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Jared Weinstein

Publications and source records attributed to Jared Weinstein.

17 recordsLinked to original sources
arXiv2024

On Hopkins' Picard group

We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$, thereby confirming a prediction made by Hopkins in the early 1990s. In fact, with the exception of the anomalous case $n=p=2$, we provide a full set of topological generators for these groups. Our arguments rely on recent advances in $p$-adic geometry to translate the problem to a computation on Drinfeld's symmetric space, which can then be solved using results of Colmez--Dospinescu--Niziol.

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arXiv2024

On the rationalization of the $K(n)$-local sphere

We compute the rational homotopy groups of the $K(n)$-local sphere for all heights $n$ and all primes $p$, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusion of the Witt vectors into the Lubin-Tate ring induces a split injection on continuous stabilizer cohomology with torsion cokernel of bounded exponent, thereby proving Hopkins' chromatic splitting conjecture and the vanishing conjecture of Beaudry-Goerss-Henn rationally. The key ingredients are the equivalence between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze-Weinstein, integral $p$-adic Hodge theory, and an integral refinement of a theorem of Tate on the Galois cohomology of non-archimedean fields.

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arXiv2022

Higher modularity of elliptic curves over function fields

We investigate a notion of "higher modularity" for elliptic curves over function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we say that $E$ is $r$-modular when there is an algebraic correspondence between a stack of $r$-legged shtukas, and the $r$-fold product of $E$ considered as an elliptic surface. The (known) case $r=1$ is analogous to the notion of modularity for elliptic curves over $\mathbf{Q}$. Our main theorem is that if $E/\mathbf{F}_q(t)$ is a nonisotrivial elliptic curve whose conductor has degree 4, then $E$ is 2-modular. Ultimately, the proof uses properties of K3 surfaces. Along the way we prove a result of independent interest: A K3 surface admits a finite morphism to a Kummer surface attached to a product of elliptic curves if and only if its Picard lattice is rationally isometric to the Picard lattice of such a Kummer surface.

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arXiv2019

The smooth locus in infinite-level Rapoport-Zink spaces

Rapoport-Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let $\mathscr{M}_{\infty}$ be an infinite-level Rapoport-Zink space of EL type, and let $\mathscr{M}_{\infty}^\circ$ be one geometrically connected component of it. We show that $\mathscr{M}_{\infty}^{\circ}$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^\infty)^{\circ}$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

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arXiv2017

On the Kottwitz conjecture for local shtuka spaces

Kottwitz's conjecture describes the contribution of a supercuspidal represention to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze's more general spaces of local shtukas. Using a new Lefschetz-Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that for an irreducible smooth representation of an inner form of $\mathrm{GL}_n$, the $L$-parameter constructed by Fargues-Scholze agrees with the usual semisimplified parameter arising from local Langlands.

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arXiv2014

$\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ as a geometric fundamental group

Let $p$ be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that the absolute Galois group of $\mathbf{Q}_p$ is the \'etale fundamental group of a certain object $Z$ which is defined over an algebraically closed field. Thus, local Galois representations correspond to local systems on $Z$. In brief, $Z$ is a (non-representable) quotient of a perfectoid space. The construction combines two themes: the fundamental curve of $p$-adic Hodge theory (due to Fargues-Fontaine) and the tilting equivalence (due to Scholze).

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arXiv2013

Geometric realization of special cases of local Langlands and Jacquet-Langlands correspondences

Let F be a non-Archimedean local field and let E be an unramified extension of F of degree n>1. To each sufficiently generic multiplicative character of E (the details are explained in the body of the paper) one can associate an irreducible n-dimensional representation of the Weil group W_F of F, which corresponds to an irreducible supercuspidal representation \pi\ of GL_n(F) via the local Langlands correspondence. In turn, via the Jacquet-Langlands correspondence, \pi\ corresponds to an irreducible representation \rho\ of the multiplicative group of the central division algebra over F with invariant 1/n. In this note we give a new geometric construction of the representations \pi\ and \rho, which is simpler than the existing algebraic approaches (in particular, the use of the Weil representation over finite fields is eliminated).

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arXiv2012

Moduli of p-divisible groups

We prove several results about p-divisible groups and Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of p-adic Hodge theory. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings. For this, we give a general classification of p-divisible groups over the ring of integers of a complete algebraically closed field in the spirit of Riemann's classification of complex abelian varieties. Another key ingredient is a full faithfulness result for the Dieudonn\'e module functor for p-divisible groups over semiperfect rings (meaning rings on which the Frobenius map is surjective).

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arXiv2012

Formal vector spaces over a local field of positive characteristic

Let $O$ be the ring of power series in one variable over a finite field, with $K$ its fraction field. We introduce the notion of a "formal $K$-vector space"; this is a certain kind of $K$-vector space object in the category of formal schemes. This concept runs parallel to the established notion of a formal $O$-module, but in many ways formal $K$-vector spaces are much simpler objects. Our main result concerns the Lubin-Tate tower, which plays a vital role in the local Langlands correspondence for $GL_n(K)$. Let $A_m$ be the complete local ring parametrizing deformations of a fixed formal $O$-module over the residue field, together with Drinfeld level $m$ structure. We show that the completion of the union of the $A_m$ has a surprisingly simple description in terms of formal $K$-vector spaces. This description shows that the generic fiber of the Lubin-Tate tower at infinite level carries the structure of a perfectoid space. As an application, we find a family of open neighborhoods of this perfectoid space whose special fibers are certain remarkable varieties over a finite field which we are able to make completely explicit. It is shown in joint work with Mitya Boyarchenko that the $\ell$-adic cohomology of these varieties realizes the local Langlands correspondence for a certain class of supercuspidal representations of $GL_n(K)$.

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arXiv2011

Maximal varieties and the local Langlands correspondence for GL(n)

The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence for GL(n) over a nonarchimedean local field. In this article we make progress towards a purely local proof of this fact. To wit, we find a family of open affinoid subsets of Lubin-Tate space at infinite level, whose cohomology realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degree n extension). A key role is played by a certain variety X, defined over a finite field, which is "maximal" in the sense that the number of rational points of X is the largest possible among varieties with the same Betti numbers as X. The variety X is derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups.

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arXiv2010

Semistable models for modular curves of arbitrary level

We produce an integral model for the modular curve $X(Np^m)$ over the ring of integers of a sufficiently ramified extension of $\mathbf{Z}_p$ whose special fiber is a {\em semistable curve} in the sense that its only singularities are normal crossings. This is done by constructing a semistable covering (in the sense of Coleman) of the supersingular part of $X(Np^m)$, which is a union of copies of a Lubin-Tate curve. In doing so we tie together nonabelian Lubin-Tate theory to the representation-theoretic point of view afforded by Bushnell-Kutzko types. For our analysis it was essential to work with the Lubin-Tate curve not at level $p^m$ but rather at infinite level. We show that the infinite-level Lubin-Tate space (in arbitrary dimension, over an arbitrary nonarchimedean local field) has the structure of a perfectoid space, which is in many ways simpler than the Lubin-Tate spaces of finite level.

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arXiv2010

On the computation of local components of a newform

We present an algorithm for computing the $p$-component of the automorphic representation arising from a cuspidal newform $f$ for a prime $p$. This is equivalent to computing the restriction to the decomposition group at $p$ of the $\ell$-adic Galois representations attached to $f$ for any $\ell\neq p$. The situation is most interesting when $p^2$ divides the level of $f$, in which case the $p$-component could be supercuspidal. In the supercuspidal case, the local component is induced from an irreducible character of a compact-mod-center subgroup of $\text{GL}_2(\mathbf{Q}_p)$; our algorithm outputs both the group and the irreducible character. We provide examples which illustrate how the local Galois representation can be completely read off from the local component.

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arXiv2010

Good reduction of affinoids on the Lubin-Tate tower

We analyze the geometry of the tower of Lubin-Tate deformation spaces, which parametrize deformations of a one-dimensional formal module of height h together with level structure. According to the conjecture of Deligne-Carayol, these spaces realize the local Langlands correspondence in their l-adic cohomology. This conjecture is now a theorem, but currently there is no purely local proof. Working in the equal characteristic case, we find a family of affinoids in the Lubin-Tate tower with good reduction equal to a rather curious nonsingular hypersurface, whose equation we present explicitly. Granting a conjecture on the L-functions of this hypersurface, we find a link between the conjecture of Deligne-Carayol and the theory of Bushnell-Kutzko types, at least for certain class of wildly ramified supercuspidal representations of small conductor.

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arXiv2009

Explicit non-abelian Lubin-Tate theory for GL(2)

Let $F$ be a non-Archimedean local field with residue field $k$ of odd characteristic, and let $B/F$ be the division algebra of rank 4. We explicitly construct a stable curve $\mathfrak{X}$ over the algebraic closure of $k$ admitting an action of $GL_2(F)\times B^\times \times W_F$ which realizes the Jacquet-Langlands correspondence and the local Langlands correspondence in its cohomology.

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arXiv2009

Hilbert modular forms with prescribed ramification

Let $K$ be a totally real field. In this article we present an asymptotic formula for the number of Hilbert modular cusp forms $f$ with given ramification at every place $v$ of $K$. When $v$ is an infinite place, this means specifying the weight of $f$ at $k$, and when $v$ is finite, this means specifying the restriction to inertia of the local Weil-Deligne representation attached to $f$ at $v$. Our formula shows that with essentially finitely many exceptions, the cusp forms of $K$ exhibit every possible sort of ramification behavior, thus generalizing a theorem of Khare and Prasad. From this fact we compute the minimal field over which a modular Jacobian becomes semi-stable.

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arXiv2009

The Local Jacquet-Langlands Correspondence Via Fourier Analysis

Let $F$ be a locally compact non-Archimedean field, and let $B/F$ be a division algebra of dimension 4. The Jacquet-Langlands correspondence provides a bijection between smooth irreducible representations of $B^\times$ of dimension $>1$ and irreducible cuspidal representations of $GL_2(F)$. We present a new construction of this bijection in which the preservation of epsilon factors is automatic.

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