Torsion Galois representations over CM fields and Hecke algebras in the derived category

We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras and apply this technique to sharpen recent results of P. Scholze

Equidistribution of frobenius eigenvalues

We study the problem of variation of Frobenius eigenvalues on the cohomology of families of local systems of algebraic curves over finite fields.This is the author accepted manuscript. The final version is available from Oxford University Press via http://dx.doi.org/10.1093/imrn/rnv02

On the GLn_{n}-eigenvariety and a conjecture of Venkatesh

Let π be a cuspidal, cohomological automorphic representation of GL$_{n}$(A). Venkatesh has suggested that there should exist a natural action of the exterior algebra of a certain motivic cohomology group on the π-part of the Betti cohomology (with rational coefficients) of the GL$_{n}$(Q)-arithmetic locally symmetric space. Venkatesh has given evidence for this conjecture by showing that its ‘l-adic realization’ is a consequence of the Taylor–Wiles formalism. We show that its ‘p-adic realization’ is related to the properties of eigenvarieties

TORSION GALOIS REPRESENTATIONS over CM FIELDS and HECKE ALGEBRAS in the DERIVED CATEGORY

We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras and apply this technique to sharpen recent results of P. Scholze.This research was partially conducted during the period that Jack Thorne served as a Clay Research Fellow. James Newton is supported by the ERC Starting grant 306326.This is the final version of the article. It first appeared from Cambridge University Press via http://dx.doi.org/10.1017/fms.2016.1

Automorphy lifting for residually reducible-adic Galois representations, II

We revisit the paper [Automorphy lifting for residually reducible$l$-adic Galois representations, J. Amer. Math. Soc. 28 (2015), 785–870] by the third author. We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.This work was partially conducted during the period that J.T. served as a Clay Research Fellow. J.T.’s work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714405). J.T. would like to thank Lue Pan for pointing out an error in [Tho15], which is addressed here in §3.1 below. P.A. was supported by Simons Foundation Collaboration Grant 527275 and NSF grant DMS- 1902155

Elliptic curves over Q∞_{∞} are modular

We show that if $\textit{p}$ is a prime, then all elliptic curves de ned over the cyclotomic $\mathbb{Z}$$_{p}$-extension of Q are modular.This work was carried out while the author served as a Clay Research Fellow

Automorphy lifting for residually reducible l-adic galois representations

In this paper we prove new automorphy lifting theorems for l-adic Galois representations over CM (by definition, a CM field is an imaginary quadratic extension of a totally real number field)fields satisfying a self-duality hypothesis. The main novelty is that we can prove lifting results for Galois representations which are residually reducible. This paper can therefore be viewed as a sequel to [Tho12], where we treated the residually irreducible case. However, there are a number of serious new obstacle

Arithmetic invariant theory and 2-descent for plane quartic curves

Given a smooth plane quartic curve C over a field $\textit{k}$ of characteristic 0, with Jacobian variety $\textit{J}$, and a marked rational point P $\in$ C($\textit{k}$), we construct a reductive group $\textit{G}$ and a $\textit{G}$-variety $\textit{X}$, together with an injection $\textit{J}$($\textit{k}$)/2$\textit{J}$($\textit{k}$) $\hookrightarrow$ $\textit{G}$($\textit{k}$)\$\textit{X}$($\textit{k}$). We do this using the Mumford theta group of the divisor 2$\Theta$ of $\textit{J}$, and a construction of Lurie which passes from Heisenberg groups to Lie algebras.This research was partially conducted during the period the author served as a Clay Research Fellow [main article]. This research is supported in part by NSF grant DMS-1161489 [appendix].This is the author accepted manuscript. The final version is available from Mathematical Sciences Publishers via https://doi.org/10.2140/ant.2016.10.137

Level-raising and symmetric power functoriality, III

© 2017. The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers Symnof a cuspidal representation of GL(2) over the adèles of F, where F is a number field. In 1978, Gelbart and Jacquet proved the existence of Sym2. After this, progress was slow, eventually leading, through the work of Kim and Shahidi, to the existence of Sym3and Sym4. In this series of articles we revisit this problem using recent progress in the deformation theory of modular Galois representations. As a consequence, our methods apply only to classical modular forms on a totally real number field; the present article proves the existence, in this "classical" case, of Sym6and Sym8