Jennifer Paulhus

Jennifer Paulhus
Grinnell College · Department of Mathematics and Statistics

Doctor of Philosophy

About

26
Publications
724
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177
Citations

Publications

Publications (26)
Preprint
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 creati...
Article
Full-text available
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. We obtain several new examples of subvarieties with this property.
Preprint
Full-text available
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.
Chapter
The automorphism group of a Riemann surface is an important object in a number of different mathematical fields. An algorithm of Thomas Breuer determines all such groups for a fixed genus given a complete classification of groups up to a sufficiently large order, but data generated from this algorithm did not include the generators of the correspon...
Chapter
The study of Riemann surfaces and the groups which act on them is a classical area of research dating back to the latter half of the 19th century. Research in this field has wide-reaching implications in geometry and topology, algebra, combinatorics, analysis, and number theory through related topics such as the study of dessins d’enfants, mapping...
Article
Suppose that $\chi_\lambda$ and $\chi_\mu$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial in $n$, constructs $\pi\in S_n$ such that $\chi_\lambda(\pi)$ is provably different from $\chi_\mu(\pi)$. In fact, we show a little more. Suppose $f = \chi_\lambda$ for some irreducible character...
Article
The topological data of a finite group G acting conformally on a compact Riemann surface is often encoded using a tuple of non-negative integers (h;m1,…,ms) called its signature, where the mi are orders of non-trivial elements in the group. There are two easily verifiable arithmetic conditions on a tuple which are necessary for it to be a signature...
Preprint
Suppose that $\chi_\lambda$ and $\chi_\mu$ are distinct irreducible characters of the symmetric group $S_n$. We show how to construct explicitly a permutation $\pi\in S_n$ such that $\chi_\lambda(\pi)$ is provably different from $\chi_\mu(\pi)$. In fact, we show a little more. Suppose $f$ is an irreducible character of $S_n$, but we do not know whi...
Preprint
Full-text available
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...
Preprint
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 whe...
Article
The size of the automorphism group of a compact Riemann surface of genus g> 1 is bounded by 84.g 1/. Curves with automorphism group of size equal to this bound are called Hurwitz curves. In many cases the automorphism group of these curves is the projective special linear group PSL.2; q/. We present a decomposition of the Jacobian varieties for all...
Article
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...
Preprint
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...
Article
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 comput...
Article
Here is an example of how to use MAGMA to compute the decomposition of the Jacobian Variety of a curve. This example demonstrates the computations needed to show that the genus 14 curve X with automorphism group PSL(2, 13) and monodromy (2, 3, 7) has J X ∼ E 14 . Details about the technique used below may be found in [3] and in [2]. The computation...
Article
Let p be an odd prime, k, A ∈ Z, p ∤ A, d = (p − 1, k), d1 = (p − 1, k − 1), s = (p − 1)/d, t = (p − 1)/d1, E the set of even residues in Zp = Z/(p), O the set of odd residues, and Nk = #{x ∈ E: Axk ∈ O}. We give several estimates of Nk, each of different strength depending on the size and parity of s, t and k. In particular we show that Nk ∼ p pro...
Article
For any prime it is possible to construct global function fields whose Jacobians have high by moving to a sufficiently large constant field extension. This was investigated in some detail by Bauer et al. in [2]. The two main results of [2] are an upper bound on the size of the field of definition of the J [ of the Jacobian, and a lower bound on the...
Article
Full-text available
Goresky and Klapper conjectured that for any prime p > 13 and any '-sequence a based on p, every pair of allowable decimations of a is cycli- cally distinct. The conjecture is essentially equivalent to the statement that the mapping x! Axd, with (d;p 1) = 1, p - A, is a permutation of the even residues (mod p) if and only if d = 1 and A 1 (mod p),...
Conference Paper
Experts recognize connections between different representations of a topic that novices often fail to use. One aspect of this is that novices may fail to realize material learned in one context, e.g. a lab assignment, is useful in a different context, e.g. homework. As we move toward teaching in a connected online world, we should also move toward...
Article
Given a fixed genus g, we would like to know the largest possible integer t such that t copies of one elliptic curve E appear in the decomposition of the Jacobian variety JX for some curve X of genus g. In this paper we find nontrivial lower bounds for t for genus up to 10. For genus 3 through 6 we demonstrate curves X such that JX Eg.
Article
We decompose the Jacobian variety of hyperelliptic curves up to genus 10, defined over an algebraically closed field of characteristic zero, with reduced automorphism group A4, S4, or A5. Among those curves is a genus 4 curve with Jacobian variety isogenous to E2 1 ◊E 2 2 and a genus 5 curve with Jacobian variety isogenous to E5 for E and Ei ellipt...

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