Related papers: On N\'eron models, divisors and modular curves

Let k be an algebraically closed field of characteristic p. Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a…

Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the…

Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D<0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $\order_{D}$ to supersingular…

Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute…

Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We…

We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree $p$, resp. $d$, onto smooth elliptic curves, with particular attention to the case $p$ prime.

The genus of projective curves discretely separates decidedly different two variable algebraic relations. So, we can focus on the connected moduli M_g of genus g curves. Yet, modern applications require a data variable (function) on such…

Consider genus g curves that admit degree d covers to an elliptic curve simply branched at 2g-2 points. Vary a branch point and the locus of such covers forms a one-parameter family W. We investigate the geometry of W by using admissible…

Let $P$ be a non-torsion point on an elliptic curve defined over a number field $K$ and consider the sequence $\{B_n\}_{n\in \mathbb{N}}$ of the denominators of $x(nP)$. We prove that every term of the sequence of the $B_n$ has a primitive…

We describe the singular locus of the compactification of the moduli space $R_{g,l}$ of curves of genus $g$ paired with an $l$-torsion point in their Jacobian. Generalising previous work for $l\le 2$, we also describe the sublocus of…

{Let $K$ be a number field, and $A_1,A_2$ abelian varieties over $K$. Let $P$ (resp. $Q$) be a non-torsion point in $ A_1(K)$ (resp. $A_2(K)$) such that for almost all places $v$ of $K$, the order of $Q$ mod $v$ divides the order of $P$ mod…

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively…

Anderson introduced a $p$-adic version of soliton theory. He then applied it to the Jacobian variety of a cyclic quotient of a Fermat curve and showed that torsion points of certain prime order lay outside of the theta divisor. In this…

We study various aspects of the behaviour of N\'eron models of semi-abelian varieties under finite extensions of the base field, with a special emphasis on wildly ramified Jacobians. In Part 1, we analyze the behaviour of the component…

Suppose that $f(x)=Ax^k$ mod $p$ is a permutation of the least residues mod $p$. With the exception of the maps $f(x)=Ax$ and $Ax^{(p+1)/2}$ mod $p$ we show that for fixed $n\geq 2$ the image of each residue class mod $n$ contains elements…

We study N\'eron models of pseudo-Abelian varieties over excellent discrete valuation rings of equal characteristic $p>0$ and generalize the notions of good reduction and semiabelian reduction to such algebraic groups. We prove that the…

Let $\mathcal{O}_K$ be a Henselian discrete valuation domain with field of fractions $K$. Assume that $\mathcal{O}_K$ has algebraically closed residue field $k$. Let $E/K$ be an elliptic curve with additive reduction. The semi-stable…

Let $\mathbb{K}$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $\mathbb{K}$, containing three points, $P_1,P_2,$ and $P_3$, such that $nP_1+P_2$, $nP_2+P_3$, and…

In this paper, we are concerned with the computation of the $p$-rank and $a$-number of singular curves and their smooth model. We consider a pair $X, X'$ of proper curves over an algebraically closed field $k$ of characteristic $p$, where…

For every $n\geq 3$, we exhibit infinitely many extremal effective divisors on the moduli space of genus one curves with $n$ marked points.