Related papers: Logarithmic Dieudonn\'e theory and overconvergent …

The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense…

We construct the Frobenius structure on a rigid connection $\mathrm{Be}_{\check{G}}$ on $\mathbb{G}_m$ for a split reductive group $\check{G}$ introduced by Frenkel-Gross. These data form a $\check{G}$-valued overconvergent $F$-isocrystal…

In this paper we define a relative rigid fundamental group, which associates to a section $p$ of a smooth and proper morphism $f:X\rightarrow S$ in characteristic $p$, a Hopf algebra in the ind-category of overconvergent $F$-isocrystals on…

By works of Kedlaya and Shiho, it is known that, for a smooth variety $\overline{X}$ over a field of positive characteristic and its simple normal crossing divisor $Z$, an overconvergent isocrystal on the compliment of $Z$ satisfying a…

We show that for a large class of finite groups G, the number of Galois extensions E/Q of group G and discriminant $|d_E|\leq y$ grows like a power of $y$ (for some specified exponent). The groups G are the regular Galois groups over Q and…

In this paper we formulate a conjecture on the relationship between the equivariant \epsilon-constants (associated to a local p-adic representation V and a finite extension of local fields L/K) and local Galois cohomology groups of a Galois…

We establish a valuative version of Grothendieck's section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the…

Let $L$ be a finite extension of $\mathbf{Q}_p$. Let $\rho_L$ be a potentially semi-stable non-crystalline $p$-adic Galois representation such that the associated $F$-semisimple Weil-Deligne representation is absolutely indecomposable. In…

We describe the mod $p^r$ pro $K$-groups $\{K_n(A/I^s)/p^r\}_s$ of a regular local $\mathbb F_p$-algebra $A$ modulo powers of a suitable ideal $I$, in terms of logarithmic Hodge-Witt groups, by proving pro analogues of the theorems of…

We study issues related to F-theory on Calabi-Yau fourfolds and its duality to heterotic theory for Calabi-Yau threefolds. We discuss principally fourfolds that are described by reflexive polyhedra and show how to read off some of the data…

Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible…

In this article we introduce the notion of projective isomonodromy, which is a special type of monodromy evolving deformation of linear differential equations, based on the example of the Darboux-Halphen equation. We give an algebraic…

Let $\mathscr{O}_K$ be a 2-adic discrete valuation ring with perfect residue field $k$. We classify $p$-divisible groups and $p$-power order finite flat group schemes over $\mathscr{O}_K$ in terms of certain Frobenius module over…

Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\bar{\rho}_{f}$ is irreducible. Under some additional hypotheses we prove that…

The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of…

In Proposition I of "Memoire sur les conditions de resolubilite des equations par radicaux", Galois established that any intermediate extension of the splitting field of a polynomial with rational coefficients is the fixed field of its…

Let $F$ be a totally real number field, $\wp$ a place of $F$ above $p$. Let $\rho$ be a $2$-dimensional $p$-adic representation of $\mathrm{Gal}(\bar{F}/F)$ which appears in the \'etale cohomology of quaternion Shimura curves (thus $\rho$…

We develop a Dieudonn\'e theory for $p$-divisible groups using sheared Witt vectors.

Fuglede's conjecture states that for a subset $\Omega$ of a locally compact abelian group $G$ with positive and finite Haar measure, there exists a subset of the dual group of $G$ which is an orthogonal basis of $L^{2}(\Omega)$ if and only…

We consider syntomic complexes of uniform F-crystals and relate them to Tamagawa number conjecture in characteristic p