相关论文: Some unlikely intersections between the Torelli lo…

We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain PEL-type Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus…

This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygon. Using this method, I give a new proof that there exist supersingular curves of genus $4$ for every…

There are open questions about which Newton polygons and Ekedahl-Oort types occur for Jacobians of smooth curves of genus $g$ in positive characteristic $p$. In this chapter, I survey the current state of knowledge about these questions. I…

Consider a smooth irreducible Hodge generic curve $S$ defined over $\bar{\Q}$ in the Torelli locus $T_g\subset \mathcal{A}_g$. We establish Zilber-Pink-type statements for such curves depending on their intersection with the boundary of the…

In this article we study the behavior of Newton polygons along $\mathbb{Z}_p$-towers of curves. Fix an ordinary curve $X$ over a finite field $\mathbb{F}_q$ of characteristic $p$. By a $\mathbb{Z}_p$-tower $X_\infty/X$ we mean a tower of…

In this paper, we show that there exist families of curves (defined over an algebraically closed field $k$ of characteristic $p >2$) whose Jacobians have interesting $p$-torsion. For example, for every $0 \leq f \leq g$, we find the…

The purpose of this article is to study Newton polygons of certain abelian $L$-functions on curves. Let $X$ be a smooth affine curve over a finite field $\mathbb{F}_q$ and let $\rho:\pi_1(X) \to \mathbb{C}_p^\times$ be a finite character of…

We survey results and open questions about the $p$-ranks and Newton polygons of Jacobians of curves in positive characteristic $p$. We prove some geometric results about the $p$-rank stratification of the moduli space of (hyperelliptic)…

By a result of Moonen, there are exactly 20 positive-dimensional families of cyclic covers of the projective line for which the Torelli image is open and dense in the associated Shimura variety. For each of these, we compute the Newton…

Let $\mathcal{A}_{g}$ be the moduli space of $g$-dimensional principally polarized abelian varieties over $\mathbb{Z}$, and let $\mathcal{T} \subset \mathcal{A}_{g}$ be a closed locus, also defined over $\mathbb{Z}$. Motivated by unlikely…

We review the Shimura-Taniyama method for computing the Newton polygon of an abelian variety with complex multiplication. We apply this method to cyclic covers of the projective line branched at three points. As an application, we produce…

Consider the locus of smooth curves of genus g and p-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the p-torsion of the Jacobians of these curves for…

Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the…

We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number…

Let $p$ denote an odd prime. In this paper, we are concerned with the $p$-divisibility of additive exponential sums associated to one variable polynomials over a finite field of characteristic $p$, and with (the very close question of)…

Consider a Jacobian elliptic surface $E \to C$ with a section $P$ of infinite order. Previous work of the first author and Urz\'ua over the complex numbers gives a bound on the number of tangencies between $P$ and a torsion section of $E$…

The moduli space $\mathcal{A}_g$ of principally polarised abelian varieties of dimension $g$, defined over an algebraically closed field of characteristic $p >0$, is studied through various stratifications. The two most prominent ones are…

We study the nef cone of self-products of a curve. When the curve is very general of genus $g>2$, we construct a nontrivial class of self-intersection 0 on the boundary of the nef cone. Up to symmetry, this is the only known nontrivial…

Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian.…

This manuscript is about abelian varieties that are Jacobians of curves. I started writing it for a lecture series at the Arizona Winter School in 2024 on abelian varieties. A longer more descriptive title might be: The Torelli locus in the…