Abstract

L-functions of degree d can be parametrized, in two different ways, by points with an attached multiplicity in d-1 dimensional Euclidean space. One approach separates the L-functions according to the shape of the Gamma-factors in the functional equation, equivalently, according to the infinity type of the underlying automorphic representation. The other approach combines all the L-functions of a given degree into a single picture in which the points, to leading order, are uniformly dense. We will describe these classifications and provide examples of several ""landscapes"" in the L-function world.

• Video
• Slides
• The landscape of L-functions

Abstract

In this joint project with Shiva Chidambaram, Edgar Costa, and Jean Kieffer, we describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface A over ℚ with geometric endomorphism ring equal to ℤ, computes all the other p.p. abelian surfaces over ℚ that are isogenous to A. This algorithm relies on explicit open image techniques for Galois representations, and we employ a combination of analytic and algebraic methods to efficiently prove or disprove the existence of isogenies. We illustrate the practicality of our algorithm by applying it to 1,538,149 isogeny classes of Jacobians of genus 2 curves.

• Video
• Slides
• Computing isogeny classes

Abstract

In this talk we present algorithms for computing geometric invariants of Hilbert modular surfaces, and describe the implementation of said algorithms. We use this implementation to compute a database of invariants, soon to be added to the LMFDB. This extends work of van der Geer to Hilbert modular groups with nontrivial level.

• Video
• Slides
• Computing Invariants Of Hilbert Modular Surfa

Abstract

We describe a generalization and improvement of Diaz y Diaz's search technique for imaginary quadratic fields with 3-rank at least 2, one of the most successful algorithms for generating many examples with relatively small discriminants, to find quadratic fields with large n-ranks for odd n >= 3. An extensive search using our new algorithm in conjunction with a variety of further practical improvements produced billions of fields with non-trivial p-rank for the primes p = 3, 5, 7, 11 and 13, and a large volume of fields with high p-ranks and unusual class group structures. Our numerical results include a field with 5-rank equal to 4 with the smallest absolute discriminant discovered to date and the first known examples of imaginary quadratic fields with 7-rank equal to 4.

• Video
• Slides
• Methods for Finding Imaginary Quadratic Field

Abstract

We describe an algorithm, based on the Selberg trace formula and explicit numerical computations of Maaß cusp forms, for computing the class groups and regulators of all real quadratic fields of discriminant D up to X in time O(X^{5/4+o(1)}), without assuming any unproven conjectures. We applied the algorithm to compute up to X=10^11 and used the output to test various implications of the Cohen-Lenstra heuristics. This is joint work with Ce Bian, Andrew Booker, Austin Docherty and Michael Jacobson.

• Unconditional computation of the class groups

Abstract

We will give an overview of the LMFDB editorial process, including the criteria we use to assess contributions (of both code and data), along with an overview of some features that have been recently added to the LMFDB, as well as a sneak preview of features that we hope to add soon. We will also highlight specific areas where we would welcome contributions from the community.

Abstract

Thanks to the collective efforts of many people, the LMFDB pages on classical modular forms are in a robust state, and have grown into an indispensable tool for research. I will lay out a roadmap for accomplishing the same with Maass forms, and describe some applications of such a database.

• Video
• Slides

Abstract

For a genus 2 curve over the rational field whose Jacobian A admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes ell for which the Galois action on ell-torsion points of A is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell's classification of maximal subgroups of the projective symplectic group of 4 x 4 matrices over the finite field of ell elements, sampling of the characteristic polynomials of Frobenius, and the Khare--Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus 2 curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such curve. This talk is based on joint work with Barinder S. Banwait, Armand Brumer, Zev Klagsbrun, Jacob Mayle, Padmavathi Srinivasan, and Isabel Vogt.

• Video
• Slides
• Computing nonsurjective primes

Abstract

Let E be an elliptic curve defined over Q. Associated to E, there is an adelic Galois representation. In this talk, I will discuss a joint work with Jacob Mayle where we consider those elliptic curves defined over Q whose adelic Galois image is as large as possible given a constraint on image modulo 2. For such curves, we give a characterization in terms of their l-adic images, compute all examples of conductor at most 500,000, precisely describe the image of their adelic Galois representation and offer an application to cyclicity problem. In this way, we generalize some foundational results on Serre curves.

• Video
• Slides
• Serre Curves Relative to Obstruction Modulo 2

Abstract

- Santiago Arango (Emory University): Frobenius distributions of abelian varieties over finite fields
- Hyun Jong Kim (University of Wisconsin-Madison): Cohen-Lenstra Heuristics and Vanishing of Zeta Functions for Trielliptic Curves over Finite Fields
- Sung Min Lee (University of Illinois at Chicago): On the congruence class bias of distribution of primes of cyclic reduction for elliptic curves
- Yongyuan Huang (University of California San Diego): Model-free Coleman Integration on Modular Curves
- Juanita Duque Rosero (Boston University): Local heights computations for quadratic Chabauty
(--10 minute break--)
- Garen Chiloyan (none): 2-adic Galois images attached to rational isogeny-torsion graphs
- Asimina Hamakiotes (University of Connecticut): Elliptic curves with complex multiplication and abelian division fields
- Sachi Hashimoto (Brown University): Towards a classification of isolated j-invariants
- Pietro Mercuri (Sapienza Università di Roma): Automorphism group of Cartan modular curves
- Ciaran Schembri (Dartmouth College): Abelian surfaces with quaternionic multiplication and their rational torsion subgroups
- Robin Visser (University of Warwick): Abelian surfaces with good reduction away from 2
- Tian Wang (University of Illinois at Chicago): Effective Serre's Open Image Theorem for elliptic curves

• Slides

Abstract

- Shiva Chidambaram (MIT): Mod-ell Galois images of Picard curves
- Edgar Costa (MIT): Abelian Varieties in the LMFDB
- David Roe (MIT): Modular curves in the LMFDB
- Manami Roy (Lafayette College): Database of finite groups in LMFDB

Abstract

Papers submitted to this conference went through a technical review with the goal of helping to improve the quality, accessibility, re-usability, and reproducibility of any submitted code and data. The authors of each LuCaNT submission were given a technical report created by a team led by Jeroen Hanselman using tools developed by the Mathematical Research Data Initiative (MaRDI). While the reports were not used to determine whether to accept a paper, some comments in those reports were included in revision suggestions.
Jeroen will join the managing editors to talk about the process and solicit feedback from authors, and the broader community.

Abstract

One major achievement in mathematics of last century is the classification of finite simple groups. Their character tables are provided in the famous big red book, the ATLAS of Finite Simple Groups; the ATLAS of Brauer Tables list the modular irreducible characters. In a joint project with Thomas Breuer and Richard Parker we computed the discriminants of the invariant quadratic forms for all even degree absolutely irreducible indicator + characters of most of the groups in these two books. For finite fields this gives the additional information which of the two possible orthogonal groups contains the image of the representation. The talk will comment on the theoretical and computational methods used to obtain these discriminants and how to work with the database of orthogonal discriminants of characters.

• Video
• Slides

Abstract

In the last few decades, many attempts have been made to extend CM theory to the setting of real quadratic fields. In this talk I will describe how to turn recent work by Darmon, Pozzi and Vonk into an efficient algorithm for computing p-adic analogues of modular units and Heegner points: Gross--Stark units and Stark--Heegner points. This builds on the framework of Darmon and Vonk's rigid meromorphic cocycles, and involves Hilbert modular forms, overconvergent elliptic modular forms, and Gauss' reduction theory of binary quadratic forms.

• Video
• Slides
• Modular algorithms

Abstract

In this talk we are going to show how we computed tables of paramodular forms of degree two and cohomological weight via a correspondence with orthogonal modular forms on quinary lattices.

• Video
• Slides
• A database of paramodular forms

Abstract

We classify extensions of function fields of curves over finite fields in which the class number does not change. This breaks up into three parts, of which we will emphasize the third in this talk. 1. Identification of a finite set of possible pairs of Weil polynomials for the two curves, and of cyclic covers consistent with this set by way of explicit class field theory (presented at ANTS-XV in August 2022). 2. Proof using 1 that the only extensions that can occur are cyclic (presented at AGC^2T in June 2023). 3. Identification of curves of genus up to 7 whose Weil polynomials are candidates for the base of the extension. This uses Mukai's explicit description of the universal curves over the various Brill-Noether strata of the moduli stacks of curves of genus 6 and 7, with some attention paid to working over a nonclosed (here finite) base field.

• Video
• Slides
• The relative class number one problem

Abstract

Building on work of Balakrishnan, Dogra, and Bianchi, we provide some improvements to the explicit quadratic Chabauty method to provably compute rational points on genus 2 bielliptic curves over Q, whose Jacobians have Mordell–Weil rank equal to 2. We complement this with a precision analysis to guarantee correct outputs. Together with the Mordell–Weil sieve, this bielliptic quadratic Chabauty method is then the main tool that we use to compute the rational points on the 411 locally solvable curves from the LMFDB which satisfy the aforementioned conditions. This is joint work with Francesca Bianchi

• Video
• Rational Points on Rank 2 Genus 2 Bielliptic

Abstract

Constructing a supersingular elliptic curve whose endomorphism ring is isomorphic to a given quaternion maximal order (one direction of the Deuring correspondence) is known to be polynomial-time assuming the generalized Riemann hypothesis [KLPT14; Wes21], but notoriously daunting in practice when not working over carefully selected base fields. In this work, we speed up the computation of the Deuring correspondence in general characteristic, i.e., without assuming any special form of the characteristic. Our algorithm follows the same overall strategy as earlier works, but we add simple (yet effective) optimizations to multiple subroutines to significantly improve the practical performance of the method. To demonstrate the impact of our improvements, we show that our implementation achieves highly practical running times even for examples of cryptographic size. One implication of these findings is that cryptographic security reductions based on KLPT-derived algorithms (such as [EHLMP18; Wes22]) have become tighter, and therefore more meaningful in practice. Another is the pure bliss of fast(er) computer algebra: We provide a Sage implementation which works for general primes and includes many necessary tools for computational number theorists’ and cryptographers’ needs when working with endomorphism rings of supersingular elliptic curves. This includes the KLPT algorithm, translation of ideals to isogenies, and finding supersingular elliptic curves with known endomorphism ring for general primes. Finally, the Deuring correspondence has recently received increased interest because of its role in the SQISign signature scheme [DeF+20]. We provide a short and self-contained summary of the state-of-the-art algorithms without going into any of the cryptographic intricacies of SQISign.

• Video
• Slides
• Deuring for the people

Abstract

- Lewis Combes (University of Sheffield): Period polynomials of Bianchi modular forms
- Pascal Molin (Université Paris Cité): LMFDB and computations with S5 modular forms
- Eric Moss (Boston College): Computing Bianchi-Maass Forms
- Tung Nguyen (Western University): On the arithmetic of Fekete polynomials of principal Dirichlet characters
- Alexey Pozdnyakov (University of Connecticut): Murmurations of Dirichlet characters
- Brandon Williams (RWTH Aachen University): Computation of vector-valued modular forms
(--10 minute break--)
- Ajmain Yamin (CUNY Graduate Center): Belyi Pairs of Complete Regular Dessins
- Mingjie Chen (University of Birmingham): Hidden Stabilizers, the Isogeny To Endomorphism Ring Problem and the Cryptanalysis of pSIDH
- Travis Morrison (Virginia Tech): Beyond the SEA (algorithm) computing the trace of a supersingular endomorphism
- James Boyd (Wolfram Institute): The Role of Computers in Mathematics Exposition: The Case of the Langlands Program
- Daniel Gordon (IDA Center for Communications Research- La Jolla): Online Math Databases on the Cheap
- Maria Sabitova (CUNY Queens College): A number theoretic classification of toroidal solenoids

• Slides

Abstract

- James Rickards (CU Boulder): Computing fundamental domains for congruence arithmetic Fuchsian groups in PARI/GP
- Eran Assaf (Dartmouth College): Paramodular forms in the LMFDB
- Juergen Klueners (Paderborn University): A database of number fields with interesting Galois groups
- Ben Breen (none): Computing with Hilbert Modular Surfaces

Abstract

Elliptic curve isogeny path finding has many applications in number theory and cryptography. For supersingular curves, this problem is known to be easy when one small endomorphism or the entire endomorphism ring are known. Unfortunately, computing the endomorphism ring, or even just finding one small endomorphism, is hard. How difficult is path finding in the presence of one (not necessarily small) endomorphism? We use the volcano structure of the oriented supersingular isogeny graph to answer this question. We give a classical algorithm for path finding that is subexponential in the degree of the endomorphism and linear in a certain class number, and a quantum algorithm for finding a smooth isogeny (and hence also a path) that is subexponential in the discriminant of the endomorphism. A crucial tool for navigating supersingular oriented isogeny volcanoes is a certain class group action on oriented elliptic curves which generalizes the well-known class group action in the setting of ordinary elliptic curves.

• Video
• Slides

Abstract

We give birational parametrizations of pairs (C,T) where T is a 5-torsion point on the Jacobian of a genus-2 curve C, possibly satisfying one or more of the following additional conditions: T = (P) - (P_0) for some points P_0, P in C with P_0 Weierstrass; C has one or more additional rational Weierstrass points; Jac(C) has real multiplication by (1+\sqrt{5})/2, and T is a \sqrt{5}-torsion point.

• Video
• Slides

Abstract

I will describe joint work with Andrew Sutherland, giving deterministic and probabilistic algorithms to decide whether a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the jinvariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. Our algorithms admit simple implementations that are asymptotically and practically faster than existing approaches

• Video
• Slides
• Computing the endomorphism ring