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"Howe, Sean"
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They live
\"A close look at John Carpenter's 1988 classic amalgam of deliberate B-movie, sci-fi, horror, anti-Yuppie agitprop\"--P. [4] of cover.
Book
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, 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.
Journal Article
Slope classicality in higher Coleman theory via highest weight vectors in completed cohomology
We give a proof of the slope classicality theorem in classical and higher Coleman theory for modular curves of 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 GL₂(ℚ
p
). The Up
operator becomes a double coset, and unitarity yields slope vanishing.
Journal Article
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.
Journal Article
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.
Journal Article
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.
Journal Article
Transcendence of the Hodge–Tate filtration
Soit C une extension algébriquement close et complète de ℚ
p
. Nous démontrons qu’un groupe p-divisible G/𝓞C
de dimension 1 est défini sur un sous-corps L ⊂ C complet pour une valuation discrète et contenant les ratios entre les périodes de Hodge–Tate si et seulement si G est de type CM et si et seulement si les ratios entre les périodes engendrent une extension de ℚ
p
de degré égal à la hauteur de la composante connexe neutre de G. C’est un analogue p-adique du résultat classique de transcendance de Schneider qui dit que, pour τ dans le demi-plan complexe supérieur, τ et j(τ) sont tous les deux algèbriques sur ℚ si et seulement si τ appartient à une extension quadratique de ℚ. Nous discutons aussi brièvement d’une conjecture qui généralise ce résultat aux shtukas à une patte.
For C a complete algebraically closed extension of ℚ
p
, we show that a one-dimensional p-divisible group G/𝓞C
can be defined over a complete discretely valued subfield L ⊂ 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 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 ℚ if and only if τ is contained in a quadratic extension of ℚ. We also briefly discuss a conjectural generalization to shtukas with one paw.
Journal Article
The relativistic \\(p\\)-adic sunscreen conjecture
We formulate a conjecture about intersections between the Banach-Colmez space \\(BC(1/2)\\) and germs of smooth rigid analytic curves at the origin in \\(A^2_C_p\\)
Paper