined, this is exactly the degree-zero locus, but over general complete algebraically closed fields topological torsion is strictly stronger than degree zero -- a first warning that the naive analogue of the complex statement requires care in its formulation. For rank at least 2 no characterization of the image is known, and Faltings expressed the hope that for curves it consists precisely of the semistable Higgs bundles of slope zero. This is the p-adic Narasimhan-Seshadri problem, and it is the subject of this problem record.

Problem Statement

Let $K$ be a finite extension of $\mathbb{Q}_p$, with algebraic closure $\bar K$ and completed algebraic closure $\mathbb{C}$, and let $X$ be a smooth, proper, geometrically connected curve over $K$ (equivalently, a smooth projective curve $X_{\mathbb{C}}$ over $\mathbb{C}$ equipped with a model over $\mathcal{O}_K$), with geometric base point $x$. By the p-adic Simpson correspondence (Faltings 2005, completed by Abbes-Gros-Tsuji), every continuous finite-dimensional $\mathbb{C}$-representation (genuine representation) of the geometric etale fundamental group $\pi_1(X_{\bar K},x)$ determines, via the intermediate category of generalized representations, a Higgs bundle on $X_{\mathbb{C}}$. Main objective -- decide Faltings' image conjecture and characterize the essential image. Determine exactly which Higgs bundles $(M,\theta)$ on $X_{\mathbb{C}}$ arise from genuine representations. Concretely, prove or refute Faltings' conjecture (formulated for curves): every semistable Higgs bundle of degree (equivalently slope) zero on $X_{\mathbb{C}}$ arises from a continuous $\mathbb{C}$-representation of $\pi_1(X_{\bar K},x)$. Through the published reformulations this is equivalent to deciding whether every semistable degree-zero Higgs bundle belongs to Xu's category of Higgs bundles with potentially strongly semistable reduction of degree zero -- here a vector bundle on the special fiber of a model is strongly semistable if all its Frobenius pullbacks are semistable, and 'potentially' means after pullback along a finite etale cover; by Xu's Theorem 1.1.7, stated for curves of genus at least 2, this category is equivalent to the genuine representations -- and to describing explicitly the open sub-$v$-stack (for the $v$-topology, a Grothendieck topology on perfectoid spaces finer than the etale topology) that the representation condition cuts out inside the moduli stack of semistable Higgs bundles (Heuer-Xu). The characterization must be intrinsic -- independent of the auxiliary choices (p-adic exponential, lifts to $A_2$) used to construct the correspondence -- and must be consistent with the rank-one benchmark: in rank one the image is the locus of topologically torsion underlying line bundles (a line bundle $L$ is topologically torsion if $L^{\otimes n!}\to 1$ as $n\to\infty$; on the Jacobian of a curve of good reduction these are exactly the points reducing to torsion points of the special fiber), which over coefficient fields isomorphic to $\mathbb{C}_p$, such as the $\mathbb{C}=\widehat{\bar K}$ fixed above, agrees with degree zero but is strictly smaller over general complete algebraically closed fields.

References

1. Gerd Faltings, 'Facsimile: A p-adic Simpson correspondence', in Ahmed Abbes, Michel Gros, Takeshi Tsuji (eds.), The p-adic Simpson Correspondence, Annals of Mathematics Studies 193, Princeton University Press, 2016, pp. 577-594. DOI: 10.1515/9781400881239-008. URL: https://doi.org/10.1515/9781400881239-008.
2. Gerd Faltings, 'A p-adic Simpson correspondence', Advances in Mathematics 198 (2005), 847-862. DOI: 10.1016/j.aim.2005.05.026. URL: https://doi.org/10.1016/j.aim.2005.05.026.
3. Ahmed Abbes, Michel Gros, Takeshi Tsuji, The p-adic Simpson Correspondence, Annals of Mathematics Studies 193, Princeton University Press, 2016. DOI: 10.1515/9781400881239. URL: https://doi.org/10.1515/9781400881239. (The volume containing the source chapter together with the editors' complete account, including Tsuji's chapter on Higgs isocrystals.)
4. Gerd Faltings, 'A p-adic Simpson Correspondence II: Small Representations', Pure and Applied Mathematics Quarterly 7 (2011), no. 4, 1241-1264. DOI: 10.4310/pamq.2011.v7.n4.a8. URL: https://doi.org/10.4310/pamq.2011.v7.n4.a8.
5. Ahmed Abbes, Michel Gros, 'Sur la correspondance de Simpson p-adique. 0 : une vue d'ensemble', arXiv:1301.1443 (2013). URL: https://arxiv.org/abs/1301.1443. (Section 1.4 contains the direct attestation that Faltings showed Higgs bundles from true representations are semistable of slope zero and formulated the hope that all semistable slope-zero Higgs bundles arise.)
6. Ahmed Abbes, Michel Gros, 'The p-adic Simpson correspondence II: functoriality by proper direct image and Hodge-Tate local systems - an overview', arXiv:2210.10536 (2022). URL: https://arxiv.org/abs/2210.10536.
7. Ahmed Abbes, Michel Gros, Takeshi Tsuji, 'Twisting Higgs Modules and Functorial Aspects of the p-adic Simpson Correspondence', arXiv:2605.02664 (2026). URL: https://arxiv.org/abs/2605.02664.
8. Christopher Deninger, Annette Werner, 'Parallel transport for vector bundles on p-adic varieties', Journal of Algebraic Geometry 29 (2020), no. 1, 1-52. DOI: 10.1090/jag/747. URL: https://doi.org/10.1090/jag/747. arXiv:1707.05121. URL: https://arxiv.org/abs/1707.05121. (The higher-dimensional refinement of their 2005 construction.)
9. Christopher Deninger, Annette Werner, 'Vector bundles on p-adic curves and parallel transport', Annales Scientifiques de l'École Normale Supérieure 38 (2005), no. 4, 553-597. DOI: 10.1016/j.ansens.2005.05.002. URL: https://doi.org/10.1016/j.ansens.2005.05.002. (The original p-adic Narasimhan-Seshadri-type parallel transport, for vector bundles with strongly semistable reduction on p-adic curves.)
10. Ruochuan Liu, Xinwen Zhu, 'Rigidity and a Riemann-Hilbert correspondence for p-adic local systems', Inventiones mathematicae 207 (2017), 291-343. DOI: 10.1007/s00222-016-0671-7. URL: https://doi.org/10.1007/s00222-016-0671-7. arXiv:1602.06282. URL: https://arxiv.org/abs/1602.06282.
11. Ben Heuer, 'A geometric p-adic Simpson correspondence in rank one', Compositio Mathematica 160 (2024), no. 7, 1433-1466. DOI: 10.1112/s0010437x24007024. URL: https://doi.org/10.1112/s0010437x24007024. arXiv:2207.13657. URL: https://arxiv.org/abs/2207.13657. (The rank-one solution via topological torsion, eliminating the exponential choice.)
12. Ben Heuer, 'A p-adic Simpson correspondence for smooth proper rigid varieties', Inventiones mathematicae (published online 2025). DOI: 10.1007/s00222-025-01321-4. URL: https://doi.org/10.1007/s00222-025-01321-4. arXiv:2307.01303. URL: https://arxiv.org/abs/2307.01303.
13. Johannes Anschütz, Ben Heuer, Arthur-César Le Bras, 'The small p-adic Simpson correspondence in terms of moduli spaces', arXiv:2312.07554. URL: https://arxiv.org/abs/2312.07554.
14. Ben Heuer, Daxin Xu, 'p-adic non-abelian Hodge theory for curves via moduli stacks', Journal of the American Mathematical Society 39 (2026), no. 3, 625-695. DOI: 10.1090/jams/1065. URL: https://doi.org/10.1090/jams/1065. arXiv:2402.01365. URL: https://arxiv.org/abs/2402.01365. (States that describing the essential image is a major open problem and solves it for rank one.)
15. Daxin Xu, 'Parallel transport for Higgs bundles over p-adic curves', Mathematische Annalen 395 (2026), no. 4, with an appendix by Tongmu He. DOI: 10.1007/s00208-026-03470-0. URL: https://doi.org/10.1007/s00208-026-03470-0. arXiv:2201.06697. URL: https://arxiv.org/abs/2201.06697. (Contains the equivalence with potentially strongly semistable reduction and the statement of Faltings' Question 1.1.10.)
16. Fabrizio Andreatta, 'On a p-adic version of Narasimhan and Seshadri's theorem', arXiv:2406.12766 (v1 June 2024; v2 withdrawn April 2025). URL: https://arxiv.org/abs/2406.12766.
17. Mao Sheng, Jiajin Zhang, Kang Zuo, 'Higgs bundles over the good reduction of a quaternionic Shimura curve', arXiv:1002.3296. URL: https://arxiv.org/abs/1002.3296. (Source of the counterexample that led to the withdrawal of Andreatta's claimed negative answer.)
18. Ben Heuer, Annette Werner, Mingjia Zhang, 'p-adic Simpson correspondences for principal bundles in abelian settings', arXiv:2308.13456. URL: https://arxiv.org/abs/2308.13456.
19. Yu Min, Yupeng Wang, 'p-adic Simpson correspondence via prismatic crystals', Journal of the European Mathematical Society (published online 2025). DOI: 10.4171/jems/1579. URL: https://doi.org/10.4171/jems/1579. arXiv:2201.08030. URL: https://arxiv.org/abs/2201.08030. (The published JEMS title carries the corrected spelling; the arXiv listing title contains the typographical error 'correpondence'.)
20. Yupeng Wang, 'A p-adic Simpson correspondence for rigid analytic varieties', Algebra & Number Theory 17 (2023), no. 8, 1453-1499. DOI: 10.2140/ant.2023.17.1453. URL: https://doi.org/10.2140/ant.2023.17.1453. arXiv:2101.00263. URL: https://arxiv.org/abs/2101.00263.
21. Jinbang Yang, Kang Zuo, 'A note on p-adic Simpson correspondence', arXiv:2012.02058. URL: https://arxiv.org/abs/2012.02058.
22. Christopher Deninger, Deepak Kamlesh, 'A remark on the vanishing of Higgs fields in the p-adic Simpson correspondence', arXiv:2508.13685 (2025). URL: https://arxiv.org/abs/2508.13685.
23. Xiangyu Pan, Jiahong Yu, 'Lift-independence problem in the p-adic Simpson correspondence for curves', arXiv:2605.29947 (2026). URL: https://arxiv.org/abs/2605.29947.