A pp-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.Comment: Comments welcome

The Primitive Comparison Theorem in characteristic pp

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.Comment: Comments welcome

GG-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

Moduli spaces in pp-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.Comment: updated now that sequel has appeared and some small correction

Identifying the greatest team and captain - A complex network approach to cricket matches

We consider all Test matches played between 1877 and 2010 and One Day International (ODI) matches played between 1971 and 2010. We form directed and weighted networks of teams and also of their captains. The success of a team (or captain) is determined by the 'quality' of wins and not on the number of wins alone. We apply the diffusion based PageRank algorithm on the networks to access the importance of wins and rank the teams and captains respectively. Our analysis identifies {\it Australia} as the best team in both forms of cricket $-$ Test and ODI. {\it Steve Waugh} is identified as the best captain in Test cricket and {\it Ricky Ponting} is the best captain in the ODI format. We also compare our ranking scheme with the existing ranking schemes which include the Reliance ICC Ranking. Our method does not depend on `external' criteria in ranking of teams (captains). The purpose of this paper is to introduce a revised ranking of cricket teams and to quantify the success of the captains

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.Comment: Version 3. Minor improvements to abstract and introductio

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

Hodge-Tate stacks and non-abelian pp-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 complete discretely valued extension of $\mathbb{Q}_p$ with $p$-finite residue 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. We deduce new results about the $p$-adic Simpson correspondence in both cases