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Organizing Committee
Arizona State University
Mount Holyoke College
University of Sydney
Abstract
This will be a one-week conference broadly focused on the topics of the L-functions and modular forms database (LMFDB), mathematical databases, computation, and number theory. The conference will include invited talks, presentations by authors of papers submitted to the conference and selected by the scientific committee following peer-review, as well as time for research and collaboration. We plan to publish a proceedings volume that will include all of the accepted papers.
The field of mathematical databases has emerged as an important area of research at the intersection of computer science and mathematics. It seeks to address questions that arise when organizing, storing, and providing access to mathematical knowledge in a structured manner. These databases are intended to be easily searchable and navigable, providing researchers, educators, and students with a convenient way to access mathematical content. There are many challenges in developing and maintaining mathematical databases, ranging from data quality and integration, to standardization, search design, updating and maintenance, and usability. In addition, many algorithmic and theoretical questions emerge naturally when considering systematic computations that must be done on a large scale.
This workshop is partially supported by the Simons Collaboration Arithmetic Geometry, Number Theory, and Computation.
Application Deadline June 18, 2025
The field of mathematical databases has emerged as an important area of research at the intersection of computer science and mathematics. It seeks to address questions that arise when organizing, storing, and providing access to mathematical knowledge in a structured manner. These databases are intended to be easily searchable and navigable, providing researchers, educators, and students with a convenient way to access mathematical content. There are many challenges in developing and maintaining mathematical databases, ranging from data quality and integration, to standardization, search design, updating and maintenance, and usability. In addition, many algorithmic and theoretical questions emerge naturally when considering systematic computations that must be done on a large scale.
This workshop is partially supported by the Simons Collaboration Arithmetic Geometry, Number Theory, and Computation.
Application Deadline June 18, 2025
The organizers of the second conference on LMFDB, Computation, and Number Theory (LuCaNT) are excited to issue a call for papers for an associated proceedings volume to be published in an open access volume of Contemporary Mathematics. We strongly encourage anyone with research related to mathematical databases or computation to submit a paper here. Papers should be at most 25 pages. Papers that are 4 pages or less, excluding references, will be considered for a special section on "short communications" that provide concise reports on the above themes. Such papers should be on novel work, and not be preliminary results to be published elsewhere.
Papers will go through a peer review process. In addition, any developed code should be placed in a publicly available repository such as GitHub to be available to the reviewers (to help improve the quality, accessibility, re-usability, and reproducibility of the code and data).
It is not required that an accepted paper be presented at the conference but from among the accepted papers a select list of speakers will be invited to give talks at the LuCaNT conference at ICERM.
The deadline for paper submissions is January 24, 2025. We anticipate final decisions on each paper by May 2025.
Graduate students will not be required to present posters but will be encouraged to submit titles and abstracts for lightning talks instead.
Papers will go through a peer review process. In addition, any developed code should be placed in a publicly available repository such as GitHub to be available to the reviewers (to help improve the quality, accessibility, re-usability, and reproducibility of the code and data).
It is not required that an accepted paper be presented at the conference but from among the accepted papers a select list of speakers will be invited to give talks at the LuCaNT conference at ICERM.
The deadline for paper submissions is January 24, 2025. We anticipate final decisions on each paper by May 2025.
Graduate students will not be required to present posters but will be encouraged to submit titles and abstracts for lightning talks instead.
This workshop is fully funded by a Simons Foundation Targeted Grant to Institutes.
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Workshop Schedule
Monday, July 07, 2025
9:00 - 9:20 AM EDT
Check In
11th Floor Collaborative Space
9:20 - 9:30 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:30 - 10:20 AM EDT
Incubating Mathematical Databases
11th Floor Lecture Hall
Speaker
David Roe, Massachusetts Institute of Technology
Session Chair
John Jones, Arizona State University
Abstract
I will speak about two new projects, one of which is gathering momentum after starting at the end of 2023 and the other is still in the design stage. MathBases is a website with a twofold mission: to provide an index of existing mathematical databases (it currently lists 116 but more will be added by the time you read this) and to provide a suite of tutorials, templates and expository articles to facilitate the creation of new databases. The Diophatine Library will be a website where users can browse families of Diophantine equations or search for a specific equation, accessing structural information, lists of solutions, heuristic and proven results on numbers of solutions, references, and code in computer algebra and formal systems for searching for and certifying solutions. Both of these projects are open source and welcome new contributors.
10:30 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:20 AM EDT
Families of $p$-adic fields.
11th Floor Lecture Hall
Speaker
David Roberts, University of Minnesota Morris
Session Chair
Sally Koutsoliotas, Bucknell University
Abstract
We improve the database of $p$-adic fields in the LMFDB by systematically using Krasner-Monge polynomials and working relatively as well as absolutely. These improvements organize $p$-adic fields into families. They thereby make long lists of fields more manageable and various theoretical structures more evident.
In particular, the database now includes all degree $n$ extensions of $\mathbb{Q}_p$, for $p<200$ and $n \leq 23$.
This is joint work with Jordi Guardia-Rubies, John W. Jones, Kevin Keating, Sebastian Pauli, and David Roe.
11:30 - 11:50 AM EDT
Sampling cubic rings
11th Floor Lecture Hall
Speaker
Fabian Gundlach, Universität Paderborn
Session Chair
Sally Koutsoliotas, Bucknell University
Abstract
Belabas showed how to construct a _list of all_ cubic integral domains S of given signature with |disc(S)| <= T in time \tilde{O}(T).
In this talk, we explain how to construct a _uniformly random_ cubic integral domain S of given signature with |disc(S)| <= T in expected time \tilde{O}(\log T).
12:00 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:20 PM EDT
A Shimura-Belyi map of degree 33, and number fields with Galois group 33T55 = ΣL2(F32)
11th Floor Lecture Hall
Speaker
Noam Elkies, Harvard University
Session Chair
Eran Assaf, Massachusetts Institute of Technology
2:30 - 2:40 PM EDT
Fine structure in landscapes of L-functions
11th Floor Lecture Hall
Speaker
Stefan Lemurell, University of Gothenburg and Chalmers University of Technology
Session Chair
Eran Assaf, Massachusetts Institute of Technology
Abstract
We will present an emerging theory of fine structure in landscapes of L-functions. Two novel phenomena will be introduced: ditch avoidance and congruence bias, demonstrating how they derive from the classical principles of zero repulsion and zero rigidity. This is joint work with David Farmer, Sally Koutsoliotas and David Roberts.
2:45 - 2:55 PM EDT
Creating a datbase of rigorous Maass Forms
11th Floor Lecture Hall
Speaker
David Lowry-Duda, ICERM & Brown University
Session Chair
Eran Assaf, Massachusetts Institute of Technology
Abstract
We describe the work leading to the new database of rigorous Maass forms in the LMFDB, as well as ongoing work to expand the database further.
3:00 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:10 PM EDT
Elliptic curve data
11th Floor Lecture Hall
Virtual Speaker
John Cremona, University of Warwick
Session Chair
Andrew Sutherland, MIT
4:10 - 5:00 PM EDT
Short Reports
Other - 11th Floor Lecture Hall
Session Chair
Andrew Sutherland, MIT
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, July 08, 2025
9:30 - 10:20 AM EDT
Do mathematicians dream of data?
11th Floor Lecture Hall
Speaker
Katja Berčič, University of Ljubljana
Session Chair
Renate Scheidler, University of Calgary
Abstract
As data becomes increasingly central to mathematics, it is reshaping how mathematicians work, reason, and communicate. This talk sketches how data and mathematics have historically intersected, examines the current landscape, and considers how things might evolve—especially in the era of computer-assisted mathematics. A truly comprehensive picture of mathematical data may be out of reach, but the goal here is to raise questions and offer perspectives that can spark further discussion.
10:30 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:20 AM EDT
The Spine of a Supersingular Ell-Isogeny Graph
11th Floor Lecture Hall
Speaker
Taha Hedayat, University of Calgary
Session Chair
Manami Roy, Lafayette College
Abstract
Supersingular ell-isogeny graphs defined over algebraically closed finite fields are used in numerous cryptographic protocols considered to be quantum-resistant. Much of the security analysis of these schemes assumes the supersingular ell-isogeny graph to have a random structure. This is a questionable assumption which may have detrimental consequences if false.
In this talk, we present the joint work with Dr. Sarah Arpin and Dr. Renate Scheidler that studies the subgraph known as the spine. We provide explicit conditions on the parameters of the supersingular ell-isogeny graph which result in detailed descriptions of the spine. As such, providing some insight into the graphical structure of the supersingular ell-isogeny graph. These findings challenge the assumption of the inherent randomness of the graph.
We also discuss numerical data collected about the spine's position within the full supersingular ell-isogeny graph. In particular, we look at the number of vertices from the spine that lie in the center of the supersingular ell-isogeny graph. Interestingly, through its wave-like pattern, this data seems to show a certain level of randomness still present in the supersingular ell-isogeny graph.
11:30 - 11:40 AM EDT
Explicit modularity of K3 surfaces with complex multiplication of large degree
11th Floor Lecture Hall
Speaker
Edgar Costa, MIT
Session Chair
Manami Roy, Lafayette College
Abstract
We consider the transcendental motive of three K3 surfaces X conjectured to have complex multiplication (CM). Under this assumption, we match these to explicit algebraic Hecke quasi-characters, and CM abelian threefolds. This provides substantial evidence that a power of A corresponds to X under the Kuga-Satake correspondence.
11:45 - 11:55 AM EDT
Some experiments on Lehmer's Conjecture for Elliptic Curves
11th Floor Lecture Hall
Speaker
Eli Orvis, University of Colorado Boulder
Session Chair
Manami Roy, Lafayette College
Abstract
In this talk we will describe a method for computing the point of smallest height over all number fields of a given degree, for a fixed elliptic curve defined over the rationals. We then describe a large computation done to collect data in the quadratic case, and investigate what these data suggest about the constants appearing in related conjectures of Lehmer and Lang.
12:00 - 2:00 PM EDT
Lunch/Free Time
2:00 - 3:15 PM EDT
Lightning Talks
11th Floor Lecture Hall
Session Chair
John Voight, University of Sydney
Abstract
- Haochen Wu (Dartmouth College): Hilbert modular forms from orthogonal modular forms on binary lattices
- Robin Visser (Charles University): Computing genus 2 curves over Q whose Jacobians have good reduction away from 2
- Ha Tran (University of Alberta): Gross lattices of supersingular elliptic curves
- Colin Jackson (University of Colorado Boulder): Homomorphisms from rational points on elliptic curves to ideal class groups
- James Rawson (University of Warwick): Wild Conductor Exponents of Plane Quartics
- Harry Spencer (University College London): Wild conductor exponents of curves
- Tamara Veenstra & Alyson Deines (Institute for Defense Analysis-Center for Communications Research): Applying Machine Learning to Maass forms in the LMFDB
- Robin Visser (Charles University): Computing genus 2 curves over Q whose Jacobians have good reduction away from 2
- Ha Tran (University of Alberta): Gross lattices of supersingular elliptic curves
- Colin Jackson (University of Colorado Boulder): Homomorphisms from rational points on elliptic curves to ideal class groups
- James Rawson (University of Warwick): Wild Conductor Exponents of Plane Quartics
- Harry Spencer (University College London): Wild conductor exponents of curves
- Tamara Veenstra & Alyson Deines (Institute for Defense Analysis-Center for Communications Research): Applying Machine Learning to Maass forms in the LMFDB
3:15 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 5:00 PM EDT
Software Demos
Other - 11th Floor Lecture Hall
Abstract
- Everett Howe (Unaffiliated): Fast low-memory enumeration of hyperelliptic curves of low genus
- Stefano Marseglia (Université Côte d'Azur): Abelian varieties over finite fields in the LMFDB
- Jacob Mayle (Wake Forest University): Rational Points on Modular Curves via Quotients
- Eray Karabiyik (Cornell University): A Classification of Modular Curves
- Håvard Damm-Johnsen (University of Gothenburg): Portraits of Number Fields
- Asimina Hamakiotes (Fordham University): Families of modular curves in the LMFDB
- Stefano Marseglia (Université Côte d'Azur): Abelian varieties over finite fields in the LMFDB
- Jacob Mayle (Wake Forest University): Rational Points on Modular Curves via Quotients
- Eray Karabiyik (Cornell University): A Classification of Modular Curves
- Håvard Damm-Johnsen (University of Gothenburg): Portraits of Number Fields
- Asimina Hamakiotes (Fordham University): Families of modular curves in the LMFDB
Wednesday, July 09, 2025
9:30 - 10:20 AM EDT
Contributing to the LMFDB: a Hands-On Welcome
Panel Discussion - 11th Floor Lecture Hall
Jennifer Paulhus, Mount Holyoke College
David Roe, Massachusetts Institute of Technology
Abstract
The vast majority of people who use the LMFDB probably don’t think a lot about what happens behind the scenes to make the pages work. While computing the data that is stored in the LMFDB is an essential part of the process, so much more happens to create the pages which show up in your browser (not to mention the BB(6) hours of human labor which go into getting those pages set up).
We will give a friendly, gentle, behind-the-scene tour of the whole process. Join us to learn how the LMFDB really works and how you too can contribute to the process. We will not assume you know anything more than how to find the LMFDB in a browser.
This session is ideal for anyone who is relatively new to the LMFDB, anyone who has contributed data in the past and has wondered how we go from raw data files to the snazzy webpage, and anyone who is just curious about how it all really works.
10:30 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:20 AM EDT
A database of finite groups in the LMFDB
11th Floor Lecture Hall
Speaker
Samuel Schiavone, Massachusetts Institute of Technology
Session Chair
David Farmer, AIM
Abstract
We have recently added a database of finite groups to the LMFDB that includes subgroups, characters, and more. We discuss the features of the database, especially those that distinguish it from other available databases of groups. The creation of the finite groups database required many large-scale computations, and we mention some mathematical problems we encountered along the way. We hope that this database can foster connections among other collections in the LMFDB, and we give a sampling of relationships between the finite groups database and some of these other collections.
11:30 - 11:50 AM EDT
The pi-Base Model for Small Semantic Databases
11th Floor Lecture Hall
Speaker
Steven Clontz, University of South Alabama
Session Chair
David Farmer, AIM
Abstract
A *small semantic database* is a mathematical database where each entry is submitted by a human contributor, and where each entry includes supporting metadata allowing software to provide some level of automated deduction and/or search. In this talk we will explore the model used by the pi-Base community database of topological counterexamples. In this model, the database tracks a category of objects and their properties, with theorems that allow for automated deduction (e.g. given the theorem P ∧ Q ⇒ R, an object tagged with properties P and ¬R is automatically tagged with ¬Q).
12:00 - 12:05 PM EDT
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:05 - 2:00 PM EDT
Lunch/Free Time
2:00 - 3:00 PM EDT
Open Collaboration Time
Other - 11th Floor Collaborative Space
3:00 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 5:00 PM EDT
Open Collaboration Time
Other - 11th Floor Collaborative Space
Thursday, July 10, 2025
9:30 - 10:20 AM EDT
Shadow line distributions
11th Floor Collaborative Space
Speaker
Jennifer Balakrishnan, Boston University
Session Chair
Alvaro Lozano-Robledo, University of Connecticut
Abstract
"Let E/Q be an elliptic curve of analytic rank 2, and let p be an odd prime of good, ordinary reduction such that the p-torsion of E(Q) is trivial. Let K be an imaginary quadratic field satisfying the Heegner hypothesis for E and such that the analytic rank of the twisted curve E^K/Q is 1. Further suppose that p splits in O_K. Under these assumptions, there is a 1-dimensional Q_p-vector space attached to the triple (E, p, K), known as the shadow line, and it can be computed using anticyclotomic p-adic heights. We describe the computation of these heights and shadow lines. Furthermore, fixing pairs (E, p) and varying K, we present some data on the distribution of these shadow lines. This is joint work with Mirela Çiperiani, Barry Mazur, and Karl Rubin.
10:30 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:20 AM EDT
Local-global principle for 11-isogenies of elliptic curves over quadratic fields
11th Floor Lecture Hall
Speaker
Stevan Gajović, Max Planck Institute for Mathematics
Session Chair
Vladimir Dokchitser, University College London
Abstract
In this talk we show that the local global principle for 11-isogenies of elliptic curves over quadratic number fields holds, proving the conjecture of Barinder Banwait and John Cremona.
11:30 - 11:50 AM EDT
Quadratic Chabauty Experiments on Genus 2 Bielliptic Modular Curves in the LMFDB
11th Floor Lecture Hall
Speaker
Kate Finnerty, Boston University
Session Chair
Vladimir Dokchitser, University College London
Abstract
We present results of computational quadratic Chabauty experiments on genus 2 bielliptic modular curves of Jacobian rank 1 and 2 that have recently been added to the LMFDB. We work over both the rationals and quadratic imaginary fields. In several cases, the experiments yielded algebraic irrational points among the set of mock rational points. We highlight several notable examples and offer a conjecture relating the level of the modular curve to the potential number fields over which these points can arise.
12:00 - 2:00 PM EDT
Lunch/Free Time
2:00 - 3:20 PM EDT
Lightning Talks
11th Floor Lecture Hall
Session Chair
John Voight, University of Sydney
Abstract
- Lewis Combes (University of Sydney): Eta expressions for (some) modular forms
- Jakab Schrettner (UCL): Reduction types of curves in p-adic families
- Meghan Lee (Wake Forest University): An algorithm for isolated j-invariants arising from X_0(n)
- Barinder Banwait (Boston University): Identifying the rational elliptic curves that provably admit infinitely many twists satisfying full BSD
- Ajmain Yamin (CUNY Graduate Center): Quaternionic analogues of Zagier's sums of binary quadratic forms
- Edwina Aylward (University College London): Reduction types of genus 2 curves
-Marcella Manivel (University of Minnesota, Twin Cities): Automorphic Hamiltonians and Functional Analytic Number Theory
- Chris Xu (Univeristy of California-San Diego): Model-free Chabauty for the modular curve 10.60.2.f.1
- Jakab Schrettner (UCL): Reduction types of curves in p-adic families
- Meghan Lee (Wake Forest University): An algorithm for isolated j-invariants arising from X_0(n)
- Barinder Banwait (Boston University): Identifying the rational elliptic curves that provably admit infinitely many twists satisfying full BSD
- Ajmain Yamin (CUNY Graduate Center): Quaternionic analogues of Zagier's sums of binary quadratic forms
- Edwina Aylward (University College London): Reduction types of genus 2 curves
-Marcella Manivel (University of Minnesota, Twin Cities): Automorphic Hamiltonians and Functional Analytic Number Theory
- Chris Xu (Univeristy of California-San Diego): Model-free Chabauty for the modular curve 10.60.2.f.1
3:20 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:20 PM EDT
Gluing Genus 1 and Genus 2 Curves over ℓ-torsion
11th Floor Lecture Hall
Speaker
Pitchayut Saengrungkongka, Massachusetts Institute of Technology
Session Chair
Everett Howe, Unaffiliated
Abstract
This talk considers the problem of producing pairs consisting of a genus 1 curve X and a genus 2 curve Y that are gluable, i.e., such that there exists a genus 3 curve Z whose Jacobian is (ℓ, ℓ, ℓ)-isogenous to Jac(X) × Jac(Y), where ℓ is a prime number. We discuss two results. First, for a fixed genus 2 curve Y, we provide a systematic way to search for a prime number ℓ and a genus 1 curve X that could produce such a genus 3 curve Z. Next, we improve a numerical algorithm to construct such a curve Z, allowing us to compute Z for ℓ up to 13. This is based on a joint work with Noah Walsh.
4:30 - 4:50 PM EDT
A census of genus 6 curves over the finite field of two elements
11th Floor Lecture Hall
Speaker
Yongyuan Huang, University of California San Diego
Session Chair
Everett Howe, Unaffiliated
Abstract
In joint work with K. Kedlaya and J. Lau, we compile a complete list of isomorphism class representatives of curves of genus 6 over the finite field of two elements. We use explicit descriptions of canonical curves in each stratum of the Brill-Noether stratification of the moduli space of genus 6 curves, due to Mukai in the generic case. In this talk, we will discuss some motivations and applications for having a census of low genus curves over various finite fields, including in the study of the moduli space of curves. We will also mention how recent work of Bergstrom-Canning-Petersen-Schmitt helps with independently verifying the completeness of our census.
Friday, July 11, 2025
9:30 - 10:20 AM EDT
The Graded Ring Database
11th Floor Lecture Hall
Virtual Speaker
Alexander Kasprzyk, University of Warwick
Session Chair
Kiran Kedlaya, University of California San Diego
Abstract
The Graded Ring Database, or GRDB, is a database of classifications in algebraic geometry. Originally built to explore the possible Hilbert series of terminal Fano 3-folds, it has expanded to include a wealth of geometric datasets. Celebrating its 25th birthday, the GRDB is positively prehistoric. In this talk I hope to explain briefly what the GRDB is, and discuss some of the lessons learnt from choices made all those years ago (some bad, some good). I also hope to give you a glimpse of what's in the pipeline for the GRDB.
10:30 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:20 AM EDT
The Kodaira dimension of Hilbert modular threefolds
11th Floor Lecture Hall
Speaker
Adam Logan, Carleton University
Session Chair
David Roberts, University of Minnesota Morris
Abstract
Following a method introduced by Thomas-Vasquez and developed by Grundman, we prove that many Hilbert modular threefolds of geometric genus 0 and 1 are of general type, and that some are of nonnegative Kodaira dimension. The new ingredient is a detailed study of the geometry and combinatorics of totally positive integral elements of a fractional ideal in a totally real number field that are minimal with respect to trace up to multiplication by totally positive units.
11:30 - 11:40 AM EDT
Mod-3 Galois images of abelian surfaces
11th Floor Lecture Hall
Speaker
Shiva Chidambaram, University of Wisconsin-Madison
Session Chair
David Roberts, University of Minnesota Morris
Abstract
This talk will discuss the images of mod-ell Galois representations associated to a principally polarized abelian surface over Q. We report on an algorithm to determine the mod-3 Galois image, with no dependence on the endomorphism ring of the surface. Moreover, we will discuss progress towards explicitly realizing all possible mod-3 Galois images.
11:45 - 11:55 AM EDT
Completely decomposable modular Jacobians
11th Floor Lecture Hall
Speaker
Andrew Sutherland, MIT
Session Chair
David Roberts, University of Minnesota Morris
Abstract
An abelian variety isogenous to a product of elliptic curves is said to be completely decomposable. In 1993 Ekedahl and Serre posed the following question: ""For which g does there exist a curve of genus g whose Jacobian is completely decomposable?"". For fields of characteristic zero this is expected to be a finite set. This question remains open, but Ekedahl and Serre produced a list of integers g for which they could exhibit a curve over C whose Jacobian is completely decomposable. Their list was later extended by others to include every g <= 55 except 38, as well as many other g <= 1297. Using the new database of modular curves in the LMFDB (and some of the algorithms used to construct it), we are able to add 13 new values of g to this list, including g=38, all of which arise for Jacobians of modular curves defined over Q.
This is joint work with Jennifer Paulhus.
12:00 - 2:00 PM EDT
Lunch/Free Time
2:00 - 5:00 PM EDT
Collaboration Time
Other - 11th Floor Collaborative Space