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Chris Hall

Publications and source records attributed to Chris Hall.

At least 19 recordsLinked to original sources
arXiv2026

Hasse-Weil Zeta Functions Modulo a Prime

Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and $\pi\colon Y\to X$ be a finite $\mathbb{F}_q$-morphism of separated $\mathbb{F}_q$-schemes of finite type. Suppose $\pi$ is generically Galois with group $G$ of prime order $r\neq p$. We determine the mod-$r$ reduction of the zeta function of $Y$ in terms of the zeta function of $X$ and the branch locus $Z\subset X$ of $\pi$. We give applications to curves and to numerators of hyperelliptic/superelliptic curves.

math.NT↗
arXiv2026

Santa Clara 3D: Digital Reconstruction and Storytelling of a Francoist Concentration Camp

This paper explores the potential of digital reconstruction and interactive storytelling to preserve historically suppressed sites. The main objective of an interdisciplinary team of data scientists from the MEMORISE project and associates of the memory association Asociacion Recuerdo y Dignidad was to preserve the memory of the Francoist Santa Clara concentration camp in Soria, Spain, through the use of digital technology. Combining archival research, 3D modelling, 360-degree photography, and web development, a prototype digital platform was created to visualise the transformation of the site across three historical phases: its origin as a convent, its use as a Francoist concentration camp, and its present-day condition. The platform allows users to navigate through spatial and temporal layers. Clickable media markers encourage exploration and interaction. Drawing on principles of participatory design, narrative visualisation, and open-ended user engagement, the project demonstrates how digital tools can support memory work, public engagement, and historical reflection. Our low-cost concept is especially adaptable to other physical sites that have been erased or forgotten.

cs.HC↗
arXiv2025

Out of the Day Job: Perspectives of Industry Practitioners in Co-Design and Delivery of Software Engineering Courses

Over more than two decades, The University of Glasgow has co-designed and delivered numerous software engineering focused courses with industry partners, covering both technical and discipline specific professional skills. Such collaborations are not unique and many of the benefits are well recognised in the literature. These include enhancing the real-world relevance of curricula, developing student professional networks ahead of graduation and easing recruitment opportunities for employers. However, there is relatively little scholarship on the perspectives of industry practitioners who participate in course design and delivery. This gap is significant, since the effort invested by practitioners is often substantial and may require ongoing support from both the industry partner and academic institution. Understanding the motivations, expectations and experiences of practitioners who engage in course delivery can guide the formation of future partnerships and ensure their long-term sustainability. We begin to address this gap by reporting on the outcomes of a retrospective conducted amongst the practitioner coauthors of this paper, with the academic coauthors acting as facilitators. All coauthors have participated in the recent co-design and delivery of software engineering courses, but we choose to focus explicitly on the perspectives of the practitioners. We report on the themes that emerged from the discussions and our resulting recommendations for future collaborations.

cs.SE↗
arXiv2025

Quantitative Analysis of Objects in Prisoner Artworks

Prisoners of Nazi concentration camps created paintings as a means to express their daily life experiences and feelings. Several thousand such paintings exist, but a quantitative analysis of them has not been carried out. We created an extensive dataset of 1,939 Holocaust prisoner artworks, and we employed an object detection framework that found 19,377 objects within these artworks. To support the quantitative and qualitative analysis of the art collection and its objects, we have developed an intuitive and interactive dashboard to promote a deeper engagement with these visual testimonies. The dashboard features various visual interfaces, e.g., a word cloud showing the detected objects and a map of artwork origins, and options for filtering. We presented the interface to domain experts, whose feedback highlights the dashboard's intuitiveness and potential for both quantitative and qualitative analysis while also providing relevant suggestions for improvement. Our project demonstrates the benefit of digital methods such as machine learning and visual analytics for Holocaust remembrance and educational purposes.

cs.HC↗
arXiv2024

Comparison of three reconstruction algorithms for low-dose phase-contrast computed tomography of the breast with synchrotron radiation

Three different computed tomography (CT) reconstruction algorithms: Filtered Back Projection (FBP), Unified Tomographic Reconstruction (UTR) and customized Simultaneous Algebraic Reconstruction Technique (cSART), have been systematically compared and evaluated using experimental data from CT scans of ten fresh mastectomy samples collected at the Imaging and Medical beamline of the Australian Synchrotron. All the scans were collected at the mean glandular dose of 2 mGy, using monochromatic X-rays with 32 keV energy, flat-panel detectors with 0.1 mm pixels and 6 meter distance between the rotation stage and the detector. Paganin's phase retrieval method was used in conjunction with all three CT reconstruction algorithms. The reconstructed images were compared in terms of the objective image quality characteristics, including spatial resolution, contrast, signal-to-noise, and contrast-to-noise ratios. The images were also evaluated by seven experienced medical imaging specialists, rating perceptible contrast, sharpness of tissue interfaces, image noise, calcification visibility and overall image quality. Of the three compared algorithms, cSART was clearly superior to UTR and FBP in terms of most measured objective image quality characteristics. At the same time, the results of the subjective quality evaluation consistently favoured the images reconstructed by FBP, followed by UTR, with cSART receiving lower scores on average. We argue that this apparent disagreement between the objective and subjective assessments of image quality can be explained by the importance assigned to image contrast in the subjective assessment, while the signal-to-noise ratio seemed to receive relatively low weighting. This study was conducted in preparation for phase-contrast breast CT imaging of live patients at Australian Synchrotron (Melbourne, Australia).

physics.med-ph↗
arXiv2020

Full field X-ray Scatter Tomography

In X-ray imaging, photons are transmitted through and absorbed by the subject, but are also scattered in significant quantities. Previous attempts to use scattered photons for biological imaging used pencil or fan beam illumination. Here we present 3D X-ray Scatter Tomography using full-field illumination. Synchrotron imaging experiments were performed of a phantom and the chest of a juvenile rat. Transmitted and scattered photons were simultaneously imaged with separate cameras; a scientific camera directly downstream of the sample stage, and a pixelated detector with a pinhole imaging system placed at 45${}^\circ$ to the beam axis. We obtained scatter tomogram feature fidelity sufficient for segmentation of the lung and major airways in the rat. The image contrast in scatter tomogram slices approached that of transmission imaging, indicating robustness to the amount of multiple scattering present in our case. This opens the possibility of augmenting full-field 2D imaging systems with additional scatter detectors to obtain complementary modes or to improve the fidelity of existing images without additional dose, potentially leading to single-shot or reduced-angle tomography or overall dose reduction for live animal studies.

physics.med-ph↗
arXiv2018

Variance of sums in arithmetic progressions of divisor functions associated with higher degree l-functions in $\mathbb{F}_q(t)$

We compute the variances of sums in arithmetic progressions of generalised k-divisor functions related to certain L-functions in $\mathbb{F}_q(t)$, in the limit as $q\to\infty$. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when $q\to\infty$, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual $k$-divisor function, when the L-function in question has degree one. They illustrate the role played by the degree of the L-functions; in particular, we find qualitatively new behaviour when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over $\mathbb{F}_q(t)$, and we illustrate them by examining in some detail the generalised $k$-divisor functions associated with the Legendre curve.

math.NT↗
arXiv2017

Coproducts of Finite Groups

We show that for any pair of non-trivial finite groups, their coproduct in the category of finite groups is not representable.

math.GR↗
arXiv2017

Variance of sums in arithmetic progressions of arithmetic functions associated with higher degree $L$-functions in $\mathbb{F}_q[t]$

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree two and higher in $\mathbb{F}_q[t]$, in the limit as $q\to\infty$. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-one $L$-functions (i.e. situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair-correlation conjecture. Our calculations apply, for example, to elliptic curves defined over $\mathbb{F}_q[t]$.

math.NT↗
arXiv2015

Ramanujan Coverings of Graphs

Let $G$ be a finite connected graph, and let $\rho$ be the spectral radius of its universal cover. For example, if $G$ is $k$-regular then $\rho=2\sqrt{k-1}$. We show that for every $r$, there is an $r$-covering (a.k.a. an $r$-lift) of $G$ where all the new eigenvalues are bounded from above by $\rho$. It follows that a bipartite Ramanujan graph has a Ramanujan $r$-covering for every $r$. This generalizes the $r=2$ case due to Marcus, Spielman and Srivastava (2013). Every $r$-covering of $G$ corresponds to a labeling of the edges of $G$ by elements of the symmetric group $S_{r}$. We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava, a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from a recent paper of them (2015). Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the $r$-th matching polynomial of $G$ to be the average matching polynomial of all $r$-coverings of $G$. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside $\left[-\rho,\rho\right]$.

math.CO↗
arXiv2015

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

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 $\mathbb{F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $\mathbb{F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb{F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^\nu +1$, and $K_d := \mathbb{F}_p(\mu_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V]^2$. When $r>2$, we prove that the "new" part of $J$ is isogenous over $\overline{\mathbb{F}_p(t)}$ to the square of a simple abelian variety of dimension $\phi(r)/2$ with endomorphism algebra $\mathbb{Z}[\mu_r]^+$. For a prime $\ell$ with $\ell \nmid pr$, we prove that $J[\ell](L)=\{0\}$ for any abelian extension $L$ of $\overline{\mathbb{F}}_p(t)$.

math.NT↗
arXiv2014

Bounded gaps between primes in number fields and function fields

The Hardy--Littlewood prime $k$-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field $\mathbb{F}_q(t)$.

math.NT↗
arXiv2013

Explicit points on the Legendre curve II

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\Fp(t)$ where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$. The rank of $E$ is given in terms of an elementary property of the subgroup of $(\Z/d\Z)^\times$ generated by $p$. We show that for many values of $d$ the rank is large. For example, if $d$ divides $2(p^f-1)$ and $2(p^f-1)/d$ is odd, then the rank is at least $d/2$. When $d=2(p^f-1)$, we exhibit explicit points generating a subgroup of $E(\Fq(t^{1/d}))$ of finite index in the "2-new" part, and we bound the index as well as the order of the "2-new" part of the Tate-Shafarevich group.

math.NT↗
arXiv2011

Characterizing Abelian Varieties by the Reductions of the Mordell-Weil Group

Let $A$ be an abelian variety defined over a number field $K$. If $\mathfrak{p}$ is a prime of $K$ of good reduction for $A$, let $A(K)_\mathfrak{p}$ denote the image of the Mordell-Weil group via reduction modulo $\mathfrak{p}$. We prove in particular that the size of $A(K)_\mathfrak{p}$, by varying $\mathfrak{p}$, encodes enough information to determine the $K$-isogeny class of $A$, provided that the following necessary condition is satisfied: $B(K)$ has positive rank for every non-trivial abelian subvariety $B$ of $A$. This is the analogue to a result by Faltings of 1983 considering instead the Hasse-Weil zeta function of the special fibers $A_\mathfrak{p}$.

math.NT↗
arXiv2011

Radical characterizations of elliptic curves

Let K be a number field, and let E be an elliptic curve over K. A famous result by Faltings of 1983 can be reformulated for elliptic curves as follows: if S is a set of primes of good reduction for E having density one, then the K-isogeny class of E is determined by the function which maps a prime in S to the size of the group of points over the residue field. In this paper, we prove that it suffices to look at the radical of the size.

math.NT↗
arXiv2011

Experimental Data for Goldfeld's Conjecture over Function Fields

This paper presents empirical evidence supporting Goldfeld's conjecture on the average analytic rank of a family of quadratic twists of a fixed elliptic curve in the function field setting. In particular, we consider representatives of the four classes of non-isogenous elliptic curves over F_q(t) with (q,6)=1 possessing two places of multiplicative reduction and one place of additive reduction. The case of q=5 provides the largest data set as well as the most convincing evidence that the average analytic rank converges to 1/2, which we also show is a lower bound following an argument of Kowalski. The data was generated via explicit computation of the L-function of these elliptic curves, and we present the key results necessary to implement an algorithm to efficiently compute the L-function of non-isotrivial elliptic curves over F_q(t) by realizing such a curve as a quadratic twist of a pullback of a `versal' elliptic curve. We also provide a reference for our open-source library ELLFF, which provides all the necessary functionality to compute such L-functions, and additional data on analytic rank distributions as they pertain to the density conjecture.

math.NT↗
arXiv2010

Expander graphs, gonality and variation of Galois representations

We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric (genus and gonality) growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in one-parameter families of abelian varieties.

math.NT↗