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Automorphy of some residually dihedral Galois representations
(Springer, 2015-05-08)
We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called ‘Taylor–Wiles hypothesis’. We apply this to the problem ...
Equidistribution of Frobenius eigenvalues
(Oxford JournalsInternational Mathematics Research Notices, 2015-02-09)
We study the problem of variation of Frobenius eigenvalues on the cohomology of families of local systems of algebraic curves over finite fields.
Level-raising and symmetric power functoriality, III
(Duke University PressDuke Mathematical Journal, 2016-12-09)
The simplest case of the Langlands functoriality principle asserts the existence of
the symmetric powers Symn of a cuspidal representation of GL.2/ over the adèles of
F , where F is a number field. In 1978, Gelbart and ...
A 2-adic automorphy lifting theorem for unitary groups over CM fields
(SpringerMathematische Zeitschrift, 2016)
We prove a ‘minimal’ type automorphy lifting theorem for 2-adic Galois representations of unitary type, over imaginary CM fields. We use this to improve an automorphy lifting theorem of Kisin for GL_2.
On the rigid cohomology of certain Shimura varieties.
(SpringerResearch in the Mathematical Sciences, 2016)
We construct the compatible system of $\textit{l}$-adic representations associated to a regular algebraic cuspidal automorphic representation of GL$_{n}$ over a CM (or totally real) field and check local-global compatibility ...
On the $\phi$-Selmer groups of the elliptic curves y$^2$ = x$^3$ - Dx
(Cambridge University PressMathematical Proceedings of the Cambridge Philosophical Society, 2016-09-09)
We study the variation of the $\phi$-Selmer groups of the elliptic curves y$^2$ = x$^3$ − Dx under quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when ...
On the GL$_{n}$-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 ...
Potential automorphy and the Leopoldt conjecture
(Johns Hopkins University PressAmerican Journal of Mathematics, 2017-10-01)
We study in this paper Hida's p-adic Hecke algebra for GL_n over a CM field F. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his ...
TORSION GALOIS REPRESENTATIONS OVER CM FIELDS AND HECKE ALGEBRAS IN THE DERIVED CATEGORY
(Cambridge University PressForum of Mathematics, Sigma, 2016-07-21)
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 ...
Arithmetic invariant theory and 2-descent for plane quartic curves
(Mathematical Sciences PublishersAlgebra & Number Theory, 2016-09-27)
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 ...