Rachel Pries

Ordinary and almost ordinary Prym varieties

Ekin Ozman, Rachel Pries

We study the $p$ -rank stratification of the moduli space of Prym varieties in characteristic $p > 0$ . For arbitrary primes $p$ and $ℓ$ with $ℓ \neq = p$ and integers $g \geq…

The de Rham cohomology of the Suzuki curves

Beth Malmskog, Rachel Pries, Colin Weir

For a natural number $m$ , let $S_{m} / F_{2}$ be the $m$ th Suzuki curve. We study the mod $2$ DieudonnÃ© module of $S_{m}$ , which gives the equivalent informat…

Positive density of primes of ordinary reduction for abelian varieties of simple signature

Victoria Cantoral FarfÃ¡n, Wanlin Li, Elena Mantovan +2

By a result of Serre, if $A$ is an elliptic curve without CM defined over a number field $L$ , then the set of primes of $L$ for which $A$ has ordinary reduction has density $1$ . Ka…

Current results on Newton polygons of curves

Rachel Pries

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 sur…

math.NT2012

The Automorphism Groups of a Family of Maximal Curves

Robert Guralnick, Beth Malmskog, Rachel Pries

The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giuliet…

math.NT2026

The Mathieu group $M_{23}$ is a Galois group over $Q$

Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee +3

Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $Q$ during 1984--1989. We complete this program b…

math.NT2008

Monodromy of the p-rank strata of the moduli space of curves

Jeff Achter, Rachel Pries

We compute the Z/\ell and \ell-adic monodromy of every irreducible component of the moduli space M_g^f of curves of genus and and p-rank f. In particular, we prove that the Z/\ell-…

math.NT2006

A short guide to p-torsion of abelian varieties in characteristic p

Rachel Pries

There are many equivalent ways to describe the p-torsion of a principally polarized abelian variety in characteristic p. We briefly explain these methods and then illustrate them f…

math.NT2013

Generic Newton polygons for curves of given p-rank

Jeff Achter, Rachel Pries

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 $p$ -ra…

math.AG2005

Equiramified deformations of covers in positive characteristic

Rachel Pries

Suppose $I ¨ $ is a wildly ramified cover of germs of curves defined over an algebraically closed field of characteristic p. We study unobstructed deformations of $I ¨ $ in equal chara…

math.DS2017

Reduction of dynatomic curves

John R. Doyle, Holly Krieger, Andrew Obus +3

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 character…

Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields

Lisa Berger, Chris Hall, RenÃ© Pannekoek +5

We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r - 1$ with affine model $y^{r} = x^{r - 1} (x + 1) (x + t)$ over the function field $F_{p} (t)$ , when $p$ is p…

math.AG2006

The integral monodromy of hyperelliptic and trielliptic curves

Jeff Achter, Rachel Pries

We compute the $\integ/\ell$ and $\integ_\ell$ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a pr…

math.NT2013

The Ekedahl-Oort type of Jacobians of Hermitian curves

Rachel Pries, Colin Weir

The Ekedahl-Oort type is a combinatorial invariant of a principally polarized abelian variety $A$ defined over an algebraically closed field of characteristic $p > 0$ . It character…

math.NT2020

Cohomology groups of Fermat curves via ray class fields of cyclotomic fields

Rachel Davis, Rachel Pries

The absolute Galois group of the cyclotomic field $K = Q (I^¶_{p})$ acts on the Ã©tale homology of the Fermat curve $X$ of exponent $p$ . We study a Galois cohomology group whic…

math.NT2019

The Supersingularity of Hurwitz Curves

Dean Bisogno, Erin Dawson, Henry Frauenhoff +5

We study when Hurwitz curves are supersingular. Specifically, we show that the curve $H_{n, ℓ} : X^{n} Y^{ℓ} + Y^{n} Z^{ℓ} + Z^{n} X^{ℓ} = 0$ , with $n$ and $ℓ$ relatively prime, is s…

Newton polygons of cyclic covers of the projective line branched at three points

Wanlin Li, Elena Mantovan, Rachel Pries +1

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 li…

math.NT2005

Hyperelliptic Curves with Prescribed $p$ -Torsion

Darren Glass, Rachel Pries

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. F…

math.NT2023

Doubly isogenous genus-2 curves with $D_{4}$ -action

Vishal Arul, Jeremy Booher, Steven R. Groen +5

We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C' are curves over…

Every $B T_{1}$ group scheme appears in a Jacobian

Rachel Pries, Douglas Ulmer

Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$ . A $B T_{1}$ group scheme over $k$ is a finite commutative group scheme which arises as th…

Supersingular curves via the Shimura--Taniyama method

Jeremy Booher, Rachel Pries

For a curve which admits an abelian cover of the projective line branched at three points, we study when its reduction to positive characteristic is supersingular. Using the method…

math.AG2009

The p-rank strata of the moduli space of hyperelliptic curves

Jeff Achter, Rachel Pries

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\…

math.AG2024

Mass formula for non-ordinary curves in one dimensional families

Renzo Cavalieri, Rachel Pries

This paper is about one dimensional families of cyclic covers of the projective line in positive characteristic. For each such family, we study the mass formula for the number of n…

math.NT2022

Realizing Artin-Schreier Covers with Minimal $a$ -numbers in Positive Characteristic

Fiona Abney-McPeek, Hugo Berg, Jeremy Booher +8

Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field of characteristic $p > 0$ and $B \subset X$ is a finite, possibly empty, set of points.…

math.NT2019

Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties

Wanlin Li, Elena Mantovan, Rachel Pries +1

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 meth…

math.NT2009

Wild cyclic-by-tame extensions

Andrew Obus, Rachel Pries

Suppose G is a semi-direct product of the form Z/p^n \rtimes Z/m where p is prime and m is relatively prime to p. Suppose K is a local field of characteristic p > 0. The main resul…

math.NT2024

The classifying element for quotients of Fermat curves

Juanita Duque-Rosero, Rachel Pries

Suppose $C$ is a cyclic Galois cover of the projective line branched at the three points $0$ , $1$ , and $\infty$ . Under a mild condition on the ramification, we determine the struct…

math.AG2017

Abhyankar's conjectures in Galois theory: Current status and future directions

David Harbater, Andrew Obus, Rachel Pries +1

In this paper, we survey the major contributions of Abhyankar to the development of the theory of fundamental groups and Galois covers in positive characteristic. We first discuss…

math.AG2008

The p-torsion of curves with large p-rank

Rachel Pries

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…

Newton Polygons Arising for Special Families of Cyclic Covers of the Projective Line

Wanlin Li, Elena Mantovan, Rachel Pries +1

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 Sh…

math.NT2022

A Mass Formula For Artin--Schreier Curves Over Finite Fields

Anne M. Ho, Rachel Pries

We study a mass formula for Artin--Schreier curves of genus $g$ defined over a finite field $k$ of characteristic $p$ . For an odd prime $p$ and for small $g$ , we determine the numb…

Fully maximal and fully minimal abelian varieties

Valentijn Karemaker, Rachel Pries

We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type o…

math.AG2010

Ekedahl-Oort strata of hyperelliptic curves in characteristic 2

Arsen Elkin, Rachel Pries

Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p = 2$ . We prove that the de Rham cohomology of $X$ decomposes int…

math.NT2015

Superspecial rank of supersingular abelian varieties and Jacobians

Jeff Achter, Rachel Pries

An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is…

math.NT2012

Families of Artin-Schreier curves with Cartier-Manin matrix of constant rank

Shawn Farnell, Rachel Pries

Let k be an algebraically closed field of characteristic p > 0. Every Artin-Schreier k-curve X has an equation of the form y^p - y = f(x) for some f(x) in k(x) such that p does not…

math.HO2021

Counting Rocks! An Introduction to Combinatorics

Henry Adams, Kelly Emmrich, Maria Gillespie +2

This textbook, "Counting Rocks!", is the written component of an interactive introduction to combinatorics at the undergraduate level. Throughout the text, we link to videos where…

Non-ordinary curves with a Prym variety of low $p$ -rank

Turku Ozlum Celik, Yara Elias, Burcin Gunes +4

If $I ¨ : Y \to X$ is an unramified double cover of a smooth curve of genus $g$ , then the Prym variety $P_{I ¨} $ is a principally polarized abelian variety of dimension $g - 1$ . When $X$ …

math.AG2005

Wildly ramified covers with large genus

Rachel Pries

We study wildly ramified G-Galois covers $I ¨ : Y \to X$ branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genu…

math.AG2010

A survey of Galois theory of curves in characteristic p

Rachel Pries, Katherine Stevenson

This survey is about Galois theory of curves in characteristic p, a topic which has inspired major research in algebraic geometry and number theory and which contains many open que…

Infinitely many primes of basic reduction for some abelian fourfolds

Wanlin Li, Elena Mantovan, Rachel Pries +1

If $E$ is an elliptic curve, defined over $Q$ or a number field having at least one real embedding, then Elkies proved that $E$ has supersingular reduction at infinitely m…

math.NT2024

Some cases of Oort's conjecture about Newton polygons

Rachel Pries

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 t…

math.NT2015

Galois action on the homology of Fermat curves

Rachel Davis, Rachel Pries, Vesna Stojanoska +1

In his paper titled "Torsion points on Fermat Jacobians, roots of circular units and relative singular homology", Anderson determines the homology of the degree $n$ Fermat curve as…

Data for Shimura varieties intersecting the Torelli locus

Wanlin Li, Elena Mantovan, Rachel Pries

For infinitely many Hurwitz spaces parametrizing cyclic covers of the projective line, we provide a method to determine the integral PEL datum of the Shimura variety that contains…

The boundary of the $p$ -rank $0$ stratum of the moduli space of cyclic covers of the projective line

Ekin Ozman, Rachel Pries, Colin Weir

We study the $p$ -rank stratification of the moduli space of cyclic degree $ℓ$ covers of the projective line in characteristic $p$ for distinct primes $p$ and $ℓ$ . The main re…

The Galois action and cohomology of a relative homology group of Fermat Curves

Rachel Davis, Rachel Pries, Vesna Stojanoska +1

For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{Q (I^¶_{p})}$ on the homology of the degree $p…

math.NT2022

The Galois action on the lower central series of the fundamental group of the Fermat curve

Rachel Davis, Rachel Pries, Kirsten Wickelgren

Information about the absolute Galois group $G_{K}$ of a number field $K$ is encoded in how it acts on the Ã©tale fundamental group $I ¨ $ of a curve $X$ defined over $K$ . In the case…

math.NT2020

Realizing Artin-Schreier covers of curves with minimal Newton polygons in positive characteristic

Jeremy Booher, Rachel Pries

Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field $k$ of characteristic $p > 0$ and $B \subset X (k)$ is a finite, possibly empty, set of p…

math.NT2015

Arithmetic of abelian varieties in Artin-Schreier extensions

Rachel Pries, Douglas Ulmer

We study abelian varieties defined over function fields of curves in positive characteristic $p$ , focusing on their arithmetic within the system of Artin-Schreier extensions. First…

math.NT2010

Alternating group covers of the affine line

Jeremy Muskat, Rachel Pries

We prove Abhyankar's Inertia Conjecture for the alternating group A_{p+2} on p+2 letters when p = 2 mod 3, by showing that every possible inertia group occurs for a (wildly ramifie…

math.NT2011

The a-numbers of Jacobians of Suzuki curves

Holley Friedlander, Derek Garton, Beth Malmskog +2

For $m \in N$ , let $S_{m}$ be the Suzuki curve defined over $F_{2^{2 m + 1}}$ . It is well-known that $S_{m}$ is supersingular, but the p-torsion group scheme of its Ja…

math.AG2025

The Torelli locus and Newton polygons

Rachel Pries

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 long…

math.NT2010

Semi-direct Galois covers of the affine line

Linda Gruendken, Laura Hall-Seelig, Bo-Hae Im +3

Let $k$ be an algebraically closed field of characteristic $p > 0$ . Let $G$ be $Z / ℓ Z$ semi-direct product $Z / pZ$ where $ℓ$ is a prime distinct from $p$ . In this paper, we stud…

math.NT2013

Newton polygons for a variant of the Kloosterman family

Rebecca Bellovin, Sharon Anne Garthwaite, Ekin Ozman +3

We study the p-adic valuations of roots of L-functions associated with certain families of exponential sums of Laurent polynomials in n variables over a finite field. The families…

On $B T_{1}$ group schemes and Fermat Jacobians

Rachel Pries, Douglas Ulmer

Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$ . A $B T_{1}$ group scheme over $k$ is a finite commutative group scheme which arises as th…

math.NT2007

Curves of given $p$ -rank with trivial automorphism group

Jeff Achter, Darren Glass, Rachel Pries

Let $k$ be an algebraically closed field of characteristic $p > 0$ . Suppose $g \geq 3$ and $0 \leq f \leq g$ . We prove there is a smooth projective $k$ -curve of genus $g$ and $p$ -ra…

math.NT2010

The p-rank stratification of Artin-Schreier curves

Rachel Pries, Hui June Zhu

We study a moduli space AS_g for Artin-Schreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification of AS_g by p-rank into strat…