arXiv subjects

Renate Scheidler

Publications and source records attributed to Renate Scheidler.

16 recordsLinked to original sources

arXiv2026

On the Geometry and Shapes of Rank 2 Log Unit Lattices

Every number field canonically gives rise to two lattices: its ring of integers and its log unit lattice. While the shapes of the former have undergone extensive research, far less is known about the shapes of the latter, referred to as unit shapes. This paper presents an in-depth analysis of the unit shapes of number fields with unit rank 2. Our first main result characterizes, in many cases, the location of unit shapes within the fundamental domain of the space of rank 2 lattice shapes in terms of the Galois group of the field's Galois closure, and determines when these unit shapes are transcendental. Next, we establish that the unit shape uniquely determines the field up to isomorphism for totally imaginary $D_6$ non-CM sextic fields; this result fails in the CM case. Finally, for certain subfamilies of $D_6$ non-CM imaginary sextics, we offer a simple sufficient condition for their log unit lattices to be orthogonal and provide lower bounds on the proportion of fields with orthogonal log unit lattice.

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arXiv2026

The Spine: A Supersingular Highway

We consider the structure of the spine of the supersingular $\ell$-isogeny graph for one of the cases which arXiv:2502.03613 was not able to fully describe, $\ell = 2$ and $p = 71, 119\pmod{120}$. We find the distance, eccentricity, and diameter functions, of the components of the spine without the non-trivial edge not defined over $\mathbb{F}_p$. Using these functions, we find the mean diameter of the spine and show how this value distinguishes the different structures of the spine. Thus, allowing us to use explicit computations to provide heuristics on the behavior of the spine's structure as $p$ varies.

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arXiv2026

Improvements to Jacobian Arithmetic in Global Function Fields

We present two improvements to arithmetic in the Jacobian of global function fields based on the approach of Hess. The first reduces the number of expensive reduction steps by optimizing for typical inputs rather than worst-case behavior, assuming the function field contains a degree-one place. The second introduces a memory-time trade-off that speeds up computations by caching frequently used intermediate results. Our asymptotic analysis and empirical experiments show that our improved algorithms are significantly faster in practice than previously published methods. To the best of our knowledge, our publicly-available software implementation of Jacobian arithmetic is the first to support unique representatives of divisor classes.

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arXiv2026

Computing Invariants of Artin-Schreier Curves

We present an algorithmic framework for computing generators for the ring of invariants of an Artin-Schreier curve. We give explicit invariants for almost all Artin-Schreier curves of genus up to~8 in standard form, and for a handful of curves of higher genus.

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arXiv2026

Computing submodules of points of general Drinfeld modules over finite fields

We present an algorithm for computing the structure of any submodule of the module of points of a Drinfeld $A$-module over a finite field, where $A$ is a function ring over $\mathbb F_q$. When the function ring is $A = \mathbb F_q[T]$, we additionally compute a Frobenius decomposition of said submodule. Our algorithms apply in particular to kernels of isogenies and torsion submodules. They are presented within the frameworks of Frobenius normal forms, presentations of modules, and Fitting ideals. They rely largely on efficient and classical linear algebra methods, combined with fast arithmetic of Ore polynomials. We analyze the complexity of our algorithms, explore optimizations, and provide an implementation in SageMath. Finally, we compute a simple invariant attached to a Drinfeld $\mathbb F_q[T]$-module that encodes all the polynomials in $\mathbb F_q[T]$ whose associated torsion is rational.

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arXiv2025

The Spine of a Supersingular $\ell$-Isogeny graph

Supersingular elliptic curve $\ell$-isogeny graphs over finite fields offer a setting for a number of quantum-resistant cryptographic protocols. The security analysis of these schemes typically assumes that these graphs behave randomly. Motivated by this debatable assertion, we explore structural properties of these graphs. We detail the behavior, governed by congruence conditions on $p$, of the $\ell$-isogeny graph over $\mathbb{F}_p$ when passing to the spine, i.e. the subgraph induced by the $\mathbb{F}_p$-vertices in the full $\ell$-isogeny graph. We describe the diameter of the spine and offer numerical data on the number of vertices, over both $\mathbb{F}_p$ and $\overline{\mathbb{F}_p}$, in the center of the $\ell$-isogeny graph. Our plots of these counts exhibit a wave-shaped pattern which supports the assertion that centers of supersingular $\ell$-isogeny graphs exhibit the same behavior as those of random $(\ell+1)$-regular graphs.

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arXiv2024

Solving norm equations in global function fields

We present two new algorithms for solving norm equations over global function fields with at least one infinite place of degree 1 and no wild ramification. The first of these is a substantial improvement of a method due to Gaál and Pohst, while the second approach uses index calculus techniques and is significantly faster asymptotically and in practice. Both algorithms incorporate compact representations of field elements which results in a significant gain in performance compared to the Gaál-Pohst approach. We provide Magma implementations, analyze the complexity of all three algorithms under varying asymptotics on the field parameters, and provide empirical data on their performance.

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arXiv2024

On invariants of Artin-Schreier curves

The main goal of this article is to expand the theory of invariants of Artin-Schreier curves by giving a complete classification in genus 3 and 4. To achieve this goal, we first establish standard forms of Artin-Schreier curves and determine all isomorphisms between curves in this form. We then compute reconstructing systems of invariants for curves in each connected component of the strata of the moduli spaces for Artin-Schreier curves of genus 3 and 4 for $p>2$.

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arXiv2022

Orientations and cycles in supersingular isogeny graphs

The paper concerns several theoretical aspects of oriented supersingular $\ell$-isogeny volcanoes and their relationship to closed walks in the supersingular $\ell$-isogeny graph. Our main result is a bijection between the rims of the union of all oriented supersingular $\ell$-isogeny volcanoes over $\overline{\mathbb{F}}_p$ (up to conjugation of the orientations), and isogeny cycles (non-backtracking closed walks which are not powers of smaller walks) of the supersingular $\ell$-isogeny graph over $\overline{\mathbb{F}}_p$. The exact proof and statement of this bijection are made more intricate by special behaviours arising from extra automorphisms and the ramification of $p$ in certain quadratic orders. We use the bijection to count isogeny cycles of given length in the supersingular $\ell$-isogeny graph exactly as a sum of class numbers of these orders, and also give an explicit upper bound by estimating the class numbers.

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arXiv2022

Orienteering with one endomorphism

In supersingular isogeny-based cryptography, the path-finding problem reduces to the endomorphism ring problem. Can path-finding be reduced to knowing just one endomorphism? It is known that a small endomorphism enables polynomial-time path-finding and endomorphism ring computation (Love-Boneh [36]). An endomorphism gives an explicit orientation of a supersingular elliptic curve. In this paper, we use the volcano structure of the oriented supersingular isogeny graph to take ascending/descending/horizontal steps on the graph and deduce path-finding algorithms to an initial curve. Each altitude of the volcano corresponds to a unique quadratic order, called the primitive order. We introduce a new hard problem of computing the primitive order given an arbitrary endomorphism on the curve, and we also provide a sub-exponential quantum algorithm for solving it. In concurrent work (Wesolowski [54]), it was shown that the endomorphism ring problem in the presence of one endomorphism with known primitive order reduces to a vectorization problem, implying path-finding algorithms. Our path-finding algorithms are more general in the sense that we don't assume the knowledge of the primitive order associated with the endomorphism.

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arXiv2020

Difference Necklaces

An $(a,b)$-difference necklace of length $n$ is a circular arrangement of the integers $0, 1, 2, \ldots , n-1$ such that any two neighbours have absolute difference $a$ or $b$. We prove that, subject to certain conditions on $a$ and $b$, such arrangements exist, and provide recurrence relations for the number of $(a,b)$-difference necklaces for $( a, b ) = ( 1, 2 )$, $( 1, 3 )$, $( 2, 3 )$ and $( 1, 4 )$. Using techniques similar to those employed for enumerating Hamiltonian cycles in certain families of graphs, we obtain these explicit recurrence relations and prove that the number of $(a,b)$-difference necklaces of length $n$ satisfies a linear recurrence relation for all permissible values $a$ and $b$. Our methods generalize to necklaces where an arbitrary number of differences is allowed.

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arXiv2016

Class number and regulator computation in cubic function fields

We present computational results on the divisor class number and the regulator of a cubic function field over a large base field. The underlying method is based on approximations of the Euler product representation of the zeta function of such a field. We give details on the implementation for purely cubic function fields of signatures $(3,1)$ and $(1, 1; 1, 2)$, operating in the ideal class group and infrastructure of the function field, respectively. Our implementation provides numerical evidence of the computational effectiveness of this algorithm. With the exception of special cases, such as purely cubic function fields defined by superelliptic curves, the examples provided are the largest divisor class numbers and regulators ever computed for a cubic function field over a large prime field. The ideas underlying the optimization of the class number algorithm can in turn be used to analyze the distribution of the zeros of the function field's zeta function. We provide a variety of data on a certain distribution of the divisor class number that verify heuristics by Katz and Sarnak on the distribution of the zeroes of the zeta function.

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arXiv2015

Zeta functions of a class of Artin-Schreier curves with many automorphisms

This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.

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arXiv2012

Computing quadratic function fields with high 3-rank via cubic field tabulation

This paper presents an algorithm for generating all imaginary and unusual discriminants up to a fixed degree bound that define a quadratic function field of positive 3-rank. Our method makes use of function field adaptations of a method due to Belabas for finding quadratic number fields of high 3-rank and of a refined function field version of a theorem due to Hasse. We provide numerical data for discriminant degree up to 11 over the finite fields $\mathbb{F}_{5}, \mathbb{F}_{7}, \mathbb{F}_{11}$ and $\mathbb{F}_{13}$. A special feature of our technique is that it produces quadratic function fields of minimal genus for any given 3-rank. Taking advantage of certain $\mathbb{F}_{q}(t)$-automorphisms in conjunction with Horner's rule for evaluating polynomials significantly speeds up our algorithm in the imaginary case; this improvement is unique to function fields and does not apply to number field tabulation. These automorphisms also account for certain divisibility properties in the number of fields found with positive 3-rank. Our numerical data mostly agrees with the predicted heuristics of Friedman-Washington and partial results on the distribution of such values due to Ellenberg-Venkatesh-Westerland for quadratic function fields over the finite field $\finfldq{q}$ where $q \equiv -1 \pmod{3}$. The corresponding data for $q \equiv 1 \pmod{3}$ does not agree closely with the previously mentioned heuristics and results, but does agree more closely with some recent number field conjectures of Malle and some work in progress on proving such conjectures for function fields due to Garton.

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arXiv2011

Tabulation of cubic function fields via polynomial binary cubic forms

We present a method for tabulating all cubic function fields over $\mathbb{F}_q(t)$ whose discriminant $D$ has either odd degree or even degree and the leading coefficient of $-3D$ is a non-square in $\mathbb{F}_{q}^*$, up to a given bound $B$ on the degree of $D$. Our method is based on a generalization of Belabas' method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires $O(B^4 q^B)$ field operations as $B \rightarrow \infty$. The algorithm, examples and numerical data for $q=5,7,11,13$ are included.

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arXiv2010

Class Number and Regulator Computation in Purely Cubic Function Fields of Unit Rank Two

We describe and give computational results of a procedure to compute the divisor class number and regulator of most purely cubic function fields of unit rank 2. Our implementation is an improvement to Pollard's Kangaroo method in infrastructures, using distribution results of class numbers as well as information on the congruence class of the divisor class number, and an adaptation that efficiently navigates these torus-shaped infrastructures. Moreover, this is the first time that an efficient "square-root" algorithm has been applied to the infrastructure of a global field of unit rank 2. With the exception of certain function fields defined by Picard curves, our examples are the largest known divisor class numbers and regulators ever computed for a function field of genus 3.

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