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Sean Howe

Publications and source records attributed to Sean Howe.

35 records · Page 2Linked to original sources
arXiv2022

The conjugate uniformization via 1-motives

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 $p$-adic field. This mirrors a mixed Hodge theory construction of the inverse uniformization map for complex semi-abelian varieties. We also highlight the geometric structure of the target of the conjugate uniformization map, which is an étale cover of a negative Banach--Colmez space in the sense of Fargues--Scholze.

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arXiv2022

Cohomological and motivic inclusion-exclusion

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 spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.

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arXiv2021

Slope classicality via completed cohomology

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 of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of $\mathrm{GL}_2(\mathbb{Q}_p)$. The $U_p$ operator becomes a double-coset, and unitarity yields the slope vanishing.

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arXiv2021

The spectral $p$-adic Jacquet-Langlands correspondence and a question of Serre

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 affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge-Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet-Langlands correspondence, that Galois representations can be attached to classical and $p$-adic quaternionic eigenforms.

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arXiv2021

Zeta statistics and Hadamard functions

We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any "natural" sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.

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arXiv2021

Overconvergent modular forms are highest weight vectors in the Hodge-Tate weight zero part of completed cohomology

We construct a $(\mathfrak{gl}_2, B(\mathbb{Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb{P}^1$, landing in the compactly supported completed $\mathbb{C}_p$-cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest weight vector for any overconvergent modular form of infinitesimal weight not equal to $1$. For classical weight $k\geq 2$, the Verma has an algebraic quotient $H^1(\mathbb{P}^1, \mathcal{O}(-k))$, and on classical forms the pairing factors through this quotient, giving a geometric description of "half" of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $\mathbb{P}^1$. Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan in arXiv:2008.07099, but the perspective here is different and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology.

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arXiv2020

A unipotent circle action on $p$-adic modular forms

Following a suggestion of Peter Scholze, we construct an action of $\hat{\mathbb{G}}_m$ on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the $p$-adic modular curve whose ring of functions is Serre's space of $p$-adic modular functions. This action is a local, $p$-adic analog of a global, archimedean action of the circle group $S^1$ on the lattice-unstable locus of the modular curve over $\mathbb{C}$. To construct the $\hat{\mathbb{G}}_m$-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates $q$; along the way we also prove a natural generalization of Dwork's equation $τ=\log q$ for extensions of $\mathbb{Q}_p/\mathbb{Z}_p$ by $μ_{p^\infty}$ valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of $\hat{\mathbb{G}}_m$ integrates the differential operator $θ$ coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and $p$-adic $L$-functions.

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arXiv2020

Transcendence of the Hodge-Tate filtration

For $C$ a complete algebraically closed extension of $\mathbb{Q}_p$, we show that a one-dimensional $p$-divisible group $G/ \mathcal{O}_C$ can be defined over a complete discretely valued subfield $L \subset C$ with Hodge-Tate period ratios contained in $L$ if and only if $G$ has CM, if and only if the period ratios generate an extension of $\mathbb{Q}_p$ of degree equal to the height of the connected part of $G$. This is a $p$-adic analog of a classical transcendence result of Schneider which states that for $τ$ in the complex upper half plane, $τ$ and $j(τ)$ are simultaneously algebraic over $\mathbb{Q}$ if and only if $τ$ is contained in a quadratic extension of $\mathbb{Q}$. We also briefly discuss a conjectural generalization to shtukas with one paw.

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arXiv2020

Motivic random variables and representation stability II: Hypersurface sections

We prove geometric and cohomological stabilization results for the universal smooth degree $d$ hypersurface section of a fixed smooth projective variety as $d$ goes to infinity. We show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed Grothendieck ring of varieties, and deduce from this the stabilization of the Hodge Euler characteristic of natural families of local systems constructed from the vanishing cohomology. We prove explicit formulas for the stable values using a probabilistic interpretation, along with the natural analogs in point counting over finite fields. We explain how these results provide new geometric examples of a weak version of representation stability for symmetric, symplectic, and orthogonal groups. This interpretation of representation stability was studied in the prequel for configuration spaces.

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arXiv2020

Motivic random variables and representation stability I: Configuration spaces

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain *motivic random variables*, and gives explicit universal formulas for the limits in terms of the exponents of a motivic Euler product for the Kapranov zeta function. The result can be thought of as a weak but explicit version of representation stability for the cohomology of ordered configuration spaces. In the sequel we find similar stability results in spaces of smooth hypersurface sections, providing new examples to be investigated through the lens of representation stability for symmetric, symplectic and orthogonal groups.

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arXiv2019

Motivic Euler products in motivic statistics

We formulate and prove an analog of Poonen's finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil-Wood, who treated the case of smooth hypersurface sections. In fact, our techniques give analogs in motivic statistics of all known results in arithmetic statistics that have been proven using Poonen's sieve, including work of Bucur-Kedlaya on complete intersections and Erman-Wood on semi-ample Bertini theorems. A key ingredient is the use of motivic Euler products, as introduced by the first author, to write down candidate motivic probabilities. We also formulate a conjecture on the uniform convergence of zeta functions that unifies motivic and arithmetic statistics for varieties over finite fields.

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arXiv2015

Asymptotics of conductors of elliptic curves over Q

In this note we study numbers which occur as conductors of elliptic curves over Q. We show, by constructing families of elliptic curves with quadratic discriminant and invoking a theorem of Iwaniec, that this set contains infinitely many almost primes. We show, assuming a strong version of the Cohen-Lenstra heuristics, that the set of prime conductors has an explicitly bounded density in the set of primes. Studying the Cremona and Stein-Watkins databases of elliptic curves we conjecture that the set of conductors should be of density zero in the set of natural numbers and that the set of prime conductors should be of density zero in the set of prime numbers.

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arXiv2014

The Log-Convex Density Conjecture and vertical surface area in warped products

We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such spaces vertical fibers are isoperimetric. Our main hypothesis is that the surface area of a fiber be a convex function of the volume it bounds. We apply our results in the specific case of $\mathbb{R}^{n}-\{0\}$ realized as the warped product $(0,\infty)\times_{r}S^{n-1}$, providing many new examples of densities where spheres about the origin are isoperimetric, including simple densities with finite volume, simple densities that at the origin are neither log-convex nor smooth, and non-simple densities. We also generalize the results of Kolesnikov and Zhdanov on large balls in $\mathbb{R}^{n}$ with increasing strictly log-convex simple density. We situate our work in relation to the Log-Convex Density Conjecture of Rosales et al. and the recent work by Morgan, Ritoré, and others on formulating a generalized log-convex density/stable spheres conjecture.

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arXiv2014

Presentations for quaternionic $S$-unit groups

The purpose of this paper is to give presentations for projective $S$-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field $\mathbb{Q}$. To our knowledge, this provides the first explicit presentations of an $S$-arithmetic lattice in a semisimple Lie group with $S$ large. In particular, we give presentations for groups acting irreducibly and cocompactly on a product of Bruhat--Tits trees. We also include some discussion and experimentation related to the congruence subgroup problem, which is open when $S$ contains at least two odd primes. In the appendix, we provide code that allows the reader to compute presentations for an arbitrary finite set $S$.

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arXiv2010

Isoperimetric problems in sectors with density

We consider the isoperimetric problem in planar sectors with density $r^{p}$, and with density $a>1$ inside the unit disk and $1$ outside. We characterize solutions as a function of sector angle. We also solve the isoperimetric problem in $\mathbb{R}^{n}$ with density $r^{p},\; p<0$.

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arXiv2009

Isoperimetric inequalities for wave fronts and a generalization of Menzin's conjecture for bicycle monodromy on surfaces of constant curvature

We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss "bicycle curves" using the generalized isoperimetric inequalities. The euclidean model of a bicycle is a unit segment AB that can move so that it remains tangent to the trajectory of point A (the rear wheel is fixed on the bicycle frame). We extend this definition to a general Riemannian manifold, and concern ourselves in particular with bicycle curves in the hyperbolic plane H^2 and on the sphere S^2. We prove results along the lines of those proven by Levi and Tabachnikov and resolve both spherical and hyperbolic versions of Menzin's conjecture, which relates the area bounded by a curve to its associated monodromy map.

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