Related papers: The de Rham and the syntomic logarithm

Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parametrised by a number k and a set Q of primes. The intuition is that two graphs G and H which are equivalent…

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm (so-called LLL) has been…

We give a short proof that the ergodic sums of $\mathcal{C}^1$ observables for a $\mathcal{C}^1$ flow on $\mathbb{T}^2$ admitting a closed transversal curve whose Poincar\'e map has constant type rotation number have growth deviating at…

Milne's correcting factor, which appears in the Zeta-value at $s=n$ of a smooth projective variety $X$ over a finite field $\mathbb{F}_q$, is the Euler characteristic of the derived de Rham cohomology of $X/\mathbb{Z}$ modulo the Hodge…

We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane ${Re}(s)>\frac12$ with $\bar{f(\bar{s})}=f(s)$ in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the…

We introduce the "sharp" (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the "sharp de Rham realization" by passing to the Lie-algebra. Over the complex numbers we then…

We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -- the prismatic site -- to a $p$-adic formal scheme. The resulting cohomology theory…

We study duality theorems for the relative logarithmic de Rham-Witt sheaves on semi-stable schemes $X$ over a local ring $\mathbb{F}_q[[t]]$, where $\mathbb{F}_q$ is a finite field. As an application, we obtain a new filtration on the…

We build a unified framework for the study of monodromy operators and weight filtrations of cohomology theories for varieties over a local field. As an application, we give a streamlined definition of Hyodo-Kato cohomology without recourse…

Building on results by Abouzahra and Lewin, McIntosh, and Kirilov we derive new functional dilogarithm equations and consequent diologarithim ladders. By showing that the ratio of a pair of sextic and cubic integrals equals a rational…

We apply Fourier analysis on finite groups to obtain simplified formulations for the Lov\'asz theta-number of a Cayley graph. We put these formulations to use by checking a few cases of a conjecture of Ellis, Friedgut, and Pilpel made in a…

In this paper consisting of two parts, we study the integral of a logarithmic differential form on a compact semi-algebraic set in R^n or C^n. In Part I, we prove the convergence of the integral when the semi-algebraic set satisfies…

We show that the factor graph and certifiable estimation paradigms, which have thus far been treated as essentially independent in the literature, can be naturally synthesized into a unified framework for certifiable factor graph…

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the…

We prove the Riemann-Roch theorem for homotopy invariant $K$-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann-Roch theorem…

We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert…

Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and…

Let $K$ be a finite extension of ${\mathbb Q}_p$ and let $X$ be Drinfel'd's symmetric space of dimension $d$ over $K$. Let $\Gamma\subset {\rm SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let…

In this note we study the deformation theory of periodic (logarithmic) Higgs-de Rham flows. Under suitable numerical assumptions, this is equivalent to the deformation theory of torsion (logarithmic) Fontaine-Faltings modules. As an…

We generalize the functorial quasi-isomorphism in \cite{Davis2011} from overconvergent Witt de-Rham cohomology to rigid cohomology on smooth varieties over a finite field $k$, dropping the quasi-projectiveness condition. We do so by…