arXiv subjects

Sean Howe

Publications and source records attributed to Sean Howe.

At least 19 recordsLinked to original sources

arXiv2026

Matrix group $\Lambda$-distributions

The $\Lambda$-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field $L$-functions. It is encoded by the $\sigma$-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the $\sigma$-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic $\Lambda$-distributions for the infinite families of Weyl reflection groups of types $B_n/C_n$ and $D_n$, complementing the previously known case of reflection groups of type $A_n$, i.e., symmetric groups. We also establish a general result relating the shapes of $\sigma$-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.

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arXiv2026

$p$-adic Maass--Shimura operators on $\mu$-ordinary Igusa varieties

For Shimura varieties of Hodge type, we optimally extend algebraic Maass--Shimura differential operators on $p$-integral nearly holomorphic automorphic forms to differential operators on $\mu$-ordinary Mantovan Igusa varieties. We then show that the rank one operators can be integrated to an action of an explicit formal group. Via $p$-adic Fourier theory, this provides a $p$-adic interpolation by extending the action of a symmetric algebra of differential operators to the algebra of functions on the Tate module of the dual $p$-divisible group. Passing to the generic fiber, we obtain an action of an explicit algebra of $p$-adic locally analytic functions, and we show that the action of the subalgebra of locally constant functions is equivalent to a natural Hecke action and thus preserves classical forms. In the ordinary case, we further show that the locally analytic action extends to nearly overconvergent automorphic forms. Our results extend, clarify, and recover prior constructions.

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arXiv2026

The relativistic $p$-adic sunscreen conjecture

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 $\mathbb{A}^2_{\mathbb{C}_p}$

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arXiv2026

Admissible pairs and $p$-adic Hodge structures III: Variation and unlikely intersection

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 particular, the families of $p$-adic Hodge structures that arise from the cohomology of certain smooth proper families. Such a variation gives rise to a cover with a global Hodge period map, and we study this cover and its period map from a differential perspective both classically and via the theory of inscription. This study is motivated by our transcendence results in Parts I and II and analogies with complex bi-algebraic geometry, and we also extend these ideas in other directions: First, we study the locus of special points on local Shimura varieties. We establish a refined version of a prediction of Rapoport-Viehmann on the density of special points, but give a robust counter-example to the local analog of the Andr\'{e}-Oort conjecture in this setting. This shuts down a broader theory of unlikely intersection but leaves open the possibility of stronger geometric transcendence results than the bi-analytic Ax--Lindemann theorem of Part II. Using the Banach-Colmez Tangent Bundles for infinite level local Shimura varieties that arise from the theory of inscription, we define precise notions of generic and exceptional intersections and then formulate an Ax--Schanuel conjecture that we expect to refine our Ax--Lindemann theorem.

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arXiv2025

The de Jong fundamental group of $\mathbb{P}^1_C$ depends on $C$ and is not always topologically countably generated

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 generated. The result and proofs generalize to any connected rigid analytic variety with a non-constant map to $\mathbb{P}^n_C$.

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arXiv2025

The de Jong fundamental group of a non-trivial abelian variety is non-abelian

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 fundamental group of any connected rigid analytic variety over $C$ admitting a non-constant map to $\mathbb{P}^n_C$ (e.g., an abelian variety) depends on $C$ and is big.

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arXiv2025

The geometric Sen morphism is the unique lift of the Kodaira--Spencer morphism

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 integrable connection. This extends previous computations in the minuscule case, and implies that the geometric Sen morphism is the derivative of the lattice Hodge period map. The computation applies, in particular, to non-minuscule period domains generalizing local Shimura varieties, furnishing new examples of towers satisfying He's stalkwise perfectoidness.

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arXiv2025

Inscription, twistors, and $p$-adic periods

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 of $p$-adic periods. In particular, we construct natural inscribed versions of the Hodge and Hodge-Tate period maps and their lattice refinements for de Rham torsors, then compute the derivatives of these period maps in terms of classical structures in $p$-adic Hodge theory. These torsors include infinite level global Shimura varieties and infinite level local Shimura varieties, and for these spaces we also give another moduli-theoretic construction of the inscribed structure; the construction in the local Shimura case applies more generally to the non-minuscule moduli of mixed characterisic local shtukas with one leg. The key new ingredients in our study of inscribed structures on $p$-adic Lie group torsors over smooth rigid varieties over a $p$-adic field are the Liu-Zhu period map, a refinement of the Hodge period map whose derivative is the geometric Sen morphism/canonical Higgs field, and a closely related exact tensor functor from $\mathbb{Q}_p$-local systems to a category of twistor bundles on the relative thickened Fargues-Fontaine curve. These new structures are only visible after passing to the inscribed setting. We also discuss some possible implications of our computations in the vein of ``differential topology for diamonds."

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arXiv2025

A cohomological smoothness conjecture for moduli of mixed characteristic local shtukas with one leg

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 morphism for the underlying diamond is cohomologically smooth over this locus and, applying the Fargues--Scholze Jacobian criterion, we prove this conjecture in the case of EL infinite level Rapoport--Zink spaces, generalizing a result of Ivanov--Weinstein in the basic case.

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arXiv2025

$p$-adic Fourier theory in families

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 formulated as isomorphisms of solid Hopf algebras over arbitrary small v-stacks, and generalize earlier constructions of Amice and Schneider--Teitelbaum. We also construct compatible integral Fourier transforms for $p$-divisible groups and their dual Tate modules. As an application, we use the Weierstrass $\wp$-function to construct a global Eisenstein measure over the $p$-adic modular curve, extending previous constructions of Katz over the ordinary locus and at CM points, and show its generic fiber, the global Eisenstein distribution, gives rise to new families of quaternionic modular forms that overconverge from profinite sets in the rigid analytic supersingular locus.

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arXiv2025

The completed Kirillov model and local-global compatibility for functions on Igusa varieties

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 of Hida's ordinary $p$-adic modular forms with the coinvariants of an action of $\tilde{\mu}_{p^\infty}$ on $\mathbb{V}_b^{\mathrm{cusp}}$. As a consequence of these results, we establish a weak local-global compatibility theorem for eigenspaces in $\mathbb{V}_b^{\mathrm{cusp}}$ associated to classical cuspidal modular forms. Based on these results, we conjecture an analog of Hida theory and an associated local-global compatibility for functions on more general Caraiani-Scholze Igusa varieties, which are natural spaces of $p$-adic automorphic forms.

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arXiv2025

Equidistribution and arithmetic $\Lambda$-distributions

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 of certain infinite sums of random variables can be computed as motivic Euler products. Combining this result with earlier generalizations of Poonen's sieve, we compute the asymptotic $\Lambda$-distributions for several natural families of function field $L$-functions and zeta functions.

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arXiv2025

Transitivity of the $\mathbb{B}^+_\mathrm{dR}$-loop group action on Schubert cells

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 topology on affinoid perfectoids, generalizing a result in the reductive case due to Fargues and Scholze.

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arXiv2025

The negative $\sigma$-moment generating function

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 symmetric power series. This isometric involution is natural as it preserves the pre-$\lambda$ ring structure on symmetric power series with pre-$\lambda$ coefficients, thus this formula provides a simple description of the $\sigma$-moment generating function of $-X$ whenever the $\sigma$-moment generating function of $X$ has a simple description using the pre-$\lambda$ structure. As an application we compute, in a natural range, the dimensions of orthogonal and symplectic group invariants in tensor products of exterior powers of their standard representations on $\mathbb{C}^n$. We also compute a generating function for stable traces of Frobenius related to the moment conjecture for prime-order function field Dirichlet characters.

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arXiv2025

Characterizing perfectoid covers of abelian varieties

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 torsors over an abelian variety, and combine this with our characterization to prove a conjecture of Rodr\'{i}guez Camargo on perfectoidness of $p$-adic Lie torsors in this case. We obtain complementary results for covers of semi-abeloid varieties, $p$-divisible rigid analytic groups, and varieties with globally generated 1-forms. Our proof of perfectoidness for covers of abelian varieties is based on results of Scholze on the canonical subgroup and holds for an arbitrary abelian variety over an algebraically closed non-archimedean extension of $\mathbb{Q}_p$. In an appendix authored by Tongmu He, an alternate proof is presented in the case of abelian varieties that can be defined over a discretely valued subfield by combining our computation of the geometric Sen morphism with previous pointwise perfectoidness and purity of perfectoidness results of He.

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arXiv2024

Random matrix statistics and zeroes of $L$-functions via probability in $\lambda$-rings

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 $\sigma$-moment generating function based on the plethystic exponential, which allows us to describe distributions and argue with independence in a way that is as simple as classical probability theory. As a first application, we use this theory to obtain a concise description of the asymptotic $\sigma$-moment generating functions describing distributions of eigenvalues of Haar random matrices in compact classical groups. Beyond the theory of probability in $\lambda$-rings, the proof uses only classical invariant theory. Using our description we reprove the results of Diaconis and Shahshahani on the joint distributions of traces of powers of matrices, and we also treat symmetric groups. Next, we use Poonen's sieve to establish equidistribution results for the zeroes of $L$-functions in some natural families: simple Dirichlet characters for $\mathbb{F}_q(x)$ and the vanishing cohomology of smooth hypersurface sections. We give concise descriptions of the asymptotic $\sigma$-moment generating functions in these families, then compare them to the associated random matrix distributions. These equidistribution results are sideways in that we fix $q$ and take the degree $d$ to infinity, as opposed to Deligne equidistribution for fixed $d$ as $q \rightarrow \infty$, and the large $d$-limits are related to explicit descriptions of stable homology with twisted coefficients.

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arXiv2023

Admissible pairs and $p$-adic Hodge structures II: The bi-analytic Ax-Lindemann theorem

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 basic case, rigid analytic subvarieties for the two distinct analytic structures induced by the Hodge and Hodge-Tate period maps and their lattice refinements. The theorem implies, in particular, that the only bi-analytic subdiamonds are special subvarieties, generalizing the bi-analytic characterization of special points given in Part I. These results suggest that there is a purely local, $p$-adic theory of bi-analytic geometry that runs in parallel to the global, archimedean theory of bi-algebraic geometry arising in the study of unlikely intersection and functional transcendence for Shimura varieties and more general period domains for variations of Hodge structure.

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arXiv2023

Admissible pairs and $p$-adic Hodge structures I: Transcendence of the de Rham lattice

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 structures in complex geometry. In this setting the filtrations of classical Hodge theory must be enriched to lattices over a complete discrete valuation ring, Fontaine's integral de Rham period ring $B^+_\mathrm{dR}$, and a pure $p$-adic Hodge structure is then a $\mathbb{Q}_p$-vector space equipped with a $B^+_\mathrm{dR}$-lattice satisfying a natural condition analogous to the transversality of the complex Hodge filtration with its conjugate. We show $p$-adic Hodge structures are equivalent to a full subcategory of basic objects in the category of admissible pairs, a toy category of cohomological motives over $C$ that is equivalent to the isogeny category of rigidified Breuil-Kisin-Fargues modules and closely related to Fontaine's $p$-adic Hodge theory over $p$-adic subfields. As an application, we characterize basic admissible pairs with complex multiplication in terms of the transcendence of $p$-adic periods. This generalizes an earlier result for one-dimensional formal groups and is an unconditional, local, $p$-adic analog of a global, archimedean characterization of CM motives over $\mathbb{C}$ conditional on the standard conjectures, the Hodge conjecture, and the Grothendieck period conjecture (known unconditionally for abelian varieties by work Cohen and Shiga and Wolfart).

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