Abstract: In this talk I will introduce relative de Rham, rigid and Hyodo-Kato realizations in the context of rigid analytic motives. In particular, I will show that these have constructibility properties. This is joint work in progress with Federico Binda and Alberto Vezzani.
Abstract: Long ago, Fontaine made precise conjectures relating p-adic étale cohomology of algebraic varieties over $p$-adic fields to their de Rham cohomology. I will report on our conjectures and results (with Wieslawa Niziol, and also Sally Gilles and Zhenghui Li) for $p$-adic analytic varieties.
Abstract: A Birch and Swinnerton-Dyer conjecture for number fields $K/\mathbb{Q}$ would assert that $$\dim V_K = \operatorname{ord}_{s=1/2} \zeta_K(s)$$ for some vector space functorially attached to $K$. Presently there is no natural candidate for the $V_K$'s. However, assuming $V_K$ is of a cohomological nature and assuming a conjecture of Serre on the vanishing order of $\zeta_K(s)$ at $s=1/2$ we show that such functors $K \mapsto V_K$ (with natural extra structures) exist and are all isomorphic. Their common automorphism group is 2-torsion and abelian.
Abstract: Given a reductive group $G$ over $\mathbb Q_p$, the $\ell$-adic local Langlands conjecture seeks a well-behaved finite-to-one map $\Pi(G) \to \Phi(G)$ from the set of irreducible smooth $G(\mathbb Q_p)$-representations to the set of Langlands parameters for $G$. Fargues--Scholze observed that this map should be a shadow of a much more structured equivalence of categories which they attach to certain non-archimedean geometric objects. In joint work with David Hansen we show that Fargues--Scholze's conjectured equivalence is surprisingly close to being proved for many groups, including $\mathrm{GL}_n$ for all $n$.
Abstract: Let $X$ be a smooth proper variety over $\mathbb C_p$, and suppose that it is equipped with a lift to a smooth scheme over the period ring $B_{\mathrm{dR}}^+$. Given a choice of an exponential map on $\mathbb C_p$, we produce an equivalence of categories between vector bundles with a t-connection on that lift, and locally free modules over the period sheaf $\mathbb B_{\mathrm{dR}}^+$ on the pro-etale site of (the analytification of) $X$. This equivalence is a deformation of the $p$-adic Simpson correspondence between Higgs bundles and pro-etale vector bundles on $X$. It may also be viewed as a partial $p$-adic analog of the construction of Simpson, associating a variation of twistor structures to any semi-simple $\C$-local system on a smooth proper variety over $\C$ ('partial' here accounts for the fact that we are only producing a a $t$-connection over a formal thickening of $X$). I hope to explain two perspectives on this result: one relying on the notion of $t$-$D$-modules twisted by the Simpson gerbe, and the other using the transmutation point of view on the categories in question, afforded by the analytic prismatization. This is joint work with Bhargav Bhatt, Ben Heuer, and Vadim Vologodsky.
Abstract: (joint with Anschütz, Le Bras, Rodriguez Camargo) Motivated by work on the $p$-adic Simpson correspondence and the Hodge-Tate stack, as well as possible applications to the locally analytic $p$-adic local Langlands correspondence, we develop a theory of analytic prismatization of rigid-analytic spaces, which roughly speaking yields a family of analytic stacks over the Fargues-Fontaine curve interpolating between analytic de Rham stacks of untilts, and the analytic Hodge-Tate stack. The comparison between v-vector bundles and twisted Higgs bundles can be generalized to a comparison between perfect complexes on the relative Fargues-Fontaine curve, and on the analytic prismatization.
