On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields

Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation \gal(\bar{K}/K) \to \gl_2(\ff_\ell). A famous theorem of Serre states that as long as $E$ has no Complex Multiplication (CM), the map \gal(\bar{K}/K) \to \gl_2(\ff_\ell) is surjective for all but finitely many $\ell$. We say that a prime number $\ell$ is exceptional (relative to the pair $(E,K)$) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of $E$. We show in particular that conditionally on the Generalized Riemann Hypothesis (GRH), the largest exceptional prime of an elliptic curve $E$ without CM is no larger than a constant (depending on $K$) times $\log N_E$, where $N_E$ is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre

A bound on the Hodge filtration of the de Rham cohomology of supervarieties

We study the relation between the Hodge filtration of the de Rham cohomology of a proper smooth supervariety $X$ and the usual Hodge filtration of the corresponding reduced variety $X_0$.Comment: 7 page

The Deligne-Mumford operad as a trivialization of the circle action

We prove that the tree-like Deligne-Mumford operad is a homotopical model for the trivialization of the circle in the higher-genus framed little discs operad. Our proof is based on a geometric argument involving nodal annuli. We use Riemann surfaces with analytically parametrized boundary as a model for higher-genus framed little discs.Comment: 52 page

Mirror symmetry and the K theory of a p-adic group

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.Cataloged from PDF version of thesis.Includes bibliographical references (pages 59-61).Let G be a split, semisimple p-adic group. We construct a derived localization functor Loc : ... from the compactified category of [BK2] associated to G to the category of equivariant sheaves on the Bruhat-Tits building whose stalks have finite-multiplicity isotypic components as representations of the stabilizer. Our construction is motivated by the "coherent-constructible correspondence" functor in toric mirror symmetry and a construction of [CCC]. We show that Loc has a number of useful properties, including the fact that the sections ... compactifying the finitely-generated representation V. We also construct a depth </= e "truncated" analogue Loc(e) which has finite-dimensional stalks, and satisfies the property RIP ... V of depth </= e. We deduce that every finitely-generated representation of G has a bounded resolution by representations induced from finite-dimensional representations of compact open subgroups, and use this to write down a set of generators for the K-theory of G in terms of K-theory of its parahoric subgroups.by Dmitry A. Vaintrob.Ph. D