Related papers: Finite descent obstruction for Hilbert modular var…

Let $G$ be a reductive algebraic group scheme defined over $\mathbb{F}_p$ and let $G_1$ denote the Frobenius kernel of $G$. To each finite-dimensional $G$-module $M$, one can define the support variety $V_{G_1}(M)$, which can be regarded as…

Making use of the recent theory of noncommutative motives, we prove that every additive invariant satisfies Galois descent. Examples include mixed complexes, Hochschild homology, cyclic homology, periodic cyclic homology, negative cyclic…

We provide evidence for this conclusion: given a finite Galois cover $f: X \rightarrow \mathbb{P}^1_\mathbb{Q}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We…

In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre…

Let ${\mathcal L}/{\mathcal K}$ be a finite Galois extension and let $X$ be an affine algebraic variety defined over ${\mathcal L}$. Weil's Galois descent theorem provides necessary and sufficient conditions for $X$ to be definable over…

We prove that the isomorphism problem for group algebras reduces to group algebras over finite extensions of the prime field. In particular, the modular isomorphism problem reduces to finite modular group algebras.

Let $E$ be a modular elliptic curve over a totally real number field $F$. We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to…

Let $A$ be an absolutely simple abelian variety without (potential) complex multiplication, defined over the number field $K$. Suppose that either $\dim A=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound $\ell_0(A,K)$…

We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modularity condition is…

Let $X$ be an algebraic surface with $\mathcal{L}$ an ample line bundle on $X$. Let $\Gamma(X, \mathcal{L})$ be the \emph{geometric monodromy} group associated to family of nonsingular curves in $X$ that are zero loci of sections of…

Let $\mathbb{F}_q$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod $p$ representations of $G_q = \text{GL}_2({\mathbb{F}_q})$ to $G_p = \text{GL}_2(\mathbb{F}_p)$. We first study the restrictions…

The Coleman-Oort conjecture says that for large $g$ there are no positive-dimensional Shimura subvarieties of $\mathsf{A}_g$ generically contained in the Jacobian locus. Counterexamples are known for $g\leq 7$. They can all be constructed…

We study the Hodge standard conjecture for varieties over finite fields admitting a CM lifting, such as abelian varieties or products of K3 surfaces. For those varieties we show that the signature predicted by the conjecture holds true…

Given a smooth, projective curve $Y$, a finite group $G$ and a positive integer $n$ we study smooth, proper families $X\to Y\times S\to S$ of Galois covers of $Y$ with Galois group isomorphic to $G$ branched in $n$ points, parameterized by…

We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such…

We give an effective version of a result reported by Serre asserting that the images of the Galois representations attached to an abelian surface with $\End(A)= \mathbb{Z}$ are as large as possible for almost every prime. Our algorithm…

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-K\"uhn. Our theorem is thus suitable for…

We determine the stable reduction at $p$ of all three point covers of the projective line with Galois group ${\rm SL}_2(p)$. As a special case, we recover the results of Deligne and Rapoport on the reduction of the modular curves $X_0(p)$…

Let $G$ be a finite group. Then there exists a first-order statement $S(G)$ in the language of rings without parameters and depending only on $G$ such that, for any field $K$, we have that $K\models S(G)$ if and only if $K$ has a Galois…

The article provides a sufficient condition for a locally finite module over the absolute Galois group of a finite field F to satisfy the Riemann Hypothesis Analogue with respect to the projective line. The condition holds for all smooth…