Related papers: An explicit Andr\'e-Oort type result for P^1(C) x …

We introduce the notion of a $(\Pi,\lambda)$-structure on a C-system and show that C-systems with $(\Pi,\lambda)$-structures are constructively equivalent to contextual categories with products of families of types. We then show how to…

In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main…

Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$…

We show that for k a perfect field of characteristic p, there exist endomorphisms of the completed algebraic closure of k((t)) which are not bijective. As a corollary, we resolve a question of Fargues and Fontaine by showing that for p a…

We determine an explicit presentation by generators and relations of the cohomology algebra $H^*(\mathbb P^2\setminus C,\mathbb C)$ of the complement to an algebraic curve $C$ in the complex projective plane $\mathbb P^2$, via the study of…

We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem…

We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the…

Let $k$ be an algebraically closed field of positive characteristic $p>0$ and $C \to {\mathbb P}^1_k$ a $p$-cyclic cover of the projective line ramified in exactly one point. We are interested in the $p$-part of the full automorphism group…

Extending the work of Freese and Cook, which develop the basic theory of calculus and power series over real associative algebras, we examine what can be said about the logarithmic functions over an algebra. In particular, we find that for…

The paper concerns a result in linear algebra motivated by ideas from tropical geometry. Let $A(t)$ be an $n \times n$ matrix whose entries are Laurent series in $t$. We show that, as $t \to 0$, logarithms of singular values of $A(t)$…

This article studies the first-order $p$-adic deformations of classical weight one newforms, relating their fourier coefficients to the $p$-adic logarithms of algebraic numbers in the field cut out by the associated projective Galois…

We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $AU_m(q)$, $ASp_{2m}(q)$, $AO_{2m+1}(q)$ and $AO^{\pm}_{2m}(q)$, where $p$ denotes the characteristic of the…

We upgrade the classical operation of \textit{isomonodromic deformations} along a path $\gamma$ to a functor $\mathbb{P}_{\gamma}$ between categories of flat connections with logarithmic singularities along a divisor $D$, which itself…

We consider rational representations of a connected linear algebraic group $\mathbb G$ over a field $k$ of positive characteristic $p > 0$. We introduce a natural extension $M \mapsto \Pi(\mathbb G)_M$ to $\mathbb G$-modules of the…

Let $\lambda \in \mathbb{Q}\setminus \{0, 1\}$ and $l \geq 2$, and denote by $C_{l,\lambda}$ the nonsingular projective algebraic curve over $\mathbb{Q}$ with affine equation given by $$y^l=x(x-1)(x-\lambda).$$ In this paper we define…

This paper is concerned with extending results from "The Geometry of 1-Based Minimal Types" by Kim and the present author. We work in the more general context of the solution set D of a regular Lascar Strong Type defined over the empty set…

Based on functional analysis, we propose an algorithm for finite-norm solutions of higher-order linear Fuchsian-type ordinary differential equations (ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only the four…

Let $E_{\lambda}$ be the Legendre family of elliptic curves with equation $Y^2=X(X-1)(X-\lambda)$. Given a curve $\mathcal{C}$, satisfying a condition on the degrees of some of its coordinates and parametrizing $m$ points $P_1, \ldots, P_m…

Inspired by a beautiful formula of Bertolini, Darmon, and Prasanna -- the oft-termed BDP formula -- we address questions about the non-vanishing of non-torsion points under $p$-adic logarithms of abelian varieties. We largely consider…

Closed form expressions for a multivector exponential and logarithm are presented in real Clifford geometric algebras Cl(p,q)when n=p+q=1 (complex and hyperbolic numbers) and n=2 (Hamilton, split and conectorine quaternions). Starting from…