Related papers: Logarithmic good reduction and the index

Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the multiplicative…

Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or,…

We prove that smooth, projective, $K$-trivial, weakly ordinary varieties over a perfect field of characteristic $p>0$ are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our…

Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a K-rational point on G of infinite order. Call n_R the number of connected components of the smallest algebraic K-subgroup of G to which R…

We study the $p$-rank stratification of the moduli space of Prym varieties in characteristic $p > 0$. For arbitrary primes $p$ and $\ell$ with $\ell \not = p$ and integers $g \geq 3$ and $0 \leq f \leq g$, the first theorem generalizes a…

Let $E$ be an elliptic curve over $\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\F_p)|$ be the order of the group of points of the…

We study the relative algebraic closure $K$ of $\bar{\mathbb{F}}_p((t))$ inside $\bar{\mathbb{F}}((t^{\mathbb{Q}}))$. We show that the supports of elements in $K$ have order type strictly less than $\omega^\omega$. We also recover a theorem…

Let K be a complete discrete valuation field of mixed characteristic (0,p) with algebraically closed residue field, and let f: Y --> P^1 be a three-point G-cover defined over K, where G has a cyclic p-Sylow subgroup P. We examine the stable…

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$. We obtain an explicit formula, as a product of local terms, for the ramification index at each cusp of a modular parametrization of $E$ by $X_0(N)$. Our formula shows that the…

Let $K$ be a number field, $\mathfrak{q}$ be an integral ideal, and $\mathrm{Cl}(\mathfrak{q})$ be the associated ray class group. Suppose $\mathrm{Cl}(\mathfrak{q})$ possesses a real exceptional character $\psi$, possibly principal, with a…

We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all…

Let $K$ be a complete discretely valued field with algebraically closed residue field and let $\mathfrak C$ be a smooth projective and geometrically connected algebraic $K$-curve of genus $g$. Assume that $g\geq 2$, so that there exists a…

Let $\mathcal{O}_K$ be the ring of integers of an algebraic number field $K$ embedded into $\mathbb{C}$. Let $X$ be a subset of the Euclidean space $\mathbb{R}^d$, and $D(X)$ be the set of the squared distances of two distinct points in…

We show that if $f\colon X \to T$ is a surjective morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic $p>0$ with geometrically integral and non-uniruled generic fiber, then $K_{X/T}$ is…

The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod…

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete…

Let A,A' be elliptic curves or abelian varieties fully of type GSp defined over a number field K. This includes principally polarized abelian varieties with geometric endomorphism ring Z and dimension 2 or odd. We compare the number of…

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 $\ell_0(A,K)$…

The dynatomic modular curves parametrize polynomial maps together with a point of period $n$. It is known that the dynatomic curves $Y_1(n)$ are smooth and irreducible in characteristic 0 for families of polynomial maps of the form $f_c(z)…

Let R be a discrete valuation ring of unequal characteristic with fraction field K which contains a primitive p^2-th root of unity. Let X be a faithfully flat R-scheme and G be a finite abstract group. Let us consider a G-torsor Y_K\to X_K…