Related papers: Serre's conjecture over F_9

This paper gives an expository account of our experiments concerning relations between modular forms for congruence subgroups of SL(3,Z) and three dimensional Galois representations. The main new result presented here is a calculation of…

We define a pro-$p$ Abelian sheaf on a modular curve of a fixed level $N \geq 5$ divisible by a prime number $p \neq 2$. Every $p$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ associated to an eigenform is obtained…

We study the possible weights of an irreducible two-dimensional mod p representation of the absolute Galois group of F which is modular in the sense of that it comes from an automorphic form on a definite quaternion algebra with centre F…

We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These…

Let $F$ be a totally real number field. We prove that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely…

We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $\rho$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a…

We prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l. This extends the results of Clozel, Harris and Taylor, and the…

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action…

Let $K$ be an imaginary quadratic field. Modular forms for GL(2) over $K$ are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over…

This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the…

Let E be a finite extension of Fp. Using Fontaine's theory of (phi,Gamma)-modules, Colmez has shown how to attach to any irreducible E-linear representation of Gal(Qpbar/Qp) an infinite dimensional smooth irreducible E-linear representation…

In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture…

We establish some new cases of Artin's conjecture. Our results apply to Galois representations over $\Q$ with image $S_5$ satisfying certain local hypotheses, the most important of which is that complex conjugation is conjugate to…

We improve the results in a previous article of Dieulefait and Manoharmayum and we deduce some stronger modularity results. In particular we deduce that any rigid Calabi-Yau threefold defined over Q with good reduction at 5 is modular and…

A number field $K$ is Hilbert-Speiser if all of its tame abelian extensions $L/K$ admit NIB (normal integral basis). It is known that $\mathbb{Q}$ is the only such field, but when we restrict $\text{Gal}(L/K)$ to be a given group $G$, the…

We prove a boundedness-theorem for families of abelian varieties with real multiplication. More generally, we study curves in Hilbert modular varieties from the point of view of the Green Griffiths-Lang conjecture claiming that entire…

For a non-CM elliptic curve $E$ defined over $\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $\rho_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to…

This is an exposition of our joint work with Kakde, Silliman, and Wang, in which we prove a version of Ribet's Lemma for $\mathrm{GL}_2$ in the residually indistinguishable case. We suppose we are given a Galois representation taking values…

We prove new cases of Fontaine-Mazur conjecture on two-dimensional Galois representations over Q when the residual representation is reducible. Our approach is via a semi-simple local-global compatibility of the completed cohomology and a…

The Iwasawa $\mu$-invariant of the Selmer group of a residually reducible Galois representation arising from a Hecke eigencuspform is studied. Furthermore, certain Iwasawa-invariants refining the $\mu$-invariant are defined and analyzed. As…