相关论文: Semistable models for modular curves and power ope…

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…

Given a one-dimensional formal group of height 2, let E be the Morava E-theory spectrum associated to its universal deformation over the Lubin-Tate ring. By computing with moduli spaces of elliptic curves, we give an explicitation for an…

We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of…

We calculate the $K(n-1)$-localized $E_n$ theory for symmetric groups, and deduce a modular interpretation of the total power operation $\psi^p_F$ on $F=L_{K(n-1)}E_n$ in terms of augmented deformations of formal groups and their subgroups.…

This expository paper introduces several ideas in chromatic homotopy theory around Morava's extraordinary E-theories. In particular, we construct various moduli problems closely related to Lubin-Tate deformation theory and study their…

We develop methods for computing the restriction map from the cohomology of the automorphism group of a height $dn$ formal group law (i.e., the height $dn$ Morava stabilizer group) to the cohomology of the automorphism group of an…

Let $E_n$ be Morava $E$-theory of height $n$. Let $R$ be a $p$-adically flat commutative ring spectrum. Then the Tate-valued Frobenius map endows $\pi_0 R$ with the structure of a $\delta$-ring. On the other hand, we may form the…

We calculate an explicit closed formula for the action of the height 2 full Morava stabilizer group on the coefficient ring of height 2 Morava E-theory. In particular, this yields an explicit, surprisingly simple closed formula for the…

We give explicit calculations of the algebraic theory of power operations for a specific Morava E-theory spectrum and its K(1)-localization. These power operations arise from the universal degree-3 isogeny of elliptic curves associated to…

Given an E-infinity ring spectrum R, with motivation from chromatic homotopy theory, we define relative effective Cartier divisors for a spectral Deligne-Mumford stack over Spet(R) and prove that, as a functor from connective R-algebras to…

We investigate a dense subgroup Gamma of the second Morava stabilizer group given by a certain group of quasi-isogenies of a supersingular elliptic curve in characteristic p. The group Gamma acts on the Bruhat-Tits building for GL_2(Q_l)…

Full level-n structures on smooth, complex curves are trivializations of the n-torsion points of their Jacobians. We give an algebraic proof that the etale cohomology of the moduli space of smooth, complex curves of genus at least 2 with…

We calculate the cohomology of the extended Morava stabilizer group of height $n$, with trivial mod $p$ coefficients, for all heights $n$ and all primes $p>>n$. The result is an exterior algebra on $n$ generators. A brief sketch of the…

We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction.…

In this paper, our aim is to find the relations amongst the cohomology classes of Brill-Noether subvarieties of the moduli space of semistable bundles over an elliptic curve. We obtain results similar to the Poincar\'e relations on a…

Given an elliptic curve with supersingular reduction at an odd prime p, Iovita and Pollack have generalised results of Kobayashi to define even and odd Coleman maps at p over Lubin-Tate extensions given by a formal group of height 1. We…

We deform the Ravenel-Wilson computation of the Morava K-homology of Eilenberg-Mac Lane spaces to obtain a similar description of their completed Morava E-homology. This yields both a cohomological description and an interpretation on the…

The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here…

We complete the description of semistable models for modular curves associated with maximal subgroups of $\mathrm{GL}_2 ({\mathbb F}_p )$ (for $p$ any prime, $p>5$). That is, in the new cases of non-split Cartan modular curves and…

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of…