Related papers: Compatibility between base change and Hecke orbits…

Let $C$ be a complex algebraic curve uniformised by a Fuchsian group $\Gamma$. In the first part of this paper we identify the automorphism group of the solenoid associated with $\Gamma$ with the Belyaev completion of its commensurator…

Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke characters for…

Let G be a finite group and V a finite-dimensional rational G-representation. We ask whether there exists a finite Galois extension L/K of number fields with Galois group G, an elliptic curve E/K, and a G-submodule of E(L) tensor Q…

In this paper, we determine mod $2$ Galois representations $\bar{\rho}_{\psi,2}:G_K:={\rm Gal}(\bar{K}/K)\longrightarrow {\rm GSp}_4(\mathbb{F}_2)$ associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic…

We study the action of the groups $H(\lambda)$ generated by the linear fractional transformations $x:z\mapsto -\frac{1}{z} \text{ and }w:z\mapsto z+\lambda$, where $\lambda$ is a positive integer, on the subsets $\mathbb…

Let $L/K$ be a Galois extension of local fields of characteristic $0$ with Galois group $G$. If $\mathcal{F}$ is a formal group over the ring of integers in $K$, one can associate to $\mathcal F$ and each positive integer $n$ a $G$-module…

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…

Let $n$ denote either a positive integer or $\infty$, let $\ell$ be a fixed prime and let $K$ be a field of characteristic different from $\ell$. In the presence of sufficiently many roots of unity in $K$, we show how to recover some of the…

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional path algebra KQ of a quiver Q. Under certain assumptions on the action of G, we show the existence of a certain kind of…

Let $A$ be an abelian variety over a number field $\mathrm E\subset \mathbb C$ and let $\mathbf G$ denote the Mumford--Tate group of $A$. After replacing $\mathrm E$ by a finite extension, the action of the absolute Galois group…

We prove a strengthening of the "reciprocity conjecture" of Khare and Wintenberger. The input to the original conjecture is an odd prime p, a CM number field F containing the pth roots of unity, and a pair of primes of the maximal totally…

One of the most significant discrete invariants of a quadratic form $\phi$ over a field $k$ is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behaviour of $\phi$ under scalar extension to…

Every Lie group $G$ carries a distinguished algebra of particularly well-behaved real-analytic mappings: The entire functions $\mathcal{E}(G)$. They were introduced for the purposes of strict deformation quantization. This paper establishes…

Let L be a Galois extension of a countable Hilbertian field K. Although L need not be Hilbertian, we prove that an abundance of large Galois subextensions of L/K are.

One of the many interesting algebraic objects associated to a given rational elliptic curve, $E$, is its full-torsion representation $\rho_E:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to\mathrm{GL}_2(\hat{\mathbf{Z}})$. Generalizing this…

We explore a natural action of Hecke operators acting on formal sums of optimal embeddings of real quadratic orders into Eichler orders. By associating an optimal embedding to its root geodesic on the corresponding Shimura curve, we can…

We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of…

Given an object over the algebraic closure Qbar of Q, there is often no reason for invariants of the corresponding holomorphic object to be preserved by the absolute Galois group Gal(Qbar/Q), and in general this is not true, although it is…

Let $G$ be a connected reductive group defined and split over a non-archimedean local field $F$. We give a new geometric proof of a special case of a recent theorem of Solleveld. Namely, we show that the class of standard Iwahori-spherical…

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension of $W(k)[\frac{1}{p}]$. We prove that the locus of potentially semi-stable $\mathrm{Gal}(\bar{K}/K)$-representations of a given…