We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.

/pdf/automorphy-for-some-l-adic-lifts-of-automorphic-mod-l-galois-1lron6w183.pdf

Automorphy for some l-adic lifts of automorphic mod l Galois representations

We prove potential modularity theorems for l-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral j -invariant defined over a totally real field.

https://www.math.ias.edu/~rtaylor/cy2fin.pdf

A Family of Calabi–Yau Varieties and Potential Automorphy II

We prove potential modularity theorems for l-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral j-invariant defined over a totally real field.

A family of Calabi-Yau varieties and potential automorphy

Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms, and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry, and more. This expository article describes their constructions and various applications in pure and applied mathematics.

https://imath.kiev.ua/~kay/pdf/lectures/cordoba/lubotzky.pdf

Expander graphs in pure and applied mathematics

Let $\mathcal{O}$
 
be an orbit in ℤ
 n
 of a finitely generated subgroup Λ of GL
 n
 (ℤ) whose Zariski closure Zcl(Λ) is suitably large (eg isomorphic to SL2) We develop a Brun combinatorial sieve for estimating the number of points on $\mathcal{O}$
 
at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings A fundamental role is played by the expansion property of the “congruence graphs” that we associate with $\mathcal{O}$
 
 This expansion property is established when Zcl(Λ)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free

https://link.springer.com/content/pdf/10.1007%2Fs00222-009-0225-3.pdf

Affine linear sieve, expanders, and sum-product

On subgroups of GL n (F p )

Affine curves in characteristic p are set theoretic complete intersections

H Clemens and C Schoen gave examples of three-folds where the group of codimension two cycles modulo algebraic equivalence has infinite rank. This paper provides yet another example of the same phenomenon.

Madhav V. Nori

Papers

On subgroups of GL n (F p )

Affine curves in characteristic p are set theoretic complete intersections

Cycles on the generic abelian threefold

The fundamental group-scheme of an abelian variety