Sean Howe

For an algebraically closed non-archimedean extension $C/\mathbb{Q}_p$, we define a Tannakian category of $p$-adic Hodge structures over $C$ that is a local, $p$-adic analog of the global, archimedean category of $\mathbb{Q}$-Hodge…

We formulate a conjecture about intersections between the Banach-Colmez space $\mathrm{BC}(1/2)$ and germs of smooth rigid analytic curves at the origin in $\mathbb{A}^2_{\mathbb{C}_p}$

We extend the relative theory of admissible pairs and $p$-adic Hodge structures introduced in Part II to allow variation in the underlying local systems of $\mathbb{Q}_p$-vector spaces and isocrystals. This extension accommodates, in…

We introduce the theory of inscribed $v$-sheaves, a differentiable extension of the theory of diamonds and $v$-sheaves with internal tangent bundles that are often relative inscribed Banach-Colmez spaces, then apply this theory to the study…

For $G$ a connected linear algebraic group over a $p$-adic field, we show that the action of $G(\mathbb{B}^+_{\mathrm{dR}})$ on each Schubert cell in the $\mathbb{B}_{\mathrm{dR}}^+$-affine Grassmannian is transitive in the \'{e}tale…

We show that the de Jong fundamental group of any non-trivial abelian variety over a complete algebraically closed extension $C/\mathbb{Q}_p$ is non-abelian. Generalizing an argument for $\mathbb{P}^1_C$, we also show that the de Jong…

For $C/\mathbb{Q}_p$ complete and algebraically closed, we show that the de Jong fundamental group $\pi_{1,\mathrm{dJ}}(\mathbb{P}^1_C)$ depends on $C$ and, if $C$ has cardinality $>2^{\mathbb{N}}$, that it is not topologically countably…

We show that the geometric Sen morphism of a de Rham torsor over a smooth rigid analytic variety over a $p$-adic field is the unique lift, along a natural map, of the Kodaira--Spencer morphism of the associated filtered torsor with…

We give a simple geometric characterization of the locus where the inscribed Banach--Colmez Tangent Spaces of moduli of mixed characteristic local shtukas with one leg and fixed determinant are connected. We conjecture that the structure…

We construct Fourier transforms relating functions and distributions on finite height $p$-divisible rigid analytic groups and objects in a dual category of $\mathbb{Z}_p$-local systems with analyticity conditions. Our Fourier transforms are…

We describe the cuspidal functions $\mathbb{V}_b^{\mathrm{cusp}}$ on the ordinary Caraiani-Scholze Igusa variety for $\mathrm{GL}_2$ as a completion of the smooth Kirillov model for classical cuspidal modular forms, and identify a variant…

For $X$ a pre-$\lambda$ random variable, we show the $\sigma$-moment generating function of $-X$ can be obtained from the $\sigma$-moment generating function of $X$ by applying the composition of the standard and degree flip involutions on…

We introduce a theory of probability in $\lambda$-rings designed to efficiently describe random variables valued in multisets of complex numbers, varieties over a field, or other similar enriched settings. A key role is played by the…

We formulate an abstract notion of equidistribution for families of $\lambda$-probability spaces parameterized by admissible $\mathbb{Z}$-sets. Under the assumption of equidistribution, we show that the $\sigma$-moment generating functions…

We reinterpret and generalize the construction of local Shimura varieties and their non-minuscule analogs by viewing them as moduli spaces of admissible pairs. Our main application is a bi-analytic Ax-Lindemann theorem comparing, in the…

We give a simple characterization of all perfectoid profinite \'{e}tale covers of abelian varieties in terms of the Hodge-Tate filtration on the $p$-adic Tate module. We also compute the geometric Sen morphism for all profinite $p$-adic Lie…

We use the $p$-divisible group attached to a 1-motive to generalize the conjugate $p$-adic uniformization of Iovita--Morrow--Zaharescu to arbitrary $p$-adic formal semi-abelian schemes or $p$-divisible groups over the ring of integers in a…

We categorify the inclusion-exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two…

We give a new proof of the slope classicality theorem in classical and higher Coleman theory for modular curves at arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding…

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an…