arXiv subjects

Jennifer Paulhus

Publications and source records attributed to Jennifer Paulhus.

8 recordsLinked to original sources

arXiv2025

Signatures of alternating group actions with non-zero quotient genus

We classify up to signature all the ways the alternating group $A_n$ can act on a compact Riemann surfaces when the quotient genus is greater than $0$. In particular, we prove that for $A_n$ with $n>6$ every potential signature for the group acting with quotient genus greater than $0$ is an actual signature. We also show that in the case of $n=5$, respectively $n=6$, the only failure is for $[1;2]$, respectively $[1;3]$. Along the way we also prove that for any finite simple non-abelian group, all potential signatures with quotient genus greater than $1$ are actual signatures.

math.GR↗

arXiv2025

Completely decomposable modular Jacobians

We use recently developed algorithms and a new database of modular curves constructed for the L-functions and Modular Forms Database to enumerate completely decomposable modular Jacobians of level N < 240. In particular, we find examples in 13 previously unknown genera of Jacobian varieties isogenous to a product of elliptic curves over Q. The new genera are: 38, 68, 75, 76, 77, 78, 113, 135, 137, 157, 159, 169, and 409.

math.NT↗

arXiv2024

Creating a dynamic database of finite groups

A database of abstract groups has been added to the L-functions and Modular Forms Database (LMFDB), available at https://www.lmfdb.org/Groups/Abstract/. We discuss the functionality of the database and what makes it distinct from other available databases of abstract groups. We describe solutions to mathematical problems we encountered while creating the database, as well as connections between the abstract groups database and other collections of objects in the LMFDB.

math.GR↗

arXiv2024

On non-normal subvarieties of the moduli space of Riemann surfaces

In this article, we consider certain irreducible subvarieties of the moduli space of compact Riemann surfaces determined by the specification of actions of finite groups. We address the general problem of determining which among them are non-normal subvarieties of the moduli space.

math.AG↗

arXiv2020

Algorithmically distinguishing irreducible characters of the symmetric group

Suppose that $χ_λ$ and $χ_μ$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial in $n$, constructs $π\in S_n$ such that $χ_λ(π)$ is provably different from $χ_μ(π)$. In fact, we show a little more. Suppose $f=χ_λ$ for some irreducible character $χ_λ$ of $S_n$, but we do not know $λ$, and we are given only oracle access to $f$. We give an algorithm that determines $λ$, using a number of queries to $f$ that is polynomial in $n$. Each query can be computed in time polynomial in $n$ by someone who knows $λ$.

math.CO↗

arXiv2019

Non-Abelian Simple Groups Act with Almost All Signatures

The topological data of a group action on a compact Riemann surface is often encoded using a tuple $(h;m_1,\dots ,m_s)$ called its signature. There are two easily verifiable arithmetic conditions on a tuple necessary for it to be a signature of some group action. In the following, we derive necessary and sufficient conditions on a group $G$ for when these arithmetic conditions are in fact sufficient to be a signature for all but finitely many tuples that satisfy them. As a consequence, we show that all non-Abelian finite simple groups exhibit this property.

math.GR↗

arXiv2016

Completely decomposable Jacobian varieties in new genera

We present a new technique to study Jacobian variety decompositions using subgroups of the automorphism group of the curve and the corresponding intermediate covers. In particular, this new method allows us to produce many new examples of genera for which there is a curve with completely decomposable Jacobian. These examples greatly extend the list given by Ekedahl and Serre of genera containing such curves, and provide more evidence for a positive answer to two questions they asked. Additionally, we produce new examples of families of curves, all of which have completely decomposable Jacobian varieties. These families relate to questions about special subvarieties in the moduli space of principally polarized abelian varieties.

math.AG↗

arXiv2015

Branching data for curves up to genus 48

An algorithm of Thomas Breuer produces complete lists of automorphism groups of curves for a fixed genus, and the code to execute this algorithm was written in the computer algebra package GAP and run by Breuer over a decade ago. For each curve $X$ of genus $g$ and a group $G$ acting on $X$, the branching data for the covering $X \to X/G$ is computed but was not record. We have added functionality to the code to output this data, and have translated the code to the computer algebra package Magma. This article provides an explanation of the code to add this branching data. Complete data from this modified program for curves up to genus 20, and data for genus up to 48 of groups $G$ so that $|G|>4(g-1)$ are available. We also include sample programs which provide a way to search the data for groups or actions with desired properties, these functions should be of use to researchers who want to find examples of group actions on lower genus curves with certain properties, or who need explicit examples of generating vectors for their research. The code also may be used to create generating vectors for higher genus curves, if the automorphism group and signature of the mapping $X \to X/G$ are already know.

math.AG↗