arXiv subjects

Ben Heuer

Publications and source records attributed to Ben Heuer.

At least 19 recordsLinked to original sources

arXiv2024

$p$-adic non-abelian Hodge theory for curves via moduli stacks

For a smooth projective curve $X$ over $\mathbb C_p$ and any reductive group $G$, we show that the moduli stack of $G$-Higgs bundles on $X$ is a twist of the moduli stack of v-topological $G$-bundles on $X_v$ in a canonical way. We explain how a choice of an exponential trivialises this twist on points. This yields a geometrisation of Faltings' $p$-adic Simpson correspondence for $X$, which we recover as a homeomorphism between the points of moduli spaces. We also show that our twisted isomorphism sends the stack of $p$-adic representations of $\pi_1(X)$ to an open substack of the stack of semi-stable Higgs bundles of degree $0$.

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arXiv2024

The relative Hodge-Tate spectral sequence for rigid analytic spaces

We construct a relative Hodge-Tate spectral sequence for any smooth proper morphism of rigid analytic spaces over a perfectoid field extension of $\mathbb Q_p$. To this end, we generalise Scholze's strategy in the absolute case by using smoothoid adic spaces. As our main additional ingredient, we prove a perfectoid version of Grothendieck's "cohomology and base-change". We also use this to prove local constancy of Hodge numbers in the rigid analytic setting, and deduce that the relative Hodge-Tate spectral sequence degenerates.

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arXiv2024

The Primitive Comparison Theorem in characteristic $p$

We prove an analogue of Scholze's Primitive Comparison Theorem for proper rigid spaces over an algebraically closed non-archimedean field $K$ of characteristic $p$. This implies a v-topological version of the Primitive Comparison Theorem for proper finite type morphisms $f:X\to Y$ of analytic adic spaces over $\mathbb Z_p$. We deduce new cases of the Proper Base Change Theorem for $p$-torsion coefficients and the K\"unneth formula in this setting.

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arXiv2023

The small $p$-adic Simpson correspondence in terms of moduli spaces

For any rigid space over a perfectoid extension of $\mathbb Q_p$ that admits a liftable smooth formal model, we construct an isomorphism between the moduli stacks of Hitchin-small Higgs bundles and Hitchin-small v-vector bundles. This constitutes a moduli-theoretic improvement of the small $p$-adic Simpson correspondence of Faltings, Abbes-Gros, Tsuji and Wang. Our construction is based on the Hodge-Tate stack of Bhatt-Lurie. We also prove an analogous correspondence in the arithmetic setting of rigid spaces of good reduction over $p$-adic fields.

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arXiv2023

$p$-adic Simpson correspondences for principal bundles in abelian settings

We explore generalizations of the $p$-adic Simpson correspondence on smooth proper rigid spaces to principal bundles under rigid group varieties $G$. For commutative $G$, we prove that such a correspondence exists if and only if the Lie group logarithm is surjective. Second, we treat the case of general $G$ when $X$ is itself an ordinary abelian variety, in which case we prove a generalisation of Faltings' ``small'' correspondence to general rigid groups. On abeloid varieties, we also prove an analog of the classical Corlette-Simpson correspondence for principal bundles under linear algebraic groups.

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arXiv2023

A $p$-adic Simpson correspondence for smooth proper rigid varieties

For any smooth proper rigid analytic space $X$ over a complete algebraically closed extension of $\mathbb Q_p$, we construct a $p$-adic Simpson correspondence: an equivalence of categories between vector bundles on Scholze's pro-\'etale site of $X$ and Higgs bundles on $X$. This generalises a result of Faltings from smooth projective curves to any higher dimension, and further to the rigid analytic setup. The strategy is new, and is based on the study of rigid analytic moduli spaces of pro-\'etale invertible sheaves on spectral varieties.

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arXiv2023

Hodge-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces

Let $X$ be a quasi-compact quasi-separated $p$-adic formal scheme that is smooth either over a perfectoid $\mathbb{Z}_p$-algebra or over some ring of integers of a $p$-adic field. We construct a fully faithful functor from perfect complexes on the Hodge-Tate stack of $X$ up to isogeny to perfect complexes on the v-site of the generic fibre of $X$. Moreover, we describe perfect complexes on the Hodge-Tate stack in terms of certain derived categories of Higgs, resp. Higgs-Sen modules. This leads to a derived $p$-adic Simpson functor.

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arXiv2022

A geometric $p$-adic Simpson correspondence in rank one

For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb Q_p$ we give a geometrisation of the $p$-adic Simpson correspondence of rank one in terms of analytic moduli spaces: The $p$-adic character variety is canonically an étale twist of the moduli space of topological torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$-line bundles: We develop a theory of topological torsion subsheaves of $v$-sheaves and apply this to the diamantine $v$-Picard functor of arXiv:2103.16557. As an application of this geometric correspondence, we study a major open question in $p$-adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles will correspond to continuous representations under the $p$-adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $π^{\acute{e}t}_1(X)\to K^\times$ in terms of moduli spaces: For projective $X$ over $K=\mathbb C_p$, it is given by Higgs line bundles with vanishing Chern classes like in complex geometry, but in general we show that the correct condition is the strictly stronger assumption that the underlying line bundle is a topological torsion element in the topological group $\mathrm{Pic}(X)$.

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arXiv2022

v-vector bundles on $p$-adic fields and Sen theory via the Hodge-Tate stack

We describe the category of continuous semilinear representations and their cohomology for the Galois group of a $p$-adic field $K$ with coefficients in a completed algebraic closure via vector bundles on the Hodge-Tate locus of the Cartier-Witt stack. This also gives a new perspective on classical Sen theory; for example it explains the appearance of an analogue of Colmez' period ring $B_{\mathrm{Sen}}$ in a geometric way.

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arXiv2022

Moduli spaces in $p$-adic non-abelian Hodge theory

We propose a new moduli-theoretic approach to the $p$-adic Simpson correspondence for a smooth proper rigid space $X$ over $\mathbb C_p$ with coefficients in any rigid analytic group $G$, in terms of a comparison of moduli stacks. For its formulation, we introduce the class of "smoothoid spaces" which are perfectoid families of smooth rigid spaces, well-suited for studying relative $p$-adic Hodge theory. For any smoothoid space $Y$, we then construct a "sheafified non-abelian Hodge correspondence", namely a canonical isomorphism \[R^1\nu_{\ast}G\xrightarrow{\sim} \mathrm{Higgs}_G\] where $\nu:Y_{v}\to Y_{et}$ is the natural morphism of sites, and where $\mathrm{Higgs}_G$ is the sheaf of isomorphism classes of $G$-Higgs bundles on $Y_{et}$. We also prove a generalisation of Faltings' local $p$-adic Simpson correspondence to $G$-bundles and to perfectoid families. We apply these results to deduce $v$-descent criteria for \'etale $G$-bundles which show that $G$-Higgs bundles on $X$ form a small $v$-stack $\mathscr Higgs_G$. As a second application, we construct an analogue of the Hitchin morphism on the Betti side: a morphism $\mathscr Bun_{G,v}\to \mathcal A_G$ from the small $v$-stack of $v$-topological $G$-bundles on $X$ to the Hitchin base. This allows us to give a conjectural reformulation of the $p$-adic Simpson correspondence for $X$ in a more geometric and more canonical way, namely in terms of a comparison of Hitchin morphisms.

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arXiv2022

$G$-torsors on perfectoid spaces

For any rigid analytic group variety $G$ over a non-archimedean field $K$ over $\mathbb Q_p$, we study $G$-torsors on adic spaces over $K$ in the $v$-topology. Our main result is that on perfectoid spaces, $G$-torsors in the \'etale and $v$-topology are equivalent. This generalises the known cases of $G=\mathbb G_a$ and $G=\mathrm{GL}_n$ due to Scholze and Kedlaya--Liu. On a general adic space $X$ over $K$, where there can be more $v$-topological $G$-torsors than \'etale ones, we show that for any open subgroup $U\subseteq G$, any $G$-torsor on $X_v$ admits a reduction of structure group to $U$ \'etale-locally on $X$. This has applications in the context of the $p$-adic Simpson correspondence: For example, we use it to show that on any adic space, generalised $\mathbb Q_p$-representations are equivalent to $v$-vector bundles.

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arXiv2022

The $p$-adic Corlette-Simpson correspondence for abeloids

For an abeloid variety $A$ over a complete algebraically closed field extension $K$ of $\mathbb Q_p$, we construct a $p$-adic Corlette-Simpson correspondence, namely an equivalence between finite-dimensional continuous $K$-linear representations of the Tate module and a certain subcategory of the Higgs bundles on $A$. To do so, our central object of study is the category of vector bundles for the $v$-topology on the diamond associated to $A$. We prove that any pro-finite-étale $v$-vector bundle can be built from pro-finite-étale $v$-line bundles and unipotent $v$-bundles. To describe the latter, we extend the theory of universal vector extensions to the $v$-topology and use this to generalise a result of Brion by relating unipotent $v$-bundles on abeloids to representations of vector groups.

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arXiv2021

Pro-étale uniformisation of abelian varieties

For an abelian variety $A$ over an algebraically closed non-archimedean field $K$ of residue characteristic $p$, we show that the isomorphism class of the pro-étale perfectoid cover $\widetilde A=\varprojlim_{[p]}A$ is locally constant as $A$ varies $p$-adically in the moduli space. This gives rise to a pro-étale uniformisation of abelian varieties as diamonds \[A^\diamond=\widetilde A/T_pA\] that works uniformly for all $A$ without any assumptions on the reduction of $A$. More generally, we determine all morphisms between pro-finite-étale covers of abeloid varieties. For example, over $\mathbb C_p$, all abeloids can be uniformised in terms of universal covers that only depend on the isogeny class of the semi-stable reduction over $\bar{\mathbb F}_p$.

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arXiv2021

Line bundles on perfectoid covers: case of good reduction

We study Picard groups and Picard functors of perfectoid spaces which are limits of rigid spaces. For sufficiently large covers that are limits of rigid spaces of good reduction, we show that the Picard functor can be represented by the special fibre. We use our results to answer several open questions about Picard groups of perfectoid spaces from the literature, for example we show that these are not always $p$-divisible. Along the way, we construct a "Hodge--Tate spectral sequence for $\mathbb G_m$" of independent interest.

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arXiv2021

Overconvergent Hilbert modular forms via perfectoid modular varieties

We give a new construction of $p$-adic overconvergent Hilbert modular forms by using Scholze's perfectoid Shimura varieties at infinite level and the Hodge--Tate period map. The definition is analytic, closely resembling that of complex Hilbert modular forms as holomorphic functions satisfying a transformation property under congruence subgroups. As a special case, we first revisit the case of elliptic modular forms, extending recent work of Chojecki, Hansen and Johansson. We then construct sheaves of geometric Hilbert modular forms, as well as subsheaves of integral modular forms, and vary our definitions in $p$-adic families. We show that the resulting spaces are isomorphic as Hecke modules to earlier constructions of Andreatta, Iovita and Pilloni. Finally, we give a new direct construction of sheaves of arithmetic Hilbert modular forms, and compare this to the construction via descent from the geometric case.

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arXiv2021

Line bundles on rigid spaces in the $v$-topology

For a smooth rigid space $X$ over a perfectoid field extension $K$ of $\mathbb Q_p$, we investigate how the $v$-Picard group of the associated diamond $X^\diamondsuit$ differs from the analytic Picard group of $X$. To this end, we construct a left-exact "Hodge--Tate logarithm" sequence \[0\to \mathrm{Pic}_{\mathrm{an}}(X)\to \mathrm{Pic}_v(X^\diamondsuit)\to H^0(X,Ω_X^1)\{-1\}.\] We deduce some analyticity criteria which have applications to $p$-adic modular forms. For algebraically closed $K$, we show that the sequence is also right-exact if $X$ is proper or one-dimensional. In contrast, we show that for the affine space $\mathbb A^n$, the image of the Hodge--Tate logarithm consists precisely of the closed differentials. It follows that up to a splitting, $v$-line bundles may be interpreted as Higgs bundles. For proper $X$, we use this to construct the $p$-adic Simpson correspondence of rank one.

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arXiv2021

Diamantine Picard functors of rigid spaces

For a connected smooth proper rigid space $X$ over a perfectoid field extension of $\mathbb Q_p$, we show that the \'etale Picard functor of $X$ defined on perfectoid test objects is the diamondification of the rigid analytic Picard functor. In particular, it is represented by a rigid analytic group variety if and only if the rigid analytic Picard functor is. Second, we study the $v$-Picard functor that parametrises line bundles in the finer $v$-topology on the diamond associated to $X$ and relate this to the rigid analytic Picard functor by a geometrisation of the multiplicative Hodge--Tate sequence. The motivation is an application to the $p$-adic Simpson correspondence, namely our results pave the way towards the first instance of a new moduli theoretic perspective.

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arXiv2020

Perfectoid covers of abelian varieties

For an abelian variety $A$ over an algebraically closed non-archimedean field of residue characteristic $p$, we show that there exists a perfectoid space which is the tilde-limit of $\varprojlim_{[p]}A$. Our proof also works for the larger class of abeloid varieties.

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